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sagemathgh-38965: adding an example of semiring
Just for the fun of it : the semiring with elements (0,1,at least 2)
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preview.
URL: sagemath#38965
Reported by: Frédéric Chapoton
Reviewer(s): Travis Scrimshaw- Loading branch information
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,249 @@ | ||
| # sage_setup: distribution = sagemath-categories | ||
| r""" | ||
| Examples of semirings | ||
| """ | ||
| # **************************************************************************** | ||
| # Copyright (C) 2024 F. Chapoton <chapoton unistra.fr> | ||
| # | ||
| # Distributed under the terms of the GNU General Public License (GPL) | ||
| # https://www.gnu.org/licenses/ | ||
| # **************************************************************************** | ||
|
|
||
| from sage.categories.semirings import Semirings | ||
| from sage.structure.element import Element | ||
| from sage.structure.parent import Parent | ||
| from sage.structure.unique_representation import UniqueRepresentation | ||
|
|
||
| # semantic : | ||
| # 0 => 0, empty; | ||
| # 1 => 1, unique; | ||
| # 2 => at least 2 | ||
| _ADD = [[0, 1, 2], [1, 2, 2], [2, 2, 2]] | ||
| _PROD = [[0, 0, 0], [0, 1, 2], [0, 2, 2]] | ||
|
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||
|
|
||
| class Ternary(Element): | ||
| """ | ||
| Elements of the ternary-logic ring. | ||
| The semantic is as follows: | ||
| - 0 -- the integer 0 | ||
| - 1 -- the integer 1 | ||
| - 2 -- some integer greater than 1 | ||
| An alternative semantic is: | ||
| - 0 -- an empty set | ||
| - 1 -- a connected set | ||
| - 2 -- a disconnected set | ||
| The same semantic works for graphs instead of sets. | ||
| """ | ||
| def __init__(self, parent, n): | ||
| """ | ||
| Initialize one element. | ||
| TESTS:: | ||
| sage: from sage.categories.examples.semirings import TernaryLogic | ||
| sage: S = TernaryLogic() | ||
| sage: S(4) | ||
| Traceback (most recent call last): | ||
| ... | ||
| ValueError: input not in (0,1,2) | ||
| """ | ||
| if n not in [0, 1, 2]: | ||
| raise ValueError("input not in (0,1,2)") | ||
| self._n = n | ||
| Element.__init__(self, parent) | ||
|
|
||
| def _repr_(self): | ||
| """ | ||
| Return the string representation. | ||
| TESTS:: | ||
| sage: from sage.categories.examples.semirings import TernaryLogic | ||
| sage: S = TernaryLogic() | ||
| sage: [S(i) for i in range(3)] | ||
| [0, 1, many] | ||
| """ | ||
| return ["0", "1", "many"][self._n] | ||
|
|
||
| def __eq__(self, other): | ||
| """ | ||
| Test for equality. | ||
| TESTS:: | ||
| sage: from sage.categories.examples.semirings import TernaryLogic | ||
| sage: S = TernaryLogic() | ||
| sage: S(1) == S(2) | ||
| False | ||
| sage: S(0) == 3 | ||
| False | ||
| """ | ||
| if not isinstance(other, Ternary): | ||
| return False | ||
| return self._n == other._n | ||
|
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||
| def __ne__(self, other): | ||
| """ | ||
| Test for non-equality. | ||
| TESTS:: | ||
| sage: from sage.categories.examples.semirings import TernaryLogic | ||
| sage: S = TernaryLogic() | ||
| sage: S(1) != S(2) | ||
| True | ||
| """ | ||
| return not (self == other) | ||
|
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||
|
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| class TernaryLogic(UniqueRepresentation, Parent): | ||
| r""" | ||
| An example of a semiring. | ||
| This class illustrates a minimal implementation of a semiring. | ||
| EXAMPLES:: | ||
| sage: S = Semirings().example(); S | ||
| An example of a semiring: the ternary-logic semiring | ||
| This is the semiring that contains 3 objects:: | ||
| sage: S.some_elements() | ||
| [0, 1, many] | ||
| The product rule is as expected:: | ||
| sage: S(1) * S(1) | ||
| 1 | ||
| sage: S(1) + S(1) | ||
| many | ||
| TESTS:: | ||
| sage: TestSuite(S).run(verbose=True) | ||
| running ._test_additive_associativity() . . . pass | ||
| running ._test_an_element() . . . pass | ||
| running ._test_associativity() . . . pass | ||
| running ._test_cardinality() . . . pass | ||
| running ._test_category() . . . pass | ||
| running ._test_construction() . . . pass | ||
| running ._test_distributivity() . . . pass | ||
| running ._test_elements() . . . | ||
| Running the test suite of self.an_element() | ||
| running ._test_category() . . . pass | ||
| running ._test_eq() . . . pass | ||
| running ._test_new() . . . pass | ||
| running ._test_nonzero_equal() . . . pass | ||
| running ._test_not_implemented_methods() . . . pass | ||
| running ._test_pickling() . . . pass | ||
| pass | ||
| running ._test_elements_eq_reflexive() . . . pass | ||
| running ._test_elements_eq_symmetric() . . . pass | ||
| running ._test_elements_eq_transitive() . . . pass | ||
| running ._test_elements_neq() . . . pass | ||
| running ._test_eq() . . . pass | ||
| running ._test_new() . . . pass | ||
| running ._test_not_implemented_methods() . . . pass | ||
| running ._test_one() . . . pass | ||
| running ._test_pickling() . . . pass | ||
| running ._test_prod() . . . pass | ||
| running ._test_some_elements() . . . pass | ||
| running ._test_zero() . . . pass | ||
| """ | ||
| def __init__(self): | ||
| r""" | ||
| The ternary-logic semiring. | ||
| EXAMPLES:: | ||
| sage: S = Semirings().example(); S | ||
| An example of a semiring: the ternary-logic semiring | ||
| """ | ||
| Parent.__init__(self, category=Semirings()) | ||
|
|
||
| def _repr_(self): | ||
| r""" | ||
| Return the string representation. | ||
| EXAMPLES:: | ||
| sage: Semirings().example()._repr_() | ||
| 'An example of a semiring: the ternary-logic semiring' | ||
| """ | ||
| return "An example of a semiring: the ternary-logic semiring" | ||
|
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||
| def summation(self, x, y): | ||
| r""" | ||
| Return the sum of ``x`` and ``y`` in the semiring as per | ||
| :meth:`Semirings.ParentMethods.summation`. | ||
| EXAMPLES:: | ||
| sage: S = Semirings().example() | ||
| sage: S(1) + S(1) | ||
| many | ||
| """ | ||
| assert x in self | ||
| assert y in self | ||
| return self(_ADD[x._n][y._n]) | ||
|
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| def one(self): | ||
| """ | ||
| Return the unit of ``self``. | ||
| EXAMPLES:: | ||
| sage: S = Semirings().example() | ||
| sage: S.one() | ||
| 1 | ||
| """ | ||
| return self(1) | ||
|
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| def product(self, x, y): | ||
| r""" | ||
| Return the product of ``x`` and ``y`` in the semiring as per | ||
| :meth:`Semirings.ParentMethods.product`. | ||
| EXAMPLES:: | ||
| sage: S = Semirings().example() | ||
| sage: S(1) * S(2) | ||
| many | ||
| """ | ||
| assert x in self | ||
| assert y in self | ||
| return self(_PROD[x._n][y._n]) | ||
|
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||
| def an_element(self): | ||
| r""" | ||
| Return an element of the semiring. | ||
| EXAMPLES:: | ||
| sage: Semirings().example().an_element() | ||
| many | ||
| """ | ||
| return self(2) | ||
|
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| def some_elements(self): | ||
| r""" | ||
| Return a list of some elements of the semiring. | ||
| EXAMPLES:: | ||
| sage: Semirings().example().some_elements() | ||
| [0, 1, many] | ||
| """ | ||
| return [self(i) for i in [0, 1, 2]] | ||
|
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| Element = Ternary | ||
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| Example = TernaryLogic |
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