Aperiodic semigroup

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.{{cn|date=October 2012}}

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.

• Monogenic semigroup
• Special classes of semigroups

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• {{cite book | last=Straubing | first=Howard | title=Finite automata, formal logic, and circuit complexity | url=https://archive.org/details/finiteautomatafo0000stra | url-access=registration | series=Progress in Theoretical Computer Science | location=Basel | publisher=Birkhäuser | year=1994 | isbn=3-7643-3719-2 | zbl=0816.68086 }}

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Category:Semigroup theory

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