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. 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==