Finite Monoids¶
-
class
sage.categories.finite_monoids.
FiniteMonoids
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_singleton
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.
-
-
class