The transitivity of primary conjugacy in regular ω-semigroups

Abstract

Abstract The conjugacy relation plays an important role in group theory and the conjugacy relation of groups has been generalized to semigroups in various methods by several authors. If a a and b b are elements of a semigroup S S , then a a is called primarily conjugate to b b if a = u v a=uv and b = v u b=vu for some u , v ∈ S 1 u,v\in {S}^{1} . In general, primary conjugacy is reflexive and symmetric, but not transitive. Finding the classes of semigroups in which the primary conjugacy is transitive is an open problem raised by Araújo et al. in the literature. In this article, among other things we prove that the primary conjugacy is transitive in regular ω \omega -semigroups.