Cancellative semigroups admitting conjugates

Abstract

We prove conjugate analogs of Mal’cev-Neumann-Taylor-Levi’s theorems. In other words, we characterize semigroups em- beddable in nilpotent groups of class n by means of conjugacy laws involving semigroup laws described by Mal’cev, Neumann and Taylor.  Moreover we prove that a cancellative semigroup admitting conjugates is embeddable in a nilpotent group of class 2 if and only if it satisfies the conjugacy law x^y^z = x^y. Also adapting Ore’s techniques we describe an exact procedure for embedding a cancellative semigroup admitting conjugates into a group.