Conjugacy in free inverse monoids

Abstract

The notion of conjugacy in groups can be extended in two ways to monoids. We keep on calling conjugacy the first version (two elements x and y are conjugate if xz=zy holds for some z), while we call transposition the second one (two elements x and y are transposed conjugate if x=uv and y=vu holds for some u,v). Using the characterization of elements in free inverse monoids due to Munn, we show that restricted to non idempotents, the relation of conjugacy is the transitive closure of the relation of transposition. Furthermore, we show that conjugacy between two elements of a free inverse monoid can be tested in linear time.