Automatic semigroups and categories

Abstract

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.

Automatic Rees Matrix Semigroups over Categories

“…In Section 2, we briefly recall the notions of generators and relations for partial algebras which we studied in [18], and the definitions of regular path languages and of automatic and prefix-automaticity introduced in [17]. We also recall some key results from [17] which will be applied in this paper. In Section 3, we prove some technical results concerning automaticity in small categories and semigroupoids, which will be needed in the sections that follow.…”

“…In this section, we briefly recall a number of definitions and results from [18] and [17]. For a more detailed introduction, the reader should consult those papers.…”

“…We recall from [17,Section 3] that a language L ⊆ X + ⊆ (X 1 ) + is regular in this sense if and only if it is regular in the usual sense as a language over the alphabet X 1 . We recall also that the set of regular path languages over X contains X + , X * and all finite path languages, and is closed under concatenation, finite intersection, finite union, complement, set difference, generation of subcategories and subsemigroupoids, and prefix-closure.…”

“…A semigroupoid admits a prefix-automatic structure if and only if it admits an automatic structure which is prefix-closed [17,Corollary 4.6]. Now let S be a semigroupoid and 0 be a new symbol not in S 1 .…”

“…Now let S be a semigroupoid and 0 be a new symbol not in S 1 . The consolidation of S is the semigroup with set of elements T = S 1 ∪ {0}, and multiplication defined by st = the S-product st if s, t ∈ S and sω = tα 0 otherwise for all s, t ∈ S. We shall need the following key results from [17]. …”