Monoids

sage.categories.monoids.Monoids

The category of (multiplicative) monoids.

A monoid is a unital semigroup, that is a set endowed with a multiplicative binary operation \(*\) which is associative and admits a unit (see Wikipedia article Monoid).

EXAMPLES:

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

sage: Monoids().axioms()
frozenset({'Associative', 'Unital'})
sage: Semigroups().Unital()
Category of monoids

sage: Monoids().example()
An example of a monoid: the free monoid generated by ('a', 'b', 'c', 'd')
sage: S = Monoids().example()
sage: x = S("aa")
sage: x^0, x^1, x^2, x^3, x^4, x^5
('', 'aa', 'aaaa', 'aaaaaa', 'aaaaaaaa', 'aaaaaaaaaa')