Aperiodic monoid

In mathematics, an aperiodic semigroup is a semigroup "S" such that for every "x" ∈ "S", there exists a nonnegative integer "n" such that"xⁿ" = "x^{n + 1}".

An aperiodic monoid is an aperiodic semigroup which is a monoid. This notion is in some sense orthogonal to that of group.

Recall that a subsemigroup "G" of a semigroup "S" is a subgroup of "S" (also called sometimes a group in "S") if there exists an idempotent "e" such that "G" is a group with identity element "e". A semigroup "S" is group-bound if some power of each element of "S" lies in some subgroup of "S". Every finite semigroup is group-bound, but a group-bound semigroup might be infinite.

A finite semigroup is aperiodic if and only if it contains no nontrivial subgroups. In terms of Green's relations, a finite semigroup is aperiodic if and only if its "H"-relation is trivial. These two characterizations extend to group-bound semigroups.

A celebrated result of algebraic automata theory due to Marcel-Paul Schützenberger asserts that a language is star-free if and only if its syntactic monoid is finite and aperiodic Schützenberger, Marcel-Paul, "On finite monoids having only trivial subgroups," "Information and Control", Vol 8 No. 2, pp. 190-194, 1965.] .

A consequence of the Krohn-Rhodes theorem is that every finite aperiodic monoid divides a wreath product of copies of the three element monoid containing an identity element and two right zeros.

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