Automatic semigroups and categories

We consider various automata-theoretic properties of semigroupoids and small categories and their relationship to the corresponding properties in semigroups and monoids. We introduce natural definitions of finite automata and regular languages over finite graphs, generalising the usual notions over finite alphabets. These allow us to introduce a definition of automaticity for semigroupoids and small categories, which generalises those introduced for semigroups by Hudson and for groupoids by Epstein. We also introduce a definition of prefix-automaticity for semigroupoids and small categories, generalising that for certain monoids introduced by Silva and Steinberg.We study the relationship between automaticity properties in a semigroupoid and in a certain associated semigroup. This allows us to extend to semigroupoids and small categories a number of results about automatic and prefix-automatic semigroups and monoids. In the course of our study, we also prove some new results about automaticity and prefix-automaticity in semigroups and monoids. These include the fact that prefix-automaticity is preserved under the taking of cofinite subsemigroups.