Conjugacy and Least Commutative Congruences in Semigroups

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.