Aperiodic semigroup

In mathematics, an aperiodic semigroup is a semigroup S such that every element is aperiodic, that is, for each x in S there exists a positive integer n such that xn = xn+1.[1] An aperiodic monoid is an aperiodic semigroup which is a monoid.

## Finite aperiodic semigroups

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A finite semigroup is aperiodic if and only if it contains no nontrivial subgroups, so a synonym used (only?) in such contexts is group-free semigroup. 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.[citation needed]

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.[2]

A consequence of the Krohn–Rhodes theorem is that every finite aperiodic monoid divides a wreath product of copies of the three-element flip-flop monoid, consisting of an identity element and two right zeros. The two-sided Krohn–Rhodes theorem alternatively characterizes finite aperiodic monoids as divisors of iterated block products of copies of the two-element semilattice.

## See also

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* Monogenic semigroup
* Special classes of semigroups

## References

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1. ^ Kilp, Mati; Knauer, Ulrich; Mikhalev, Alexander V. (2000). Monoids, Acts and Categories: With Applications to Wreath Products and Graphs. A Handbook for Students and Researchers. De Gruyter Expositions in Mathematics. Vol. 29\. Walter de Gruyter. p. 29\. ISBN 3110812908. Zbl 0945.20036.
2. ^ Schützenberger, Marcel-Paul, "On finite monoids having only trivial subgroups," Information and Control, Vol 8 No. 2, pp. 190–194, 1965.

* Straubing, Howard (1994). Finite automata, formal logic, and circuit complexity. Progress in Theoretical Computer Science. Basel: Birkhäuser. ISBN 3-7643-3719-2. Zbl 0816.68086.

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