CANCELLATIVITY IS UNDECIDABLE FOR AUTOMATIC SEMIGROUPS

Abstract

Campbell, Robertson, Ruskuc, and Thomas [CRRT] generalized the concept of an automatic structure from groups [ECH+] to semigroups. Epstein et al. [ECH+] asked whether the conjugacy and isomorphism problems were soluble for automatic groups. [These problems are known to be undecidable for general finitely presented groups [LS, p.]; however, the proofs of these facts rely on the word problem being undecidable in general, which is not the case in automatic groups and semigroups [CRRT, Corollary .].] For automatic semigroups, one can ask many more questions: Is a given automatic semigroup cancellative, a monoid, a group, inverse, free, completely simple, completely -simple, or a Clifford semigroup? Does a given element in an automatic monoid have a right or left inverse? There are known algorithms to determine whether an automatic semigroup is a monoid or group [Cai, Section .]; whether it is free on some generating set [Cai, Section .]; whether it is completely simple or completely -simple [KO, Section ]; and whether it is a Clifford semigroup [CR]. It is not yet knownwhether one can decide if an automatic semigroup is inverse. It is undecidable whether a given element of an automatic semigroup has a right inverse [KO, Theorem .]; it is decidable whether it has a left inverse [KO, Proposition .].