Monoids

class sage.categories.monoids.Monoids(s=None)

Bases: sage.categories.category.Category

The category of (multiplicative) monoids, i.e. semigroups with a unit.

EXAMPLES:

sage: Monoids()
Category of monoids
sage: Monoids().super_categories()
[Category of semigroups]
sage: Monoids().all_super_categories()
[Category of monoids,
 Category of semigroups,
 Category of magmas,
 Category of sets,
 Category of sets with partial maps,
 Category of objects]

TESTS:

sage: C = Monoids()
sage: TestSuite(C).run()

class Algebras(base_category, base_ring)

Bases: sage.categories.algebra_functor.AlgebrasCategory

ParentMethods

alias of Algebras.ParentMethods

extra_super_categories()

EXAMPLES:

sage: Monoids().Algebras(QQ).extra_super_categories()
[Category of algebras with basis over Rational Field]
sage: Monoids().Algebras(QQ).super_categories()
[Category of semigroup algebras over Rational Field]

sage: Monoids().example().algebra(ZZ).categories()
[Category of monoid algebras over Integer Ring,
 Category of semigroup algebras over Integer Ring,
 Category of algebras with basis over Integer Ring,
 ...
 Category of objects]

class Monoids.CartesianProducts(category, *args)

Bases: sage.categories.cartesian_product.CartesianProductsCategory

The category of monoids constructed as cartesian products of monoids

ParentMethods

alias of CartesianProducts.ParentMethods

extra_super_categories()

A cartesian product of monoids is endowed with a natural monoid structure.

EXAMPLES:

sage: Monoids().CartesianProducts().extra_super_categories()
[Category of monoids]
sage: Monoids().CartesianProducts().super_categories()
[Category of monoids, Category of Cartesian products of semigroups]

class Monoids.ElementMethods

is_one()

Returns whether self is the one of the monoid

The default implementation, is to compare with self.one().

TESTS:

sage: S = Monoids().example()
sage: S.one().is_one()
True
sage: S("aa").is_one()
False

class Monoids.ParentMethods

one(*args, **kwds)

Returns the one of the monoid, that is the unique neutral element for .

EXAMPLES:

sage: M = Monoids().example(); M
An example of a monoid: the free monoid generated by ('a', 'b', 'c', 'd')
sage: M.one()
''

one_element()

Backward compatibility alias for one().

TESTS:

sage: S = Monoids().example()
sage: S.one_element()
''

prod(args)

n-ary product

INPUT:

• args – a list (or iterable) of elements of self

Returns the product of the elements in args, as an element of self.

EXAMPLES:

sage: S = Monoids().example()
sage: S.prod([S('a'), S('b')])
'ab'

class Monoids.Subquotients(category, *args)

Bases: sage.categories.subquotients.SubquotientsCategory

ParentMethods

alias of Subquotients.ParentMethods

Monoids.super_categories(*args, **kwds)

Returns a list of the immediate super categories of self.

EXAMPLES:

sage: Monoids().super_categories()
[Category of semigroups]