In mathematics, an automatic semigroup is a finitely generated semigroup equipped with several regular languages over an alphabet representing a generating set.
One of these languages determines "canonical forms" for the elements of the semigroup, the other languages determine if two canonical forms represent elements that differ by multiplication by a generator.
However, if an automatic semigroup has an identity, then it has an automatic structure with respect to any generating set (Duncan et al. 1999).
Kambites & Otto (2006) showed that it is undecidable whether an element of an automatic monoid possesses a right inverse.
Cain (2006) proved that both cancellativity and left-cancellativity are undecidable for automatic semigroups.
On the other hand, right-cancellativity is decidable for automatic semigroups (Silva & Steinberg 2004).
Automatic structures for groups have an elegant geometric characterization called the fellow traveller property (Epstein et al. 1992, ch.
Automatic structures for semigroups possess the fellow traveller property but are not in general characterized by it (Campbell et al. 2001).