Regular Gröbner Bases

In Ufnarovski (1989), the concept of automaton algebras is introduced. These are quotients of the non-commutative polynomial ring where the defining ideal allows some Gröbner basis with a regular set of leading words. However, nothing is reflected concerning the whole structure of the Gröbner basis (except of course for monomial algebras). In this paper we introduce the concept of regular Gröbner bases and bi-automaton algebras. Regular Gröbner bases consist of (pure difference) binomials which can be represented, in an appropriate way, as regular sets. The corresponding finite automata allow us to perform reduction with respect to such (in general) infinite bases. In particular, this enables us to do computations in any factor algebra where the defining ideal admits a regular Gröbner basis; we call such an algebra bi-automaton. We show that most examples of automaton (binomial) algebras given by Ufnarovski are bi-automaton. Moreover, we construct automata showing that all subalgebras of the free algebra generated by a finite set of words are bi-automaton. As a consequence, we find that all algebras allowing a finite SAGBI basis are automaton. Since a great part of the motivation for our work is to be able to compute normal forms with respect to binomial ideals, we describe how regular Gröbner bases, or more precisely the corresponding automata, can be used to perform reduction. In the last section we indicate how a prediction algorithm for regular sets, recently developed by the authors, can be used to find regular Gröbner bases. The subject of this paper has natural applications in the theory of term rewriting in monoid (and group) rings. If we associate the binomial u − v to every rewrite rule u → v in the semi-Thue system defining a monoid, then the congruence generated by the semi-Thue system corresponds to the ideal generated by the binomials. It is easy to see that every monoid can be presented by a semi-Thue system, so being able to handle binomial ideals allows us to do computations in monoid rings. The correspondence between binomial Gröbner bases and term rewriting systems has been investigated in