Bases: sage.categories.category_types.Category_over_base_ring
The category of algebras over a given base ring.
An algebra over a ring is a module over
endowed with a
bilinear multiplication.
Warning
MagmaticAlgebras will eventually replace the current Algebras for consistency with e.g. Wikipedia article Algebras which assumes neither associativity nor the existence of a unit (see trac ticket #15043).
EXAMPLES:
sage: from sage.categories.magmatic_algebras import MagmaticAlgebras
sage: C = MagmaticAlgebras(ZZ); C
Category of magmatic algebras over Integer Ring
sage: C.super_categories()
[Category of additive commutative additive associative additive unital distributive magmas and additive magmas,
Category of modules over Integer Ring]
TESTS:
sage: TestSuite(C).run()
alias of AssociativeAlgebras
Return a family of generators of this algebra.
EXAMPLES:
sage: F = AlgebrasWithBasis(QQ).example(); F
An example of an algebra with basis: the free algebra on the generators ('a', 'b', 'c') over Rational Field
sage: F.algebra_generators()
Family (B[word: a], B[word: b], B[word: c])
alias of UnitalAlgebras
Bases: sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring
TESTS:
sage: C = Modules(ZZ).FiniteDimensional(); C
Category of finite dimensional modules over Integer Ring
sage: type(C)
<class 'sage.categories.modules.Modules.FiniteDimensional_with_category'>
sage: type(C).__base__.__base__
<class 'sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring'>
sage: TestSuite(C).run()
The product of the algebra, as per Magmas.ParentMethods.product()
By default, this is implemented using one of the following methods, in the specified order:
EXAMPLES:
sage: A = AlgebrasWithBasis(QQ).example()
sage: a, b, c = A.algebra_generators()
sage: A.product(a + 2*b, 3*c)
3*B[word: ac] + 6*B[word: bc]
The product of the algebra on the basis (optional).
INPUT:
Return the product of the two corresponding basis elements indexed by i and j.
If implemented, product() is defined from it by bilinearity.
EXAMPLES:
sage: A = AlgebrasWithBasis(QQ).example()
sage: Word = A.basis().keys()
sage: A.product_on_basis(Word("abc"),Word("cba"))
B[word: abccba]
EXAMPLES:
sage: from sage.categories.magmatic_algebras import MagmaticAlgebras
sage: MagmaticAlgebras(ZZ).super_categories()
[Category of additive commutative additive associative additive unital distributive magmas and additive magmas, Category of modules over Integer Ring]
sage: from sage.categories.additive_semigroups import AdditiveSemigroups
sage: MagmaticAlgebras(ZZ).is_subcategory((AdditiveSemigroups() & Magmas()).Distributive())
True