Commutativity in free inverse monoids

Abstract

This article characterizes the solutions of the commutativity equation xy = yx in free inverse monoids. The main result implies the following interesting property that is the natural generalization to free inverse monoids of the solutions of the same equation in free monoids. Let x and y be non-idempotent elements of a free inverse monoid such that xy = yx. Then there exist some elements χ and z such that x and y are conjugate by χ to some positive powers of z, namely xχ = χz n and yχ = χz m , with n, m⩾ 1. We also show that the centralizer of a given non-idempotent element is a rational, non-recognizable subset of the free inverse monoid.