On identities of Rees quotients of free inverse semigroups defined by positive relators

David Easdown and Lev Shneerson

Abstract

We give a complete description of Rees quotients of free inverse semigroups given by positive relators that satisfy nontrivial identities, including identities in signature with involution. They are finitely presented in the class of all inverse semigroups. Those that satisfy a nontrivial semigroup identity have polynomial growth and can be given by an irredundant presentation with at most four relators. Those that satisfy a nontrivial identity in signature with involution, but which do not satisfy a nontrivial semigroup identity, have exponential growth and fall within two infinite families of finite presentations with two generators. The first family involves an unbounded number of relators, and the other involves presentations with at most four relators of unbounded length. We give a new sufficient condition for which a finite set $X$ of reduced words over an alphabet $A\cup A^{-1}$ freely generates a free inverse subsemigroup of $F{I}_A$ and use it in our proofs.

Keywords: free inverse semigroup, Rees quotient, inverse semigroup presentation, growth, identities.

AMS Subject Classification: Primary 20M18.