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Aperiodic semigroup
In mathematics, an aperiodic semigroup is a semigroup ''S'' such that every element ''x'' ∈ ''S'' is aperiodic, that is, for each ''x'' there exists a positive integer ''n'' such that ''x''''n'' = ''x''''n'' + 1. An aperiodic monoid is an aperiodic semigroup which is a monoid.
== Finite aperiodic semigroups ==
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.
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 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.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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