Automatic presentations for semigroups.
A structure \(\mathcal A=(A,R_1,\dots,R_n)\) consists of a set \(A\), called the domain of \(\mathcal A\), and there exists \(r_i\geq 1\) such that \(R_i\) is a subset of \(A^{r_i}\) (\(1\leq i\leq n\)). A structure \(\mathcal A\) has an automatic presentation over an alphabet, if it satisfies certain conditions (see Section 2). A structure that admits an automatic presentation is said to be FA-presentable. The paper under review is concerned with FA-presentable semigroups. First, the authors prove that all finitely generated commutative semigroups are FA-presentable (Theorem 6.1). Then they observe that any finitely generated subsemigroup of a semigroup admitting an automatic presentation has polynomial growth (Theorem 7.4). After that the authors consider one-relation semigroups and characterize one-relation semigroup presentations that define FA-presentable semigroups (Proposition 9.1). Later they give a complete classification of finitely generated FA-presentable cancellative semigroups and prove that a finitely generated cancellative semigroup is FA-presentable if and only if it embeds into a virtually Abelian group (Theorem 10.1). Finally, they consider the relationship between the class of FA-presentable semigroups and automatic semigroups (Proposition 11.2).
finitely generated cancellative semigroups
scientific article; zbMATH DE number 5639681
scientific article