Given a semigroup S and \(s,t \in S\) , write \(s \sim _p^1 t\) if \(s=pr\) and \(t=rp\) , for some \(p,r \in S \cup \{1\}\) . This relation, known as “primary conjugacy”, along with its transitive closure \(\sim _p\) , has been extensively used and studied in many fields of algebra. This paper is devoted to a natural generalization, defined by \(s \sim _s^1 t\) whenever \(s=p_1\cdots p_{n}\) and \(t=p_{f(1)}\cdots p_{f(n)}\) , for some \(p_1, \dots , p_n \in S \cup \{1\}\) and permutation f of \(\{1, \dots , n\}\) , together with its transitive closure \(\sim _s\) . The relation \(\sim _s\) is the congruence generated by either \(\sim _p^1\) or \(\sim _p\) , and is moreover the least commutative congruence on any semigroup. We explore general properties of \(\sim _s\) , discuss it in the context of groups and rings, compare it to other semigroup conjugacy relations, and fully describe its equivalence classes in free, Rees matrix, graph inverse, and various transformation semigroups.
