| 1 | """ |
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| 2 | Coercion via Construction Functors |
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| 3 | """ |
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| 4 | from functor import Functor, IdentityFunctor_generic |
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| 5 | from basic import * |
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| 6 | |
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| 7 | from sage.structure.parent import CoercionException |
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| 8 | |
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| 9 | # TODO, think through the rankings, and override pushout where necessary. |
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| 10 | |
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| 11 | class ConstructionFunctor(Functor): |
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| 12 | """ |
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| 13 | Base class for construction functors. |
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| 14 | |
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| 15 | A construction functor is a functorial algebraic construction, |
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| 16 | such as the construction of a matrix ring over a given ring |
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| 17 | or the fraction field of a given ring. |
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| 18 | |
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| 19 | In addition to the class :class:`~sage.categories.functor.Functor`, |
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| 20 | construction functors provide rules for combining and merging |
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| 21 | constructions. This is an important part of Sage's coercion model, |
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| 22 | namely the pushout of two constructions: When a polynomial ``p`` in |
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| 23 | a variable ``x`` with integer coefficients is added to a rational |
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| 24 | number ``q``, then Sage finds that the parents ``ZZ['x']`` and |
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| 25 | ``QQ`` are obtained from ``ZZ`` by applying a polynomial ring |
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| 26 | construction respectively the fraction field construction. Each |
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| 27 | construction functor has an attribute ``rank``, and the rank of |
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| 28 | the polynomial ring construction is higher than the rank of the |
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| 29 | fraction field construction. This means that the pushout of ``QQ`` |
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| 30 | and ``ZZ['x']``, and thus a common parent in which ``p`` and ``q`` |
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| 31 | can be added, is ``QQ['x']``, since the construction functor with |
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| 32 | a lower rank is applied first. |
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| 33 | |
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| 34 | :: |
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| 35 | |
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| 36 | sage: F1, R = QQ.construction() |
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| 37 | sage: F1 |
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| 38 | FractionField |
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| 39 | sage: R |
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| 40 | Integer Ring |
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| 41 | sage: F2, R = (ZZ['x']).construction() |
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| 42 | sage: F2 |
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| 43 | Poly[x] |
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| 44 | sage: R |
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| 45 | Integer Ring |
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| 46 | sage: F3 = F2.pushout(F1) |
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| 47 | sage: F3 |
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| 48 | Poly[x](FractionField(...)) |
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| 49 | sage: F3(R) |
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| 50 | Univariate Polynomial Ring in x over Rational Field |
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| 51 | sage: from sage.categories.pushout import pushout |
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| 52 | sage: P.<x> = ZZ[] |
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| 53 | sage: pushout(QQ,P) |
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| 54 | Univariate Polynomial Ring in x over Rational Field |
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| 55 | sage: ((x+1) + 1/2).parent() |
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| 56 | Univariate Polynomial Ring in x over Rational Field |
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| 57 | |
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| 58 | When composing two construction functors, they are sometimes |
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| 59 | merged into one, as is the case in the Quotient construction:: |
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| 60 | |
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| 61 | sage: Q15, R = (ZZ.quo(15*ZZ)).construction() |
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| 62 | sage: Q15 |
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| 63 | QuotientFunctor |
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| 64 | sage: Q35, R = (ZZ.quo(35*ZZ)).construction() |
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| 65 | sage: Q35 |
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| 66 | QuotientFunctor |
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| 67 | sage: Q15.merge(Q35) |
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| 68 | QuotientFunctor |
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| 69 | sage: Q15.merge(Q35)(ZZ) |
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| 70 | Ring of integers modulo 5 |
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| 71 | """ |
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| 72 | def __mul__(self, other): |
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| 73 | """ |
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| 74 | Compose construction functors to a composit construction functor, unless one of them is the identity. |
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| 75 | |
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| 76 | NOTE: |
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| 77 | |
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| 78 | The product is in functorial notation, i.e., when applying the product to an object |
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| 79 | then the second factor is applied first. |
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| 80 | |
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| 81 | TESTS:: |
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| 82 | |
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| 83 | sage: from sage.categories.pushout import IdentityConstructionFunctor |
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| 84 | sage: I = IdentityConstructionFunctor() |
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| 85 | sage: F = QQ.construction()[0] |
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| 86 | sage: P = ZZ['t'].construction()[0] |
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| 87 | sage: F*P |
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| 88 | FractionField(Poly[t](...)) |
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| 89 | sage: P*F |
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| 90 | Poly[t](FractionField(...)) |
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| 91 | sage: (F*P)(ZZ) |
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| 92 | Fraction Field of Univariate Polynomial Ring in t over Integer Ring |
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| 93 | sage: I*P is P |
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| 94 | True |
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| 95 | sage: F*I is F |
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| 96 | True |
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| 97 | |
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| 98 | """ |
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| 99 | if not isinstance(self, ConstructionFunctor) and not isinstance(other, ConstructionFunctor): |
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| 100 | raise CoercionException, "Non-constructive product" |
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| 101 | if isinstance(other,IdentityConstructionFunctor): |
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| 102 | return self |
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| 103 | if isinstance(self,IdentityConstructionFunctor): |
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| 104 | return other |
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| 105 | return CompositeConstructionFunctor(other, self) |
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| 106 | |
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| 107 | def pushout(self, other): |
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| 108 | """ |
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| 109 | Composition of two construction functors, ordered by their ranks. |
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| 110 | |
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| 111 | NOTE: |
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| 112 | |
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| 113 | - This method seems not to be used in the coercion model. |
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| 114 | |
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| 115 | - By default, the functor with smaller rank is applied first. |
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| 116 | |
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| 117 | TESTS:: |
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| 118 | |
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| 119 | sage: F = QQ.construction()[0] |
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| 120 | sage: P = ZZ['t'].construction()[0] |
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| 121 | sage: F.pushout(P) |
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| 122 | Poly[t](FractionField(...)) |
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| 123 | sage: P.pushout(F) |
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| 124 | Poly[t](FractionField(...)) |
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| 125 | |
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| 126 | """ |
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| 127 | if self.rank > other.rank: |
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| 128 | return self * other |
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| 129 | else: |
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| 130 | return other * self |
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| 131 | |
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| 132 | def __cmp__(self, other): |
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| 133 | """ |
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| 134 | Equality here means that they are mathematically equivalent, though they may have |
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| 135 | specific implementation data. This method will usually be overloaded in subclasses. |
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| 136 | by default, only the types of the functors are compared. Also see the \code{merge} |
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| 137 | function. |
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| 138 | |
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| 139 | TESTS:: |
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| 140 | |
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| 141 | sage: from sage.categories.pushout import IdentityConstructionFunctor |
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| 142 | sage: I = IdentityConstructionFunctor() |
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| 143 | sage: F = QQ.construction()[0] |
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| 144 | sage: P = ZZ['t'].construction()[0] |
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| 145 | sage: I == F # indirect doctest |
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| 146 | False |
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| 147 | sage: I == I # indirect doctest |
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| 148 | True |
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| 149 | |
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| 150 | """ |
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| 151 | return cmp(type(self), type(other)) |
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| 152 | |
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| 153 | def __str__(self): |
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| 154 | """ |
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| 155 | NOTE: |
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| 156 | |
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| 157 | By default, it returns the name of the construction functor's class. |
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| 158 | Usually, this method will be overloaded. |
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| 159 | |
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| 160 | TEST:: |
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| 161 | |
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| 162 | sage: F = QQ.construction()[0] |
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| 163 | sage: F # indirect doctest |
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| 164 | FractionField |
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| 165 | sage: Q = ZZ.quo(2).construction()[0] |
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| 166 | sage: Q # indirect doctest |
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| 167 | QuotientFunctor |
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| 168 | |
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| 169 | """ |
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| 170 | s = str(type(self)) |
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| 171 | import re |
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| 172 | return re.sub("<.*'.*\.([^.]*)'>", "\\1", s) |
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| 173 | |
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| 174 | def __repr__(self): |
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| 175 | """ |
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| 176 | NOTE: |
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| 177 | |
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| 178 | By default, it returns the name of the construction functor's class. |
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| 179 | Usually, this method will be overloaded. |
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| 180 | |
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| 181 | TEST:: |
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| 182 | |
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| 183 | sage: F = QQ.construction()[0] |
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| 184 | sage: F # indirect doctest |
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| 185 | FractionField |
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| 186 | sage: Q = ZZ.quo(2).construction()[0] |
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| 187 | sage: Q # indirect doctest |
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| 188 | QuotientFunctor |
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| 189 | |
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| 190 | """ |
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| 191 | return str(self) |
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| 192 | |
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| 193 | def merge(self, other): |
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| 194 | """ |
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| 195 | Merge ``self`` with another construction functor, or return None. |
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| 196 | |
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| 197 | NOTE: |
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| 198 | |
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| 199 | The default is to merge only if the two functors coincide. But this |
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| 200 | may be overloaded for subclasses, such as the quotient functor. |
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| 201 | |
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| 202 | EXAMPLES:: |
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| 203 | |
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| 204 | sage: F = QQ.construction()[0] |
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| 205 | sage: P = ZZ['t'].construction()[0] |
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| 206 | sage: F.merge(F) |
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| 207 | FractionField |
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| 208 | sage: F.merge(P) |
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| 209 | sage: P.merge(F) |
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| 210 | sage: P.merge(P) |
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| 211 | Poly[t] |
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| 212 | |
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| 213 | """ |
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| 214 | if self == other: |
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| 215 | return self |
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| 216 | else: |
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| 217 | return None |
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| 218 | |
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| 219 | def commutes(self, other): |
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| 220 | """ |
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| 221 | Determine whether ``self`` commutes with another construction functor. |
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| 222 | |
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| 223 | NOTE: |
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| 224 | |
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| 225 | By default, ``False`` is returned in all cases (even if the two |
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| 226 | functors are the same, since in this case :meth:`merge` will apply |
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| 227 | anyway). So far there is no construction functor that overloads |
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| 228 | this method. Anyway, this method only becomes relevant if two |
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| 229 | construction functors have the same rank. |
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| 230 | |
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| 231 | EXAMPLES:: |
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| 232 | |
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| 233 | sage: F = QQ.construction()[0] |
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| 234 | sage: P = ZZ['t'].construction()[0] |
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| 235 | sage: F.commutes(P) |
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| 236 | False |
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| 237 | sage: P.commutes(F) |
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| 238 | False |
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| 239 | sage: F.commutes(F) |
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| 240 | False |
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| 241 | |
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| 242 | """ |
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| 243 | return False |
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| 244 | |
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| 245 | def expand(self): |
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| 246 | """ |
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| 247 | Decompose ``self`` into a list of construction functors. |
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| 248 | |
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| 249 | NOTE: |
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| 250 | |
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| 251 | The default is to return the list only containing ``self``. |
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| 252 | |
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| 253 | EXAMPLE:: |
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| 254 | |
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| 255 | sage: F = QQ.construction()[0] |
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| 256 | sage: F.expand() |
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| 257 | [FractionField] |
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| 258 | sage: Q = ZZ.quo(2).construction()[0] |
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| 259 | sage: Q.expand() |
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| 260 | [QuotientFunctor] |
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| 261 | sage: P = ZZ['t'].construction()[0] |
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| 262 | sage: FP = F*P |
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| 263 | sage: FP.expand() |
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| 264 | [FractionField, Poly[t]] |
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| 265 | """ |
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| 266 | return [self] |
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| 267 | |
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| 268 | |
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| 269 | class CompositeConstructionFunctor(ConstructionFunctor): |
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| 270 | """ |
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| 271 | A Construction Functor composed by other Construction Functors. |
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| 272 | |
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| 273 | INPUT: |
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| 274 | |
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| 275 | ``F1, F2,...``: A list of Construction Functors. The result is the |
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| 276 | composition ``F1`` followed by ``F2`` followed by ... |
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| 277 | |
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| 278 | EXAMPLES:: |
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| 279 | |
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| 280 | sage: from sage.categories.pushout import CompositeConstructionFunctor |
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| 281 | sage: F = CompositeConstructionFunctor(QQ.construction()[0],ZZ['x'].construction()[0],QQ.construction()[0],ZZ['y'].construction()[0]) |
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| 282 | sage: F |
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| 283 | Poly[y](FractionField(Poly[x](FractionField(...)))) |
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| 284 | sage: F == loads(dumps(F)) |
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| 285 | True |
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| 286 | sage: F == CompositeConstructionFunctor(*F.all) |
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| 287 | True |
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| 288 | sage: F(GF(2)['t']) |
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| 289 | Univariate Polynomial Ring in y over Fraction Field of Univariate Polynomial Ring in x over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 2 (using NTL) |
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| 290 | |
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| 291 | """ |
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| 292 | |
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| 293 | def __init__(self, *args): |
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| 294 | """ |
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| 295 | TESTS:: |
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| 296 | |
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| 297 | sage: from sage.categories.pushout import CompositeConstructionFunctor |
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| 298 | sage: F = CompositeConstructionFunctor(QQ.construction()[0],ZZ['x'].construction()[0],QQ.construction()[0],ZZ['y'].construction()[0]) |
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| 299 | sage: F |
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| 300 | Poly[y](FractionField(Poly[x](FractionField(...)))) |
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| 301 | sage: F == CompositeConstructionFunctor(*F.all) |
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| 302 | True |
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| 303 | |
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| 304 | """ |
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| 305 | |
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| 306 | def _apply_functor_to_morphism(self, f): |
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| 307 | """ |
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| 308 | Apply the functor to an object of ``self``'s domain. |
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| 309 | |
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| 310 | TESTS:: |
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| 311 | |
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| 312 | sage: from sage.categories.pushout import CompositeConstructionFunctor |
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| 313 | sage: F = CompositeConstructionFunctor(QQ.construction()[0],ZZ['x'].construction()[0],QQ.construction()[0],ZZ['y'].construction()[0]) |
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| 314 | sage: R.<a,b> = QQ[] |
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| 315 | sage: f = R.hom([a+b, a-b]) |
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| 316 | sage: F(f) # indirect doctest |
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| 317 | Ring ... |
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| 318 | """ |
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| 319 | |
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| 320 | def _apply_functor(self, R): |
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| 321 | """ |
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| 322 | Apply the functor to an object of ``self``'s domain. |
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| 323 | |
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| 324 | TESTS:: |
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| 325 | |
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| 326 | sage: from sage.categories.pushout import CompositeConstructionFunctor |
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| 327 | sage: F = CompositeConstructionFunctor(QQ.construction()[0],ZZ['x'].construction()[0],QQ.construction()[0],ZZ['y'].construction()[0]) |
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| 328 | sage: R.<a,b> = QQ[] |
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| 329 | sage: F(R) # indirect doctest |
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| 330 | Univariate Polynomial Ring in y over Fraction Field of Univariate Polynomial Ring in x over Fraction Field of Multivariate Polynomial Ring in a, b over Rational Field |
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| 331 | |
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| 332 | """ |
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| 333 | for c in self.all: |
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| 334 | R = c(R) |
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| 335 | return R |
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| 336 | |
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| 337 | def __cmp__(self, other): |
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| 338 | """ |
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| 339 | TESTS:: |
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| 340 | |
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| 341 | sage: from sage.categories.pushout import CompositeConstructionFunctor |
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| 342 | sage: F = CompositeConstructionFunctor(QQ.construction()[0],ZZ['x'].construction()[0],QQ.construction()[0],ZZ['y'].construction()[0]) |
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| 343 | sage: F == loads(dumps(F)) # indirect doctest |
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| 344 | True |
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| 345 | |
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| 346 | """ |
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| 347 | if isinstance(other, CompositeConstructionFunctor): |
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| 348 | return cmp(self.all, other.all) |
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| 349 | else: |
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| 350 | return cmp(type(self), type(other)) |
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| 351 | |
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| 352 | def __mul__(self, other): |
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| 353 | """ |
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| 354 | Compose construction functors to a composit construction functor, unless one of them is the identity. |
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| 355 | |
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| 356 | NOTE: |
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| 357 | |
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| 358 | The product is in functorial notation, i.e., when applying the product to an object |
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| 359 | then the second factor is applied first. |
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| 360 | |
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| 361 | EXAMPLES:: |
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| 362 | |
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| 363 | sage: from sage.categories.pushout import CompositeConstructionFunctor |
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| 364 | sage: F1 = CompositeConstructionFunctor(QQ.construction()[0],ZZ['x'].construction()[0]) |
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| 365 | sage: F2 = CompositeConstructionFunctor(QQ.construction()[0],ZZ['y'].construction()[0]) |
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| 366 | sage: F1*F2 |
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| 367 | Poly[x](FractionField(Poly[y](FractionField(...)))) |
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| 368 | |
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| 369 | """ |
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| 370 | if isinstance(self, CompositeConstructionFunctor): |
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| 371 | all = [other] + self.all |
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| 372 | elif isinstance(other,IdentityConstructionFunctor): |
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| 373 | return self |
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| 374 | else: |
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| 375 | all = other.all + [self] |
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| 376 | return CompositeConstructionFunctor(*all) |
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| 377 | |
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| 378 | def __str__(self): |
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| 379 | """ |
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| 380 | TESTS:: |
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| 381 | |
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| 382 | sage: from sage.categories.pushout import CompositeConstructionFunctor |
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| 383 | sage: F = CompositeConstructionFunctor(QQ.construction()[0],ZZ['x'].construction()[0],QQ.construction()[0],ZZ['y'].construction()[0]) |
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| 384 | sage: F # indirect doctest |
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| 385 | Poly[y](FractionField(Poly[x](FractionField(...)))) |
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| 386 | |
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| 387 | """ |
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| 388 | s = "..." |
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| 389 | for c in self.all: |
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| 390 | s = "%s(%s)" % (c,s) |
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| 391 | return s |
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| 392 | |
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| 393 | def expand(self): |
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| 394 | """ |
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| 395 | Return expansion of a CompositeConstructionFunctor. |
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| 396 | |
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| 397 | NOTE: |
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| 398 | |
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| 399 | The product over the list of components, as returned by |
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| 400 | the ``expand()`` method, is equal to ``self``. |
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| 401 | |
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| 402 | EXAMPLES:: |
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| 403 | |
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| 404 | sage: from sage.categories.pushout import CompositeConstructionFunctor |
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| 405 | sage: F = CompositeConstructionFunctor(QQ.construction()[0],ZZ['x'].construction()[0],QQ.construction()[0],ZZ['y'].construction()[0]) |
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| 406 | sage: F |
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| 407 | Poly[y](FractionField(Poly[x](FractionField(...)))) |
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| 408 | sage: prod(F.expand()) == F |
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| 409 | True |
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| 410 | |
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| 411 | """ |
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| 412 | return list(reversed(self.all)) |
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| 413 | |
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| 414 | |
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| 415 | class IdentityConstructionFunctor(ConstructionFunctor): |
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| 416 | """ |
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| 417 | A construction functor that is the identity functor. |
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| 418 | |
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| 419 | TESTS:: |
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| 420 | |
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| 421 | sage: from sage.categories.pushout import IdentityConstructionFunctor |
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| 422 | sage: I = IdentityConstructionFunctor() |
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| 423 | """ |
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| 424 | rank = -100 |
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| 425 | |
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| 426 | def __init__(self): |
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| 427 | """ |
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| 428 | TESTS:: |
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| 429 | |
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| 430 | sage: from sage.categories.pushout import IdentityConstructionFunctor |
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| 431 | sage: I = IdentityConstructionFunctor() |
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| 432 | sage: IdentityFunctor(Sets()) == I |
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| 433 | True |
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| 434 | sage: I(RR) is RR |
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| 435 | True |
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| 436 | |
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| 437 | """ |
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| 438 | ConstructionFunctor.__init__(self, Sets(), Sets()) |
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| 439 | |
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| 440 | def _apply_functor(self, x): |
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| 441 | """ |
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| 442 | Return the argument unaltered. |
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| 443 | |
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| 444 | TESTS:: |
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| 445 | |
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| 446 | sage: from sage.categories.pushout import IdentityConstructionFunctor |
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| 447 | sage: I = IdentityConstructionFunctor() |
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| 448 | sage: I(RR) is RR # indirect doctest |
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| 449 | True |
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| 450 | """ |
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| 451 | return x |
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| 452 | |
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| 453 | def _apply_functor_to_morphism(self, f): |
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| 454 | """ |
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| 455 | Return the argument unaltered. |
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| 456 | |
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| 457 | TESTS:: |
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| 458 | |
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| 459 | sage: from sage.categories.pushout import IdentityConstructionFunctor |
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| 460 | sage: I = IdentityConstructionFunctor() |
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| 461 | sage: f = ZZ['t'].hom(['x'],QQ['x']) |
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| 462 | """ |
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| 463 | return f |
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| 464 | |
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| 465 | def __cmp__(self, other): |
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| 466 | """ |
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| 467 | TESTS:: |
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| 468 | |
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| 469 | sage: from sage.categories.pushout import IdentityConstructionFunctor |
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| 470 | sage: I = IdentityConstructionFunctor() |
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| 471 | sage: I == IdentityFunctor(Sets()) # indirect doctest |
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| 472 | True |
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| 473 | sage: I == QQ.construction()[0] |
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| 474 | False |
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| 475 | |
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| 476 | """ |
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| 477 | |
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| 478 | def __mul__(self, other): |
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| 479 | """ |
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| 480 | Compose construction functors to a composit construction functor, unless one of them is the identity. |
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| 481 | |
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| 482 | NOTE: |
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| 483 | |
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| 484 | The product is in functorial notation, i.e., when applying the product to an object |
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| 485 | then the second factor is applied first. |
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| 486 | |
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| 487 | TESTS:: |
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| 488 | |
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| 489 | sage: from sage.categories.pushout import IdentityConstructionFunctor |
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| 490 | sage: I = IdentityConstructionFunctor() |
|---|
| 491 | sage: F = QQ.construction()[0] |
|---|
| 492 | sage: P = ZZ['t'].construction()[0] |
|---|
| 493 | sage: I*F is F # indirect doctest |
|---|
| 494 | True |
|---|
| 495 | sage: F*I is F |
|---|
| 496 | True |
|---|
| 497 | sage: I*P is P |
|---|
| 498 | True |
|---|
| 499 | sage: P*I is P |
|---|
| 500 | True |
|---|
| 501 | |
|---|
| 502 | """ |
|---|
| 503 | if isinstance(self, IdentityConstructionFunctor): |
|---|
| 504 | return other |
|---|
| 505 | else: |
|---|
| 506 | return self |
|---|
| 507 | |
|---|
| 508 | |
|---|
| 509 | class PolynomialFunctor(ConstructionFunctor): |
|---|
| 510 | """ |
|---|
| 511 | Construction functor for univariate polynomial rings. |
|---|
| 512 | |
|---|
| 513 | EXAMPLE:: |
|---|
| 514 | |
|---|
| 515 | sage: P = ZZ['t'].construction()[0] |
|---|
| 516 | sage: P(GF(3)) |
|---|
| 517 | Univariate Polynomial Ring in t over Finite Field of size 3 |
|---|
| 518 | sage: P == loads(dumps(P)) |
|---|
| 519 | True |
|---|
| 520 | sage: R.<x,y> = GF(5)[] |
|---|
| 521 | sage: f = R.hom([x+2*y,3*x-y],R) |
|---|
| 522 | sage: P(f)((x+y)*P(R).0) |
|---|
| 523 | (-x + y)*t |
|---|
| 524 | |
|---|
| 525 | By trac ticket #9944, the construction functor distinguishes sparse and |
|---|
| 526 | dense polynomial rings. Before, the following example failed:: |
|---|
| 527 | |
|---|
| 528 | sage: R.<x> = PolynomialRing(GF(5), sparse=True) |
|---|
| 529 | sage: F,B = R.construction() |
|---|
| 530 | sage: F(B) is R |
|---|
| 531 | True |
|---|
| 532 | sage: S.<x> = PolynomialRing(ZZ) |
|---|
| 533 | sage: R.has_coerce_map_from(S) |
|---|
| 534 | False |
|---|
| 535 | sage: S.has_coerce_map_from(R) |
|---|
| 536 | False |
|---|
| 537 | sage: S.0 + R.0 |
|---|
| 538 | 2*x |
|---|
| 539 | sage: 1+1 |
|---|
| 540 | 2 |
|---|
| 541 | sage: 1+1 |
|---|
| 542 | 2 |
|---|
| 543 | """ |
|---|
| 544 | rank = 9 |
|---|
| 545 | |
|---|
| 546 | def __init__(self, var, multi_variate=False, sparse=False): |
|---|
| 547 | """ |
|---|
| 548 | TESTS:: |
|---|
| 549 | |
|---|
| 550 | sage: from sage.categories.pushout import PolynomialFunctor |
|---|
| 551 | sage: P = PolynomialFunctor('x') |
|---|
| 552 | sage: P(GF(3)) |
|---|
| 553 | Univariate Polynomial Ring in x over Finite Field of size 3 |
|---|
| 554 | |
|---|
| 555 | There is an optional parameter ``multi_variate``, but |
|---|
| 556 | apparently it is not used:: |
|---|
| 557 | |
|---|
| 558 | sage: Q = PolynomialFunctor('x',multi_variate=True) |
|---|
| 559 | sage: Q(ZZ) |
|---|
| 560 | Univariate Polynomial Ring in x over Integer Ring |
|---|
| 561 | sage: Q == P |
|---|
| 562 | True |
|---|
| 563 | |
|---|
| 564 | """ |
|---|
| 565 | from rings import Rings |
|---|
| 566 | Functor.__init__(self, Rings(), Rings()) |
|---|
| 567 | self.var = var |
|---|
| 568 | self.multi_variate = multi_variate |
|---|
| 569 | self.sparse = sparse |
|---|
| 570 | |
|---|
| 571 | def _apply_functor(self, R): |
|---|
| 572 | """ |
|---|
| 573 | Apply the functor to an object of ``self``'s domain. |
|---|
| 574 | |
|---|
| 575 | TEST:: |
|---|
| 576 | |
|---|
| 577 | sage: P = ZZ['x'].construction()[0] |
|---|
| 578 | sage: P(GF(3)) # indirect doctest |
|---|
| 579 | Univariate Polynomial Ring in x over Finite Field of size 3 |
|---|
| 580 | |
|---|
| 581 | """ |
|---|
| 582 | from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing |
|---|
| 583 | return PolynomialRing(R, self.var, sparse=self.sparse) |
|---|
| 584 | |
|---|
| 585 | def __cmp__(self, other): |
|---|
| 586 | """ |
|---|
| 587 | TESTS:: |
|---|
| 588 | |
|---|
| 589 | sage: from sage.categories.pushout import MultiPolynomialFunctor |
|---|
| 590 | sage: Q = MultiPolynomialFunctor(('x',),'lex') |
|---|
| 591 | sage: P = ZZ['x'].construction()[0] |
|---|
| 592 | sage: P |
|---|
| 593 | Poly[x] |
|---|
| 594 | sage: Q |
|---|
| 595 | MPoly[x] |
|---|
| 596 | sage: P == Q |
|---|
| 597 | True |
|---|
| 598 | sage: P == loads(dumps(P)) |
|---|
| 599 | True |
|---|
| 600 | sage: P == QQ.construction()[0] |
|---|
| 601 | False |
|---|
| 602 | """ |
|---|
| 603 | |
|---|
| 604 | def merge(self, other): |
|---|
| 605 | """ |
|---|
| 606 | EXAMPLE:: |
|---|
| 607 | |
|---|
| 608 | sage: P = ZZ['x'].construction()[0] |
|---|
| 609 | sage: Q = ZZ['y','x'].construction()[0] |
|---|
| 610 | sage: P.merge(Q) |
|---|
| 611 | sage: P.merge(P) is P |
|---|
| 612 | True |
|---|
| 613 | |
|---|
| 614 | """ |
|---|
| 615 | if isinstance(other, MultiPolynomialFunctor): |
|---|
| 616 | return other.merge(self) |
|---|
| 617 | elif self == other: |
|---|
| 618 | # i.e., they only differ in sparsity |
|---|
| 619 | if not self.sparse: |
|---|
| 620 | return self |
|---|
| 621 | return other |
|---|
| 622 | else: |
|---|
| 623 | return None |
|---|
| 624 | |
|---|
| 625 | def __str__(self): |
|---|
| 626 | """ |
|---|
| 627 | TEST:: |
|---|
| 628 | |
|---|
| 629 | sage: P = ZZ['x'].construction()[0] |
|---|
| 630 | sage: P # indirect doctest |
|---|
| 631 | Poly[x] |
|---|
| 632 | |
|---|
| 633 | """ |
|---|
| 634 | return "Poly[%s]" % self.var |
|---|
| 635 | |
|---|
| 636 | class MultiPolynomialFunctor(ConstructionFunctor): |
|---|
| 637 | """ |
|---|
| 638 | A constructor for multivariate polynomial rings. |
|---|
| 639 | |
|---|
| 640 | EXAMPLES:: |
|---|
| 641 | |
|---|
| 642 | sage: P.<x,y> = ZZ[] |
|---|
| 643 | sage: F = P.construction()[0]; F |
|---|
| 644 | MPoly[x,y] |
|---|
| 645 | sage: A.<a,b> = GF(5)[] |
|---|
| 646 | sage: F(A) |
|---|
| 647 | Multivariate Polynomial Ring in x, y over Multivariate Polynomial Ring in a, b over Finite Field of size 5 |
|---|
| 648 | sage: f = A.hom([a+b,a-b],A) |
|---|
| 649 | sage: F(f) |
|---|
| 650 | Ring endomorphism of Multivariate Polynomial Ring in x, y over Multivariate Polynomial Ring in a, b over Finite Field of size 5 |
|---|
| 651 | Defn: Induced from base ring by |
|---|
| 652 | Ring endomorphism of Multivariate Polynomial Ring in a, b over Finite Field of size 5 |
|---|
| 653 | Defn: a |--> a + b |
|---|
| 654 | b |--> a - b |
|---|
| 655 | sage: F(f)(F(A)(x)*a) |
|---|
| 656 | (a + b)*x |
|---|
| 657 | |
|---|
| 658 | """ |
|---|
| 659 | |
|---|
| 660 | rank = 9 |
|---|
| 661 | |
|---|
| 662 | def __init__(self, vars, term_order): |
|---|
| 663 | """ |
|---|
| 664 | EXAMPLES:: |
|---|
| 665 | |
|---|
| 666 | sage: F = sage.categories.pushout.MultiPolynomialFunctor(['x','y'], None) |
|---|
| 667 | sage: F(CC) |
|---|
| 668 | Multivariate Polynomial Ring in x, y over Complex Field with 53 bits of precision |
|---|
| 669 | """ |
|---|
| 670 | Functor.__init__(self, Rings(), Rings()) |
|---|
| 671 | self.vars = vars |
|---|
| 672 | self.term_order = term_order |
|---|
| 673 | |
|---|
| 674 | def _apply_functor(self, R): |
|---|
| 675 | """ |
|---|
| 676 | Apply the functor to an object of ``self``'s domain. |
|---|
| 677 | |
|---|
| 678 | EXAMPLES:: |
|---|
| 679 | |
|---|
| 680 | sage: R.<x,y,z> = QQ[] |
|---|
| 681 | sage: F = R.construction()[0]; F |
|---|
| 682 | MPoly[x,y,z] |
|---|
| 683 | sage: F(RR) # indirect doctest |
|---|
| 684 | Multivariate Polynomial Ring in x, y, z over Real Field with 53 bits of precision |
|---|
| 685 | sage: F(RR) # indirect doctest |
|---|
| 686 | Multivariate Polynomial Ring in x, y, z over Real Field with 53 bits of precision |
|---|
| 687 | sage: F(RR) # indirect doctest |
|---|
| 688 | Multivariate Polynomial Ring in x, y, z over Real Field with 53 bits of precision |
|---|
| 689 | sage: F(RR) # indirect doctest |
|---|
| 690 | Multivariate Polynomial Ring in x, y, z over Real Field with 53 bits of precision |
|---|
| 691 | sage: F(RR) # indirect doctest |
|---|
| 692 | Multivariate Polynomial Ring in x, y, z over Real Field with 53 bits of precision |
|---|
| 693 | """ |
|---|
| 694 | from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing |
|---|
| 695 | return PolynomialRing(R, self.vars) |
|---|
| 696 | |
|---|
| 697 | def __cmp__(self, other): |
|---|
| 698 | """ |
|---|
| 699 | EXAMPLES:: |
|---|
| 700 | |
|---|
| 701 | sage: F = ZZ['x,y,z'].construction()[0] |
|---|
| 702 | sage: G = QQ['x,y,z'].construction()[0] |
|---|
| 703 | sage: F == G |
|---|
| 704 | True |
|---|
| 705 | sage: G == loads(dumps(G)) |
|---|
| 706 | True |
|---|
| 707 | sage: G = ZZ['x,y'].construction()[0] |
|---|
| 708 | sage: F == G |
|---|
| 709 | False |
|---|
| 710 | """ |
|---|
| 711 | c = cmp(type(self), type(other)) |
|---|
| 712 | if c == 0: |
|---|
| 713 | c = cmp(self.vars, other.vars) or cmp(self.term_order, other.term_order) |
|---|
| 714 | elif isinstance(other, PolynomialFunctor): |
|---|
| 715 | c = cmp(self.vars, (other.var,)) |
|---|
| 716 | return c |
|---|
| 717 | |
|---|
| 718 | def __mul__(self, other): |
|---|
| 719 | """ |
|---|
| 720 | If two MPoly functors are given in a row, form a single MPoly functor |
|---|
| 721 | with all of the variables. |
|---|
| 722 | |
|---|
| 723 | EXAMPLES:: |
|---|
| 724 | |
|---|
| 725 | sage: F = sage.categories.pushout.MultiPolynomialFunctor(['x','y'], None) |
|---|
| 726 | sage: G = sage.categories.pushout.MultiPolynomialFunctor(['t'], None) |
|---|
| 727 | sage: G*F |
|---|
| 728 | MPoly[x,y,t] |
|---|
| 729 | """ |
|---|
| 730 | if isinstance(other,IdentityConstructionFunctor): |
|---|
| 731 | return self |
|---|
| 732 | if isinstance(other, MultiPolynomialFunctor): |
|---|
| 733 | if self.term_order != other.term_order: |
|---|
| 734 | raise CoercionException, "Incompatible term orders (%s,%s)." % (self.term_order, other.term_order) |
|---|
| 735 | if set(self.vars).intersection(other.vars): |
|---|
| 736 | raise CoercionException, "Overlapping variables (%s,%s)" % (self.vars, other.vars) |
|---|
| 737 | return MultiPolynomialFunctor(other.vars + self.vars, self.term_order) |
|---|
| 738 | elif isinstance(other, CompositeConstructionFunctor) \ |
|---|
| 739 | and isinstance(other.all[-1], MultiPolynomialFunctor): |
|---|
| 740 | return CompositeConstructionFunctor(other.all[:-1], self * other.all[-1]) |
|---|
| 741 | else: |
|---|
| 742 | return CompositeConstructionFunctor(other, self) |
|---|
| 743 | |
|---|
| 744 | def merge(self, other): |
|---|
| 745 | """ |
|---|
| 746 | Merge ``self`` with another construction functor, or return None. |
|---|
| 747 | |
|---|
| 748 | EXAMPLES:: |
|---|
| 749 | |
|---|
| 750 | sage: F = sage.categories.pushout.MultiPolynomialFunctor(['x','y'], None) |
|---|
| 751 | sage: G = sage.categories.pushout.MultiPolynomialFunctor(['t'], None) |
|---|
| 752 | sage: F.merge(G) is None |
|---|
| 753 | True |
|---|
| 754 | sage: F.merge(F) |
|---|
| 755 | MPoly[x,y] |
|---|
| 756 | """ |
|---|
| 757 | if self == other: |
|---|
| 758 | return self |
|---|
| 759 | else: |
|---|
| 760 | return None |
|---|
| 761 | |
|---|
| 762 | def expand(self): |
|---|
| 763 | """ |
|---|
| 764 | Decompose ``self`` into a list of construction functors. |
|---|
| 765 | |
|---|
| 766 | EXAMPLES:: |
|---|
| 767 | |
|---|
| 768 | sage: F = QQ['x,y,z,t'].construction()[0]; F |
|---|
| 769 | MPoly[x,y,z,t] |
|---|
| 770 | sage: F.expand() |
|---|
| 771 | [MPoly[t], MPoly[z], MPoly[y], MPoly[x]] |
|---|
| 772 | |
|---|
| 773 | Now an actual use case:: |
|---|
| 774 | |
|---|
| 775 | sage: R.<x,y,z> = ZZ[] |
|---|
| 776 | sage: S.<z,t> = QQ[] |
|---|
| 777 | sage: x+t |
|---|
| 778 | x + t |
|---|
| 779 | sage: parent(x+t) |
|---|
| 780 | Multivariate Polynomial Ring in x, y, z, t over Rational Field |
|---|
| 781 | sage: T.<y,s> = QQ[] |
|---|
| 782 | sage: x + s |
|---|
| 783 | Traceback (most recent call last): |
|---|
| 784 | ... |
|---|
| 785 | TypeError: unsupported operand parent(s) for '+': 'Multivariate Polynomial Ring in x, y, z over Integer Ring' and 'Multivariate Polynomial Ring in y, s over Rational Field' |
|---|
| 786 | sage: R = PolynomialRing(ZZ, 'x', 500) |
|---|
| 787 | sage: S = PolynomialRing(GF(5), 'x', 200) |
|---|
| 788 | sage: R.gen(0) + S.gen(0) |
|---|
| 789 | 2*x0 |
|---|
| 790 | """ |
|---|
| 791 | if len(self.vars) <= 1: |
|---|
| 792 | return [self] |
|---|
| 793 | else: |
|---|
| 794 | return [MultiPolynomialFunctor((x,), self.term_order) for x in reversed(self.vars)] |
|---|
| 795 | |
|---|
| 796 | def __str__(self): |
|---|
| 797 | """ |
|---|
| 798 | TEST:: |
|---|
| 799 | |
|---|
| 800 | sage: QQ['x,y,z,t'].construction()[0] |
|---|
| 801 | MPoly[x,y,z,t] |
|---|
| 802 | """ |
|---|
| 803 | return "MPoly[%s]" % ','.join(self.vars) |
|---|
| 804 | |
|---|
| 805 | |
|---|
| 806 | |
|---|
| 807 | class InfinitePolynomialFunctor(ConstructionFunctor): |
|---|
| 808 | """ |
|---|
| 809 | A Construction Functor for Infinite Polynomial Rings (see :mod:`~sage.rings.polynomial.infinite_polynomial_ring`). |
|---|
| 810 | |
|---|
| 811 | AUTHOR: |
|---|
| 812 | |
|---|
| 813 | -- Simon King |
|---|
| 814 | |
|---|
| 815 | This construction functor is used to provide uniqueness of infinite polynomial rings as parent structures. |
|---|
| 816 | As usual, the construction functor allows for constructing pushouts. |
|---|
| 817 | |
|---|
| 818 | Another purpose is to avoid name conflicts of variables of the to-be-constructed infinite polynomial ring with |
|---|
| 819 | variables of the base ring, and moreover to keep the internal structure of an Infinite Polynomial Ring as simple |
|---|
| 820 | as possible: If variables `v_1,...,v_n` of the given base ring generate an *ordered* sub-monoid of the monomials |
|---|
| 821 | of the ambient Infinite Polynomial Ring, then they are removed from the base ring and merged with the generators |
|---|
| 822 | of the ambient ring. However, if the orders don't match, an error is raised, since there was a name conflict |
|---|
| 823 | without merging. |
|---|
| 824 | |
|---|
| 825 | EXAMPLES:: |
|---|
| 826 | |
|---|
| 827 | sage: A.<a,b> = InfinitePolynomialRing(ZZ['t']) |
|---|
| 828 | sage: A.construction() |
|---|
| 829 | [InfPoly{[a,b], "lex", "dense"}, |
|---|
| 830 | Univariate Polynomial Ring in t over Integer Ring] |
|---|
| 831 | sage: type(_[0]) |
|---|
| 832 | <class 'sage.categories.pushout.InfinitePolynomialFunctor'> |
|---|
| 833 | sage: B.<x,y,a_3,a_1> = PolynomialRing(QQ, order='lex') |
|---|
| 834 | sage: B.construction() |
|---|
| 835 | (MPoly[x,y,a_3,a_1], Rational Field) |
|---|
| 836 | sage: A.construction()[0]*B.construction()[0] |
|---|
| 837 | InfPoly{[a,b], "lex", "dense"}(MPoly[x,y](...)) |
|---|
| 838 | |
|---|
| 839 | Apparently the variables `a_1,a_3` of the polynomial ring are merged with the variables |
|---|
| 840 | `a_0, a_1, a_2, ...` of the infinite polynomial ring; indeed, they form an ordered sub-structure. |
|---|
| 841 | However, if the polynomial ring was given a different ordering, merging would not be allowed, |
|---|
| 842 | resulting in a name conflict:: |
|---|
| 843 | |
|---|
| 844 | sage: A.construction()[0]*PolynomialRing(QQ,names=['x','y','a_3','a_1']).construction()[0] |
|---|
| 845 | Traceback (most recent call last): |
|---|
| 846 | ... |
|---|
| 847 | CoercionException: Incompatible term orders lex, degrevlex |
|---|
| 848 | |
|---|
| 849 | In an infinite polynomial ring with generator `a_\\ast`, the variable `a_3` will always be greater |
|---|
| 850 | than the variable `a_1`. Hence, the orders are incompatible in the next example as well:: |
|---|
| 851 | |
|---|
| 852 | sage: A.construction()[0]*PolynomialRing(QQ,names=['x','y','a_1','a_3'], order='lex').construction()[0] |
|---|
| 853 | Traceback (most recent call last): |
|---|
| 854 | ... |
|---|
| 855 | CoercionException: Overlapping variables (('a', 'b'),['a_1', 'a_3']) are incompatible |
|---|
| 856 | |
|---|
| 857 | Another requirement is that after merging the order of the remaining variables must be unique. |
|---|
| 858 | This is not the case in the following example, since it is not clear whether the variables `x,y` |
|---|
| 859 | should be greater or smaller than the variables `b_\\ast`:: |
|---|
| 860 | |
|---|
| 861 | sage: A.construction()[0]*PolynomialRing(QQ,names=['a_3','a_1','x','y'], order='lex').construction()[0] |
|---|
| 862 | Traceback (most recent call last): |
|---|
| 863 | ... |
|---|
| 864 | CoercionException: Overlapping variables (('a', 'b'),['a_3', 'a_1']) are incompatible |
|---|
| 865 | |
|---|
| 866 | Since the construction functors are actually used to construct infinite polynomial rings, the following |
|---|
| 867 | result is no surprise:: |
|---|
| 868 | |
|---|
| 869 | sage: C.<a,b> = InfinitePolynomialRing(B); C |
|---|
| 870 | Infinite polynomial ring in a, b over Multivariate Polynomial Ring in x, y over Rational Field |
|---|
| 871 | |
|---|
| 872 | There is also an overlap in the next example:: |
|---|
| 873 | |
|---|
| 874 | sage: X.<w,x,y> = InfinitePolynomialRing(ZZ) |
|---|
| 875 | sage: Y.<x,y,z> = InfinitePolynomialRing(QQ) |
|---|
| 876 | |
|---|
| 877 | `X` and `Y` have an overlapping generators `x_\\ast, y_\\ast`. Since the default lexicographic order is |
|---|
| 878 | used in both rings, it gives rise to isomorphic sub-monoids in both `X` and `Y`. They are merged in the |
|---|
| 879 | pushout, which also yields a common parent for doing arithmetic:: |
|---|
| 880 | |
|---|
| 881 | sage: P = sage.categories.pushout.pushout(Y,X); P |
|---|
| 882 | Infinite polynomial ring in w, x, y, z over Rational Field |
|---|
| 883 | sage: w[2]+z[3] |
|---|
| 884 | w_2 + z_3 |
|---|
| 885 | sage: _.parent() is P |
|---|
| 886 | True |
|---|
| 887 | """ |
|---|
| 888 | |
|---|
| 889 | def __init__(self, gens, order, implementation): |
|---|
| 890 | """ |
|---|
| 891 | TEST:: |
|---|
| 892 | |
|---|
| 893 | sage: 1+1 |
|---|
| 894 | 2 |
|---|
| 895 | sage: 1+1 |
|---|
| 896 | 2 |
|---|
| 897 | """ |
|---|
| 898 | |
|---|
| 899 | def _apply_functor_to_morphism(self, f): |
|---|
| 900 | """ |
|---|
| 901 | Morphisms for inifinite polynomial rings are not implemented yet. |
|---|
| 902 | |
|---|
| 903 | TEST:: |
|---|
| 904 | |
|---|
| 905 | sage: P.<x,y> = QQ[] |
|---|
| 906 | sage: R.<alpha> = InfinitePolynomialRing(P) |
|---|
| 907 | sage: 1+1 |
|---|
| 908 | 2 |
|---|
| 909 | sage: 1+1 |
|---|
| 910 | 2 |
|---|
| 911 | sage: 1+1 |
|---|
| 912 | 2 |
|---|
| 913 | """ |
|---|
| 914 | |
|---|
| 915 | def _apply_functor(self, R): |
|---|
| 916 | """ |
|---|
| 917 | Apply the functor to an object of ``self``'s domain. |
|---|
| 918 | |
|---|
| 919 | TEST:: |
|---|
| 920 | |
|---|
| 921 | sage: F = sage.categories.pushout.InfinitePolynomialFunctor(['a','b','x'],'degrevlex','sparse'); F |
|---|
| 922 | InfPoly{[a,b,x], "degrevlex", "sparse"} |
|---|
| 923 | sage: F(QQ['t']) # indirect doctest |
|---|
| 924 | Infinite polynomial ring in a, b, x over Univariate Polynomial Ring in t over Rational Field |
|---|
| 925 | |
|---|
| 926 | """ |
|---|
| 927 | from sage.rings.polynomial.infinite_polynomial_ring import InfinitePolynomialRing |
|---|
| 928 | return InfinitePolynomialRing(R, self._gens, order=self._order, implementation=self._imple) |
|---|
| 929 | |
|---|
| 930 | def __str__(self): |
|---|
| 931 | """ |
|---|
| 932 | TEST:: |
|---|
| 933 | |
|---|
| 934 | sage: F = sage.categories.pushout.InfinitePolynomialFunctor(['a','b','x'],'degrevlex','sparse'); F # indirect doctest |
|---|
| 935 | InfPoly{[a,b,x], "degrevlex", "sparse"} |
|---|
| 936 | |
|---|
| 937 | """ |
|---|
| 938 | return 'InfPoly{[%s], "%s", "%s"}'%(','.join(self._gens), self._order, self._imple) |
|---|
| 939 | |
|---|
| 940 | def __cmp__(self, other): |
|---|
| 941 | """ |
|---|
| 942 | TEST:: |
|---|
| 943 | |
|---|
| 944 | sage: F = sage.categories.pushout.InfinitePolynomialFunctor(['a','b','x'],'degrevlex','sparse'); F # indirect doctest |
|---|
| 945 | InfPoly{[a,b,x], "degrevlex", "sparse"} |
|---|
| 946 | sage: F == sage.categories.pushout.InfinitePolynomialFunctor(['a','b','x'],'deglex','sparse') |
|---|
| 947 | False |
|---|
| 948 | |
|---|
| 949 | """ |
|---|
| 950 | c = cmp(type(self), type(other)) |
|---|
| 951 | if c == 0: |
|---|
| 952 | c = cmp(self._gens, other._gens) or cmp(self._order, other._order) or cmp(self._imple, other._imple) |
|---|
| 953 | return c |
|---|
| 954 | |
|---|
| 955 | def __mul__(self, other): |
|---|
| 956 | """ |
|---|
| 957 | Compose construction functors to a composit construction functor, unless one of them is the identity. |
|---|
| 958 | |
|---|
| 959 | NOTE: |
|---|
| 960 | |
|---|
| 961 | The product is in functorial notation, i.e., when applying the product to an object |
|---|
| 962 | then the second factor is applied first. |
|---|
| 963 | |
|---|
| 964 | TESTS:: |
|---|
| 965 | |
|---|
| 966 | sage: F1 = QQ['a','x_2','x_1','y_3','y_2'].construction()[0]; F1 |
|---|
| 967 | MPoly[a,x_2,x_1,y_3,y_2] |
|---|
| 968 | sage: F2 = InfinitePolynomialRing(QQ, ['x','y'],order='degrevlex').construction()[0]; F2 |
|---|
| 969 | InfPoly{[x,y], "degrevlex", "dense"} |
|---|
| 970 | sage: F3 = InfinitePolynomialRing(QQ, ['x','y'],order='degrevlex',implementation='sparse').construction()[0]; F3 |
|---|
| 971 | InfPoly{[x,y], "degrevlex", "sparse"} |
|---|
| 972 | sage: F2*F1 |
|---|
| 973 | InfPoly{[x,y], "degrevlex", "dense"}(Poly[a](...)) |
|---|
| 974 | sage: F3*F1 |
|---|
| 975 | InfPoly{[x,y], "degrevlex", "sparse"}(Poly[a](...)) |
|---|
| 976 | sage: F4 = sage.categories.pushout.FractionField() |
|---|
| 977 | sage: F2*F4 |
|---|
| 978 | InfPoly{[x,y], "degrevlex", "dense"}(FractionField(...)) |
|---|
| 979 | |
|---|
| 980 | """ |
|---|
| 981 | if isinstance(other,IdentityConstructionFunctor): |
|---|
| 982 | return self |
|---|
| 983 | if isinstance(other, self.__class__): # |
|---|
| 984 | INT = set(self._gens).intersection(other._gens) |
|---|
| 985 | if INT: |
|---|
| 986 | # if there is overlap of generators, it must only be at the ends, so that |
|---|
| 987 | # the resulting order after the merging is unique |
|---|
| 988 | if other._gens[-len(INT):] != self._gens[:len(INT)]: |
|---|
| 989 | raise CoercionException, "Overlapping variables (%s,%s) are incompatible" % (self._gens, other._gens) |
|---|
| 990 | OUTGENS = list(other._gens) + list(self._gens[len(INT):]) |
|---|
| 991 | else: |
|---|
| 992 | OUTGENS = list(other._gens) + list(self._gens) |
|---|
| 993 | # the orders must coincide |
|---|
| 994 | if self._order != other._order: |
|---|
| 995 | return CompositeConstructionFunctor(other, self) |
|---|
| 996 | # the implementations must coincide |
|---|
| 997 | if self._imple != other._imple: |
|---|
| 998 | return CompositeConstructionFunctor(other, self) |
|---|
| 999 | return InfinitePolynomialFunctor(OUTGENS, self._order, self._imple) |
|---|
| 1000 | |
|---|
| 1001 | # Polynomial Constructor |
|---|
| 1002 | # Idea: We merge into self, if the polynomial functor really provides a substructure, |
|---|
| 1003 | # even respecting the order. Note that, if the pushout is computed, only *one* variable |
|---|
| 1004 | # will occur in the polynomial constructor. Hence, any order is fine, which is exactly |
|---|
| 1005 | # what we need in order to have coercion maps for different orderings. |
|---|
| 1006 | if isinstance(other, MultiPolynomialFunctor) or isinstance(other, PolynomialFunctor): |
|---|
| 1007 | if isinstance(other, MultiPolynomialFunctor): |
|---|
| 1008 | othervars = other.vars |
|---|
| 1009 | else: |
|---|
| 1010 | othervars = [other.var] |
|---|
| 1011 | OverlappingGens = [] ## Generator names of variable names of the MultiPolynomialFunctor |
|---|
| 1012 | ## that can be interpreted as variables in self |
|---|
| 1013 | OverlappingVars = [] ## The variable names of the MultiPolynomialFunctor |
|---|
| 1014 | ## that can be interpreted as variables in self |
|---|
| 1015 | RemainingVars = [x for x in othervars] |
|---|
| 1016 | IsOverlap = False |
|---|
| 1017 | BadOverlap = False |
|---|
| 1018 | for x in othervars: |
|---|
| 1019 | if x.count('_') == 1: |
|---|
| 1020 | g,n = x.split('_') |
|---|
| 1021 | if n.isdigit(): |
|---|
| 1022 | if g.isalnum(): # we can interprete x in any InfinitePolynomialRing |
|---|
| 1023 | if g in self._gens: # we can interprete x in self, hence, we will not use it as a variable anymore. |
|---|
| 1024 | RemainingVars.pop(RemainingVars.index(x)) |
|---|
| 1025 | IsOverlap = True # some variables of other can be interpreted in self. |
|---|
| 1026 | if OverlappingVars: |
|---|
| 1027 | # Is OverlappingVars in the right order? |
|---|
| 1028 | g0,n0 = OverlappingVars[-1].split('_') |
|---|
| 1029 | i = self._gens.index(g) |
|---|
| 1030 | i0 = self._gens.index(g0) |
|---|
| 1031 | if i<i0: # wrong order |
|---|
| 1032 | BadOverlap = True |
|---|
| 1033 | if i==i0 and int(n)>int(n0): # wrong order |
|---|
| 1034 | BadOverlap = True |
|---|
| 1035 | OverlappingVars.append(x) |
|---|
| 1036 | else: |
|---|
| 1037 | if IsOverlap: # The overlap must be on the right end of the variable list |
|---|
| 1038 | BadOverlap = True |
|---|
| 1039 | else: |
|---|
| 1040 | if IsOverlap: # The overlap must be on the right end of the variable list |
|---|
| 1041 | BadOverlap = True |
|---|
| 1042 | else: |
|---|
| 1043 | if IsOverlap: # The overlap must be on the right end of the variable list |
|---|
| 1044 | BadOverlap = True |
|---|
| 1045 | else: |
|---|
| 1046 | if IsOverlap: # The overlap must be on the right end of the variable list |
|---|
| 1047 | BadOverlap = True |
|---|
| 1048 | |
|---|
| 1049 | if BadOverlap: # the overlapping variables appear in the wrong order |
|---|
| 1050 | raise CoercionException, "Overlapping variables (%s,%s) are incompatible" % (self._gens, OverlappingVars) |
|---|
| 1051 | if len(OverlappingVars)>1: # multivariate, hence, the term order matters |
|---|
| 1052 | if other.term_order.name()!=self._order: |
|---|
| 1053 | raise CoercionException, "Incompatible term orders %s, %s" % (self._order, other.term_order.name()) |
|---|
| 1054 | # ok, the overlap is fine, we will return something. |
|---|
| 1055 | if RemainingVars: # we can only partially merge other into self |
|---|
| 1056 | if len(RemainingVars)>1: |
|---|
| 1057 | return CompositeConstructionFunctor(MultiPolynomialFunctor(RemainingVars,term_order=other.term_order), self) |
|---|
| 1058 | return CompositeConstructionFunctor(PolynomialFunctor(RemainingVars[0]), self) |
|---|
| 1059 | return self |
|---|
| 1060 | return CompositeConstructionFunctor(other, self) |
|---|
| 1061 | |
|---|
| 1062 | def merge(self,other): |
|---|
| 1063 | """ |
|---|
| 1064 | Merge two construction functors of infinite polynomial rings, regardless of monomial order and implementation. |
|---|
| 1065 | |
|---|
| 1066 | The purpose is to have a pushout (and thus, arithmetic) even in cases when the parents are isomorphic as |
|---|
| 1067 | rings, but not as ordered rings. |
|---|
| 1068 | |
|---|
| 1069 | EXAMPLES:: |
|---|
| 1070 | |
|---|
| 1071 | sage: X.<x,y> = InfinitePolynomialRing(QQ,implementation='sparse') |
|---|
| 1072 | sage: Y.<x,y> = InfinitePolynomialRing(QQ,order='degrevlex') |
|---|
| 1073 | sage: X.construction() |
|---|
| 1074 | [InfPoly{[x,y], "lex", "sparse"}, Rational Field] |
|---|
| 1075 | sage: Y.construction() |
|---|
| 1076 | [InfPoly{[x,y], "degrevlex", "dense"}, Rational Field] |
|---|
| 1077 | sage: Y.construction()[0].merge(Y.construction()[0]) |
|---|
| 1078 | InfPoly{[x,y], "degrevlex", "dense"} |
|---|
| 1079 | sage: y[3] + X(x[2]) |
|---|
| 1080 | x_2 + y_3 |
|---|
| 1081 | sage: _.parent().construction() |
|---|
| 1082 | [InfPoly{[x,y], "degrevlex", "dense"}, Rational Field] |
|---|
| 1083 | |
|---|
| 1084 | """ |
|---|
| 1085 | # Merging is only done if the ranks of self and other are the same. |
|---|
| 1086 | # It may happen that other is a substructure of self up to the monomial order |
|---|
| 1087 | # and the implementation. And this is when we want to merge, in order to |
|---|
| 1088 | # provide multiplication for rings with different term orderings. |
|---|
| 1089 | if not isinstance(other, InfinitePolynomialFunctor): |
|---|
| 1090 | return None |
|---|
| 1091 | if set(other._gens).issubset(self._gens): |
|---|
| 1092 | return self |
|---|
| 1093 | return None |
|---|
| 1094 | try: |
|---|
| 1095 | OUT = self*other |
|---|
| 1096 | # The following happens if "other" has the same order type etc. |
|---|
| 1097 | if not isinstance(OUT, CompositeConstructionFunctor): |
|---|
| 1098 | return OUT |
|---|
| 1099 | except CoercionException: |
|---|
| 1100 | pass |
|---|
| 1101 | if isinstance(other,InfinitePolynomialFunctor): |
|---|
| 1102 | # We don't require that the orders coincide. This is a difference to self*other |
|---|
| 1103 | # We only merge if other's generators are an ordered subset of self's generators |
|---|
| 1104 | for g in other._gens: |
|---|
| 1105 | if g not in self._gens: |
|---|
| 1106 | return None |
|---|
| 1107 | # The sequence of variables is part of the ordering. It must coincide in both rings |
|---|
| 1108 | Ind = [self._gens.index(g) for g in other._gens] |
|---|
| 1109 | if sorted(Ind)!=Ind: |
|---|
| 1110 | return None |
|---|
| 1111 | # OK, other merges into self. Now, chose the default dense implementation, |
|---|
| 1112 | # unless both functors refer to the sparse implementation |
|---|
| 1113 | if self._imple != other._imple: |
|---|
| 1114 | return InfinitePolynomialFunctor(self._gens, self._order, 'dense') |
|---|
| 1115 | return self |
|---|
| 1116 | return None |
|---|
| 1117 | |
|---|
| 1118 | def expand(self): |
|---|
| 1119 | """ |
|---|
| 1120 | Decompose the functor `F` into sub-functors, whose product returns `F`. |
|---|
| 1121 | |
|---|
| 1122 | EXAMPLES:: |
|---|
| 1123 | |
|---|
| 1124 | sage: F = InfinitePolynomialRing(QQ, ['x','y'],order='degrevlex').construction()[0]; F |
|---|
| 1125 | InfPoly{[x,y], "degrevlex", "dense"} |
|---|
| 1126 | sage: F.expand() |
|---|
| 1127 | [InfPoly{[y], "degrevlex", "dense"}, InfPoly{[x], "degrevlex", "dense"}] |
|---|
| 1128 | sage: F = InfinitePolynomialRing(QQ, ['x','y','z'],order='degrevlex').construction()[0]; F |
|---|
| 1129 | InfPoly{[x,y,z], "degrevlex", "dense"} |
|---|
| 1130 | sage: F.expand() |
|---|
| 1131 | [InfPoly{[z], "degrevlex", "dense"}, |
|---|
| 1132 | InfPoly{[y], "degrevlex", "dense"}, |
|---|
| 1133 | InfPoly{[x], "degrevlex", "dense"}] |
|---|
| 1134 | sage: prod(F.expand())==F |
|---|
| 1135 | True |
|---|
| 1136 | |
|---|
| 1137 | """ |
|---|
| 1138 | if len(self._gens)==1: |
|---|
| 1139 | return [self] |
|---|
| 1140 | return [InfinitePolynomialFunctor((x,), self._order, self._imple) for x in reversed(self._gens)] |
|---|
| 1141 | |
|---|
| 1142 | |
|---|
| 1143 | |
|---|
| 1144 | class MatrixFunctor(ConstructionFunctor): |
|---|
| 1145 | """ |
|---|
| 1146 | A construction functor for matrices over rings. |
|---|
| 1147 | |
|---|
| 1148 | EXAMPLES:: |
|---|
| 1149 | |
|---|
| 1150 | sage: MS = MatrixSpace(ZZ,2, 3) |
|---|
| 1151 | sage: F = MS.construction()[0]; F |
|---|
| 1152 | MatrixFunctor |
|---|
| 1153 | sage: MS = MatrixSpace(ZZ,2) |
|---|
| 1154 | sage: F = MS.construction()[0]; F |
|---|
| 1155 | MatrixFunctor |
|---|
| 1156 | sage: P.<x,y> = QQ[] |
|---|
| 1157 | sage: R = F(P); R |
|---|
| 1158 | Full MatrixSpace of 2 by 2 dense matrices over Multivariate Polynomial Ring in x, y over Rational Field |
|---|
| 1159 | sage: f = P.hom([x+y,x-y],P); F(f) |
|---|
| 1160 | Ring endomorphism of Full MatrixSpace of 2 by 2 dense matrices over Multivariate Polynomial Ring in x, y over Rational Field |
|---|
| 1161 | Defn: Induced from base ring by |
|---|
| 1162 | Ring endomorphism of Multivariate Polynomial Ring in x, y over Rational Field |
|---|
| 1163 | Defn: x |--> x + y |
|---|
| 1164 | y |--> x - y |
|---|
| 1165 | sage: M = R([x,y,x*y,x+y]) |
|---|
| 1166 | sage: F(f)(M) |
|---|
| 1167 | [ x + y x - y] |
|---|
| 1168 | [x^2 - y^2 2*x] |
|---|
| 1169 | |
|---|
| 1170 | """ |
|---|
| 1171 | rank = 10 |
|---|
| 1172 | |
|---|
| 1173 | def __init__(self, nrows, ncols, is_sparse=False): |
|---|
| 1174 | """ |
|---|
| 1175 | TEST:: |
|---|
| 1176 | |
|---|
| 1177 | sage: from sage.categories.pushout import MatrixFunctor |
|---|
| 1178 | sage: F = MatrixFunctor(2,3) |
|---|
| 1179 | sage: F == MatrixSpace(ZZ,2,3).construction()[0] |
|---|
| 1180 | True |
|---|
| 1181 | sage: F.codomain() |
|---|
| 1182 | Category of commutative additive groups |
|---|
| 1183 | sage: R = MatrixSpace(ZZ,2,2).construction()[0] |
|---|
| 1184 | sage: R.codomain() |
|---|
| 1185 | Category of rings |
|---|
| 1186 | sage: F(ZZ) |
|---|
| 1187 | Full MatrixSpace of 2 by 3 dense matrices over Integer Ring |
|---|
| 1188 | sage: F(ZZ) in F.codomain() |
|---|
| 1189 | True |
|---|
| 1190 | sage: R(GF(2)) |
|---|
| 1191 | Full MatrixSpace of 2 by 2 dense matrices over Finite Field of size 2 |
|---|
| 1192 | sage: R(GF(2)) in R.codomain() |
|---|
| 1193 | True |
|---|
| 1194 | """ |
|---|
| 1195 | if nrows == ncols: |
|---|
| 1196 | Functor.__init__(self, Rings(), Rings()) # Algebras() takes a base ring |
|---|
| 1197 | else: |
|---|
| 1198 | # Functor.__init__(self, Rings(), MatrixAlgebras()) # takes a base ring |
|---|
| 1199 | Functor.__init__(self, Rings(), CommutativeAdditiveGroups()) # not a nice solution, but the best we can do. |
|---|
| 1200 | self.nrows = nrows |
|---|
| 1201 | self.ncols = ncols |
|---|
| 1202 | self.is_sparse = is_sparse |
|---|
| 1203 | |
|---|
| 1204 | def _apply_functor(self, R): |
|---|
| 1205 | """ |
|---|
| 1206 | Apply the functor to an object of ``self``'s domain. |
|---|
| 1207 | |
|---|
| 1208 | TEST: |
|---|
| 1209 | |
|---|
| 1210 | The following is a test against a bug discussed at ticket #8800 |
|---|
| 1211 | |
|---|
| 1212 | sage: F = MatrixSpace(ZZ,2,3).construction()[0] |
|---|
| 1213 | sage: F(RR) # indirect doctest |
|---|
| 1214 | Full MatrixSpace of 2 by 3 dense matrices over Real Field with 53 bits of precision |
|---|
| 1215 | sage: F(RR) in F.codomain() |
|---|
| 1216 | True |
|---|
| 1217 | |
|---|
| 1218 | """ |
|---|
| 1219 | from sage.matrix.matrix_space import MatrixSpace |
|---|
| 1220 | return MatrixSpace(R, self.nrows, self.ncols, sparse=self.is_sparse) |
|---|
| 1221 | |
|---|
| 1222 | def __cmp__(self, other): |
|---|
| 1223 | """ |
|---|
| 1224 | TEST:: |
|---|
| 1225 | |
|---|
| 1226 | sage: F = MatrixSpace(ZZ,2,3).construction()[0] |
|---|
| 1227 | sage: F == loads(dumps(F)) |
|---|
| 1228 | True |
|---|
| 1229 | sage: F == MatrixSpace(ZZ,2,2).construction()[0] |
|---|
| 1230 | False |
|---|
| 1231 | |
|---|
| 1232 | """ |
|---|
| 1233 | c = cmp(type(self), type(other)) |
|---|
| 1234 | if c == 0: |
|---|
| 1235 | c = cmp((self.nrows, self.ncols), (other.nrows, other.ncols)) |
|---|
| 1236 | return c |
|---|
| 1237 | |
|---|
| 1238 | def merge(self, other): |
|---|
| 1239 | """ |
|---|
| 1240 | Merging is only happening if both functors are matrix functors of the same dimension. |
|---|
| 1241 | The result is sparse if and only if both given functors are sparse. |
|---|
| 1242 | |
|---|
| 1243 | EXAMPLE:: |
|---|
| 1244 | |
|---|
| 1245 | sage: F1 = MatrixSpace(ZZ,2,2).construction()[0] |
|---|
| 1246 | sage: F2 = MatrixSpace(ZZ,2,3).construction()[0] |
|---|
| 1247 | sage: F3 = MatrixSpace(ZZ,2,2,sparse=True).construction()[0] |
|---|
| 1248 | sage: F1.merge(F2) |
|---|
| 1249 | sage: F1.merge(F3) |
|---|
| 1250 | MatrixFunctor |
|---|
| 1251 | sage: F13 = F1.merge(F3) |
|---|
| 1252 | sage: F13.is_sparse |
|---|
| 1253 | False |
|---|
| 1254 | sage: F1.is_sparse |
|---|
| 1255 | False |
|---|
| 1256 | sage: F3.is_sparse |
|---|
| 1257 | True |
|---|
| 1258 | sage: F3.merge(F3).is_sparse |
|---|
| 1259 | True |
|---|
| 1260 | |
|---|
| 1261 | """ |
|---|
| 1262 | if self != other: |
|---|
| 1263 | return None |
|---|
| 1264 | else: |
|---|
| 1265 | return MatrixFunctor(self.nrows, self.ncols, self.is_sparse and other.is_sparse) |
|---|
| 1266 | |
|---|
| 1267 | class LaurentPolynomialFunctor(ConstructionFunctor): |
|---|
| 1268 | """ |
|---|
| 1269 | Construction functor for Laurent polynomial rings. |
|---|
| 1270 | |
|---|
| 1271 | EXAMPLES:: |
|---|
| 1272 | |
|---|
| 1273 | sage: L.<t> = LaurentPolynomialRing(ZZ) |
|---|
| 1274 | sage: F = L.construction()[0] |
|---|
| 1275 | sage: F |
|---|
| 1276 | LaurentPolynomialFunctor |
|---|
| 1277 | sage: F(QQ) |
|---|
| 1278 | Univariate Laurent Polynomial Ring in t over Rational Field |
|---|
| 1279 | sage: K.<x> = LaurentPolynomialRing(ZZ) |
|---|
| 1280 | sage: F(K) |
|---|
| 1281 | Univariate Laurent Polynomial Ring in t over Univariate Laurent Polynomial Ring in x over Integer Ring |
|---|
| 1282 | sage: P.<x,y> = ZZ[] |
|---|
| 1283 | sage: f = P.hom([x+2*y,3*x-y],P) |
|---|
| 1284 | sage: F(f) |
|---|
| 1285 | Ring endomorphism of Univariate Laurent Polynomial Ring in t over Multivariate Polynomial Ring in x, y over Integer Ring |
|---|
| 1286 | Defn: Induced from base ring by |
|---|
| 1287 | Ring endomorphism of Multivariate Polynomial Ring in x, y over Integer Ring |
|---|
| 1288 | Defn: x |--> x + 2*y |
|---|
| 1289 | y |--> 3*x - y |
|---|
| 1290 | sage: F(f)(x*F(P).gen()^-2+y*F(P).gen()^3) |
|---|
| 1291 | (3*x - y)*t^3 + (x + 2*y)*t^-2 |
|---|
| 1292 | |
|---|
| 1293 | """ |
|---|
| 1294 | rank = 9 |
|---|
| 1295 | |
|---|
| 1296 | def __init__(self, var, multi_variate=False): |
|---|
| 1297 | """ |
|---|
| 1298 | INPUT: |
|---|
| 1299 | |
|---|
| 1300 | - ``var``, a string or a list of strings |
|---|
| 1301 | - ``multi_variate``, optional bool, default ``False`` if ``var`` is a string |
|---|
| 1302 | and ``True`` otherwise: If ``True``, application to a Laurent polynomial |
|---|
| 1303 | ring yields a multivariate Laurent polynomial ring. |
|---|
| 1304 | |
|---|
| 1305 | TESTS:: |
|---|
| 1306 | |
|---|
| 1307 | sage: from sage.categories.pushout import LaurentPolynomialFunctor |
|---|
| 1308 | sage: F1 = LaurentPolynomialFunctor('t') |
|---|
| 1309 | sage: F2 = LaurentPolynomialFunctor('s', multi_variate=True) |
|---|
| 1310 | sage: F3 = LaurentPolynomialFunctor(['s','t']) |
|---|
| 1311 | sage: F1(F2(QQ)) |
|---|
| 1312 | Univariate Laurent Polynomial Ring in t over Univariate Laurent Polynomial Ring in s over Rational Field |
|---|
| 1313 | sage: F2(F1(QQ)) |
|---|
| 1314 | Multivariate Laurent Polynomial Ring in t, s over Rational Field |
|---|
| 1315 | sage: F3(QQ) |
|---|
| 1316 | Multivariate Laurent Polynomial Ring in s, t over Rational Field |
|---|
| 1317 | |
|---|
| 1318 | """ |
|---|
| 1319 | Functor.__init__(self, Rings(), Rings()) |
|---|
| 1320 | if not isinstance(var, (basestring,tuple,list)): |
|---|
| 1321 | raise TypeError, "variable name or list of variable names expected" |
|---|
| 1322 | self.var = var |
|---|
| 1323 | self.multi_variate = multi_variate or not isinstance(var, basestring) |
|---|
| 1324 | |
|---|
| 1325 | def __cmp__(self, other): |
|---|
| 1326 | """ |
|---|
| 1327 | TESTS:: |
|---|
| 1328 | |
|---|
| 1329 | sage: from sage.categories.pushout import LaurentPolynomialFunctor |
|---|
| 1330 | sage: F1 = LaurentPolynomialFunctor('t') |
|---|
| 1331 | sage: F2 = LaurentPolynomialFunctor('t', multi_variate=True) |
|---|
| 1332 | sage: F3 = LaurentPolynomialFunctor(['s','t']) |
|---|
| 1333 | sage: F1 == F2 |
|---|
| 1334 | True |
|---|
| 1335 | sage: F1 == loads(dumps(F1)) |
|---|
| 1336 | True |
|---|
| 1337 | sage: F1 == F3 |
|---|
| 1338 | False |
|---|
| 1339 | sage: F1 == QQ.construction()[0] |
|---|
| 1340 | False |
|---|
| 1341 | |
|---|
| 1342 | """ |
|---|
| 1343 | c = cmp(type(self), type(other)) |
|---|
| 1344 | if c == 0: |
|---|
| 1345 | c = cmp(self.var, other.var) |
|---|
| 1346 | return c |
|---|
| 1347 | |
|---|
| 1348 | def merge(self, other): |
|---|
| 1349 | """ |
|---|
| 1350 | Two Laurent polynomial construction functors merge if the variable names coincide. |
|---|
| 1351 | The result is multivariate if one of the arguments is multivariate. |
|---|
| 1352 | |
|---|
| 1353 | EXAMPLE:: |
|---|
| 1354 | |
|---|
| 1355 | sage: from sage.categories.pushout import LaurentPolynomialFunctor |
|---|
| 1356 | sage: F1 = LaurentPolynomialFunctor('t') |
|---|
| 1357 | sage: F2 = LaurentPolynomialFunctor('t', multi_variate=True) |
|---|
| 1358 | sage: F1.merge(F2) |
|---|
| 1359 | LaurentPolynomialFunctor |
|---|
| 1360 | sage: F1.merge(F2)(LaurentPolynomialRing(GF(2),'a')) |
|---|
| 1361 | Multivariate Laurent Polynomial Ring in a, t over Finite Field of size 2 |
|---|
| 1362 | sage: F1.merge(F1)(LaurentPolynomialRing(GF(2),'a')) |
|---|
| 1363 | Univariate Laurent Polynomial Ring in t over Univariate Laurent Polynomial Ring in a over Finite Field of size 2 |
|---|
| 1364 | |
|---|
| 1365 | """ |
|---|
| 1366 | if self == other or isinstance(other, PolynomialFunctor) and self.var == other.var: |
|---|
| 1367 | return LaurentPolynomialFunctor(self.var, (self.multi_variate or other.multi_variate)) |
|---|
| 1368 | else: |
|---|
| 1369 | return None |
|---|
| 1370 | |
|---|
| 1371 | |
|---|
| 1372 | class VectorFunctor(ConstructionFunctor): |
|---|
| 1373 | """ |
|---|
| 1374 | A construction functor for free modules over commutative rings. |
|---|
| 1375 | |
|---|
| 1376 | EXAMPLE:: |
|---|
| 1377 | |
|---|
| 1378 | sage: F = (ZZ^3).construction()[0] |
|---|
| 1379 | sage: F |
|---|
| 1380 | VectorFunctor |
|---|
| 1381 | sage: F(GF(2)['t']) |
|---|
| 1382 | Ambient free module of rank 3 over the principal ideal domain Univariate Polynomial Ring in t over Finite Field of size 2 (using NTL) |
|---|
| 1383 | |
|---|
| 1384 | |
|---|
| 1385 | """ |
|---|
| 1386 | rank = 10 # ranking of functor, not rank of module. |
|---|
| 1387 | # This coincides with the rank of the matrix construction functor, but this is OK since they can not both be applied in any order |
|---|
| 1388 | |
|---|
| 1389 | |
|---|
| 1390 | def pushout_lattice(R, S): |
|---|
| 1391 | r""" |
|---|
| 1392 | Given a pair of Objects $R$ and $S$, try and construct a |
|---|
| 1393 | reasonable object $Y$ and return maps such that |
|---|
| 1394 | canonically $R \leftarrow Y \rightarrow S$. |
|---|
| 1395 | |
|---|
| 1396 | ALGORITHM: |
|---|
| 1397 | |
|---|
| 1398 | This is based on the model that arose from much discussion at Sage Days 4. |
|---|
| 1399 | Going up the tower of constructions of $R$ and $S$ (e.g. the reals |
|---|
| 1400 | come from the rationals come from the integers) try and find a |
|---|
| 1401 | common parent, and then try and fill in a lattice with these |
|---|
| 1402 | two towers as sides with the top as the common ancestor and |
|---|
| 1403 | the bottom will be the desired ring. |
|---|
| 1404 | |
|---|
| 1405 | See the code for a specific worked-out example. |
|---|
| 1406 | |
|---|
| 1407 | EXAMPLES:: |
|---|
| 1408 | |
|---|
| 1409 | sage: from sage.categories.pushout import pushout_lattice |
|---|
| 1410 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1411 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1412 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1413 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1414 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1415 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1416 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1417 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1418 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1419 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1420 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1421 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1422 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1423 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1424 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1425 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1426 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1427 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1428 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1429 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1430 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1431 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1432 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1433 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1434 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1435 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1436 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1437 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1438 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1439 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1440 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1441 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1442 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1443 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1444 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1445 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1446 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1447 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1448 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1449 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1450 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1451 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1452 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1453 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1454 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1455 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1456 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1457 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1458 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1459 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1460 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1461 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1462 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1463 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1464 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1465 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1466 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1467 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1468 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1469 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1470 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1471 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1472 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1473 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1474 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1475 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1476 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1477 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1478 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1479 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1480 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1481 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1482 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1483 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1484 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1485 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1486 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1487 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1488 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1489 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1490 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1491 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1492 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1493 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1494 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1495 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1496 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1497 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1498 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1499 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1500 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1501 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1502 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1503 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1504 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1505 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1506 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1507 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1508 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1509 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1510 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1511 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1512 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1513 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1514 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1515 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1516 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1517 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1518 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1519 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1520 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1521 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1522 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1523 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1524 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1525 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1526 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1527 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1528 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1529 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1530 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1531 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1532 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1533 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1534 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1535 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1536 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1537 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1538 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1539 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1540 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1541 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1542 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1543 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1544 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1545 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1546 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1547 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1548 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1549 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1550 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1551 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1552 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1553 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1554 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1555 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1556 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1557 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1558 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1559 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1560 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1561 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1562 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1563 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1564 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1565 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1566 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1567 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1568 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1569 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1570 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1571 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1572 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1573 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1574 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1575 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1576 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1577 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1578 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1579 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1580 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1581 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1582 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1583 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1584 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1585 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1586 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1587 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1588 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1589 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1590 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1591 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1592 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1593 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1594 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1595 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1596 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1597 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1598 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1599 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1600 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1601 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1602 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1603 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1604 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1605 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1606 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1607 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1608 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1609 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1610 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1611 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1612 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1613 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1614 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1615 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1616 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1617 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1618 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1619 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1620 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1621 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1622 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1623 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1624 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1625 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1626 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1627 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1628 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1629 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1630 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1631 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1632 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1633 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1634 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1635 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1636 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1637 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1638 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1639 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1640 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1641 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1642 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1643 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1644 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1645 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1646 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1647 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1648 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1649 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1650 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1651 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1652 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1653 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1654 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1655 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1656 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1657 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1658 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1659 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1660 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1661 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1662 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1663 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1664 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1665 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1666 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1667 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1668 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1669 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1670 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1671 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1672 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1673 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1674 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1675 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1676 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1677 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1678 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1679 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1680 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1681 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1682 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1683 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1684 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1685 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1686 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1687 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1688 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1689 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1690 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1691 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1692 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1693 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1694 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1695 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1696 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1697 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1698 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1699 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1700 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1701 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1702 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1703 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1704 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1705 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1706 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1707 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1708 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1709 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1710 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1711 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1712 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1713 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1714 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1715 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1716 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1717 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1718 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1719 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1720 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1721 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1722 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1723 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1724 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1725 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1726 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1727 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1728 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1729 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1730 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1731 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1732 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1733 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1734 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1735 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1736 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1737 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1738 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1739 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1740 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1741 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1742 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1743 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1744 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1745 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1746 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1747 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1748 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1749 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1750 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1751 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1752 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1753 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1754 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1755 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1756 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1757 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1758 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1759 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1760 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1761 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1762 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1763 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1764 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1765 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1766 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1767 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1768 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1769 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1770 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1771 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1772 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1773 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1774 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1775 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1776 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1777 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1778 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1779 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1780 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1781 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1782 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1783 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1784 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1785 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1786 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1787 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1788 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1789 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1790 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1791 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1792 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1793 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1794 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1795 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1796 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1797 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1798 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1799 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1800 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1801 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1802 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1803 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1804 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1805 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1806 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1807 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1808 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1809 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1810 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1811 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1812 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1813 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1814 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1815 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1816 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1817 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1818 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1819 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1820 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1821 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1822 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1823 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1824 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1825 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1826 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1827 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1828 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1829 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1830 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1831 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1832 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1833 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1834 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1835 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1836 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1837 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1838 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1839 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1840 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1841 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1842 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1843 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1844 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1845 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1846 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1847 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1848 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1849 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1850 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1851 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1852 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1853 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1854 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1855 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1856 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1857 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1858 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1859 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1860 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1861 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1862 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1863 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1864 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1865 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1866 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1867 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1868 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1869 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1870 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1871 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1872 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1873 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1874 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1875 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1876 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1877 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1878 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1879 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1880 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1881 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1882 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1883 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1884 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1885 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1886 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1887 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1888 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1889 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1890 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1891 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1892 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1893 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1894 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1895 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1896 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1897 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1898 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1899 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1900 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1901 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1902 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1903 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1904 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1905 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1906 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1907 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1908 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1909 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1910 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1911 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1912 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1913 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1914 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1915 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1916 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1917 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1918 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1919 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1920 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1921 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1922 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1923 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1924 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1925 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1926 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1927 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1928 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1929 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1930 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1931 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1932 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1933 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1934 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1935 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1936 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1937 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1938 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1939 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1940 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1941 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1942 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1943 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1944 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1945 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1946 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1947 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1948 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1949 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1950 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1951 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1952 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1953 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1954 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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| 1955 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1956 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1957 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1958 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1959 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1960 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1961 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1962 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1963 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1964 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1965 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1966 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1967 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1968 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1969 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1970 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1971 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1972 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1973 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1974 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1975 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1976 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1977 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1978 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1979 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1980 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
|---|
| 1981 | """ |
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