Given a semigroup S and elements s,t in S, write s~t if s=pr and t=rp, for some p,r in S (with 1 adjoined). This relation, and its transitive closure, both known as "primary conjugacy", has been extensively used and studied in connection with many algebraic objects, including groups, rings, and C*-algebras. It is the standard tool for either measuring or forcing commutativity. We discuss a natural generalization, defined by s~t whenever s=p_{1}...p_{n} and t=p_{f(1)}...p_{f(n)}, for some p_{1},...,p_{n} in S (with 1 adjoined) and permutation f of {1,...,n}, together with its transitive closure, which we call the "permutation" relation. The permutation relation is actually the congruence generated by the primary conjugacy, and is the least commutative congruence on any semigroup. We explore general properties of the permutation relation, discuss it in the context of groups and rings, compare it to various known semigroup conjugacy relations, and describe its equivalence classes in different semigroups.
Conjugacy and least commutative congruences in semigroups Sponsored by the Meyer Fund
Conjugacy and least commutative congruences in semigroups