# Finite Monoids¶

class sage.categories.finite_monoids.FiniteMonoids(base_category)

The category of finite (multiplicative) monoids.

A finite monoid is a finite sets endowed with an associative unital binary operation $$*$$.

EXAMPLES:

sage: FiniteMonoids()
Category of finite monoids
sage: FiniteMonoids().super_categories()
[Category of monoids, Category of finite semigroups]

TESTS:

sage: TestSuite(FiniteMonoids()).run()
class ElementMethods
pseudo_order()

Returns the pair $$[k, j]$$ with $$k$$ minimal and $$0\leq j <k$$ such that self^k == self^j.

Note that $$j$$ is uniquely determined.

EXAMPLES:

sage: M = FiniteMonoids().example(); M
An example of a finite multiplicative monoid: the integers modulo 12

sage: x = M(2)
sage: [ x^i for i in range(7) ]
[1, 2, 4, 8, 4, 8, 4]
sage: x.pseudo_order()
[4, 2]

sage: x = M(3)
sage: [ x^i for i in range(7) ]
[1, 3, 9, 3, 9, 3, 9]
sage: x.pseudo_order()
[3, 1]

sage: x = M(4)
sage: [ x^i for i in range(7) ]
[1, 4, 4, 4, 4, 4, 4]
sage: x.pseudo_order()
[2, 1]

sage: x = M(5)
sage: [ x^i for i in range(7) ]
[1, 5, 1, 5, 1, 5, 1]
sage: x.pseudo_order()
[2, 0]

TODO: more appropriate name? see, for example, Jean-Eric Pin’s lecture notes on semigroups.

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