Monoids¶
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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')