Coverage for local/lib/python2.7/site-packages/sage/categories/semirings.py : 0%

r""" 

Semirngs 

""" 

from __future__ import absolute_import 

#***************************************************************************** 

# Copyright (C) 2010 Nicolas Borie <nicolas.borie@math.u-psud.fr> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# http://www.gnu.org/licenses/ 

#****************************************************************************** 

from sage.categories.category_with_axiom import CategoryWithAxiom 

from .magmas_and_additive_magmas import MagmasAndAdditiveMagmas 

class Semirings(CategoryWithAxiom): 

""" 

The category of semirings. 

A semiring `(S,+,*)` is similar to a ring, but without the 

requirement that each element must have an additive inverse. In 

other words, it is a combination of a commutative additive monoid 

`(S,+)` and a multiplicative monoid `(S,*)`, where `*` distributes 

over `+`. 

.. SEEALSO:: 

:wikipedia:`Semiring` 

EXAMPLES:: 

sage: Semirings() 

Category of semirings 

sage: Semirings().super_categories() 

[Category of associative additive commutative additive associative additive unital distributive magmas and additive magmas, 

Category of monoids] 

sage: sorted(Semirings().axioms()) 

['AdditiveAssociative', 'AdditiveCommutative', 'AdditiveUnital', 'Associative', 'Distributive', 'Unital'] 

sage: Semirings() is (CommutativeAdditiveMonoids() & Monoids()).Distributive() 

True 

sage: Semirings().AdditiveInverse() 

Category of rings 

TESTS:: 

sage: TestSuite(Semirings()).run() 

""" 

_base_category_class_and_axiom = (MagmasAndAdditiveMagmas.Distributive.AdditiveAssociative.AdditiveCommutative.AdditiveUnital.Associative, "Unital")