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() |
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491 | sage: F = QQ.construction()[0] |
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492 | sage: P = ZZ['t'].construction()[0] |
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493 | sage: I*F is F # indirect doctest |
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494 | True |
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495 | sage: F*I is F |
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496 | True |
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497 | sage: I*P is P |
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498 | True |
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499 | sage: P*I is P |
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500 | True |
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501 | |
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502 | """ |
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503 | if isinstance(self, IdentityConstructionFunctor): |
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504 | return other |
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505 | else: |
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506 | return self |
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507 | |
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508 | |
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509 | class PolynomialFunctor(ConstructionFunctor): |
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510 | """ |
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511 | Construction functor for univariate polynomial rings. |
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512 | |
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513 | EXAMPLE:: |
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514 | |
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515 | sage: P = ZZ['t'].construction()[0] |
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516 | sage: P(GF(3)) |
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517 | Univariate Polynomial Ring in t over Finite Field of size 3 |
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518 | sage: P == loads(dumps(P)) |
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519 | True |
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520 | sage: R.<x,y> = GF(5)[] |
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521 | sage: f = R.hom([x+2*y,3*x-y],R) |
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522 | sage: P(f)((x+y)*P(R).0) |
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523 | (-x + y)*t |
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524 | |
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525 | By trac ticket #9944, the construction functor distinguishes sparse and |
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526 | dense polynomial rings. Before, the following example failed:: |
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527 | |
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528 | sage: R.<x> = PolynomialRing(GF(5), sparse=True) |
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529 | sage: F,B = R.construction() |
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530 | sage: F(B) is R |
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531 | True |
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532 | sage: S.<x> = PolynomialRing(ZZ) |
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533 | sage: R.has_coerce_map_from(S) |
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534 | False |
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535 | sage: S.has_coerce_map_from(R) |
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536 | False |
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537 | sage: S.0 + R.0 |
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538 | 2*x |
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539 | sage: 1+1 |
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540 | 2 |
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541 | sage: 1+1 |
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542 | 2 |
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543 | """ |
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544 | rank = 9 |
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545 | |
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546 | def __init__(self, var, multi_variate=False, sparse=False): |
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547 | """ |
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548 | TESTS:: |
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549 | |
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550 | sage: from sage.categories.pushout import PolynomialFunctor |
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551 | sage: P = PolynomialFunctor('x') |
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552 | sage: P(GF(3)) |
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553 | Univariate Polynomial Ring in x over Finite Field of size 3 |
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554 | |
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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'])) |
---|
1820 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1821 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1822 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1823 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1824 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1825 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1826 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1827 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1828 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1829 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1830 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1831 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1832 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1833 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1834 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1835 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1836 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1837 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1838 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1839 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1840 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1841 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1842 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1843 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1844 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1845 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1846 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1847 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1848 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1849 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1850 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1851 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1852 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1853 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1854 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1855 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1856 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1857 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1858 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1859 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1860 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1861 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1862 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1863 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1864 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1865 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1866 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1867 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1868 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1869 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1870 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1871 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1872 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1873 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1874 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1875 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1876 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1877 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1878 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1879 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1880 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1881 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1882 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1883 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1884 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1885 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1886 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1887 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1888 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1889 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1890 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1891 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1892 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1893 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1894 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1895 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1896 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1897 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1898 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1899 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1900 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1901 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1902 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1903 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1904 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1905 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1906 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1907 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1908 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1909 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1910 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1911 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1912 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1913 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1914 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1915 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1916 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1917 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1918 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1919 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1920 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1921 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1922 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1923 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1924 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1925 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1926 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1927 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1928 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1929 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1930 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1931 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1932 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1933 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1934 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1935 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1936 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
---|
1937 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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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'])) |
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1942 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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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'])) |
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1945 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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1946 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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1947 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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1948 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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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'])) |
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1952 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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1953 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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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'])) |
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1956 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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1957 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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1958 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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1959 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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1960 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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1961 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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1962 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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1963 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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1964 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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1965 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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1966 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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1967 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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1968 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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1969 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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1970 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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1971 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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1972 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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1973 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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1974 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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1975 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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1976 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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1977 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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1978 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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1979 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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1980 | sage: A, B = pushout_lattice(Qp(7), Frac(ZZ['x'])) |
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1981 | """ |
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