Unification in Commutative Semigroups

Abstract

Unification is one of the basic concepts of automated theorem proving. It concerns such questions as finding solutions of finite sets of equations, determining if every solution comes from a most general solution, and if so, determining how many most general solutions are needed to generate all solutions. These solutions given in terms of substitutions are called, more formally, unifiers. The unification Ž . type of a variery equational class of algebras is defined according to the cardinality or existence of minimal complete sets of most general unifiers. Of particular interest, from a computational point of view, are varieties of groups and semigroups. So far the problem has been considered mainly for particular varieties. In this paper we determine unification types for all varieties of commutative semigroups. In particular, we prove that for commutative semigroups the unification problem is solvable in the very strong sense that there is an algorithm which for any two finite sets ⌺ and ⌺ of semigroup equations produces the minimal 1 2 complete set of the most general unifiers of ⌺ over the variety of commutative 1 semigroups generated by ⌺ . It seems that this is the first so general decidability 2 result in the area.

Permutability Class of a Semigroup

“…In particular, we hope that it may allow us to extend some results and methods of [9] to a class of noncommutative varieties of semigroups, and to approach problems of unification as in [11]. Note that every class of semigroups of a given permutability class is, in fact, an equational class alias variety.…”