Finite semigroups¶
-
sage.categories.finite_semigroups.
FiniteSemigroups
¶ The category of finite (multiplicative) semigroups.
A finite semigroup is a
finite set
endowed with an associative binary operation \(*\).Warning
Finite semigroups in Sage used to be automatically endowed with an
enumerated set
structure; the default enumeration is then obtained by iteratively multiplying the semigroup generators. This forced any finite semigroup to either implement an enumeration, or provide semigroup generators; this was often inconvenient.Instead, finite semigroups that provide a distinguished finite set of generators with
semigroup_generators()
should now explicitly declare themselves in the category offinitely generated semigroups
:sage: Semigroups().FinitelyGenerated() Category of finitely generated semigroups
This is a backward incompatible change.
EXAMPLES:
sage: C = FiniteSemigroups(); C Category of finite semigroups sage: C.super_categories() [Category of semigroups, Category of finite sets] sage: sorted(C.axioms()) ['Associative', 'Finite'] sage: C.example() An example of a finite semigroup: the left regular band generated by ('a', 'b', 'c', 'd')