A finite semigroup is aperiodic if and only if it contains no nontrivial subgroups, so a synonym used (only?)
In terms of Green's relations, a finite semigroup is aperiodic if and only if its H-relation is trivial.
[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.