Quasi commutative semigroups

Abstract

A semigroup S is called left (right) quasi commutative if, for every a, b Δ S, there is a positive integer r such that ab = b r a (ab = ba r ). A semigroup S is called σ-reflexive if ab Δ H implies ba Δ H for every a, b Δ S and every subsemigroup H of S. In this chapter it is proved that the left quasi commutative semigroups, the right quasi commutative semigroups and the σ-reflexive semigroups are the same. They are called quasi commutative semigroups. As a quasi commutative semnigroup is also weakly commutative, they are semilattice of archimedean semigroups. As the commutative archimedean semnigroups are describen in Chapter 3, here is considered only the non-commutative case. It is proved that a semigroup is a non-commutative quasi commutative archimedean semigroup containing at least one idempotent element if and only if it is an ideal extension of a hamiltonian group by a commutative nil semigroup. At the end of the chapter, the least weakly separative congruence of a quasi commutative semigroup is constructed. It is shown that, on a quasi commutative semigroup S, σ defined by a σ b (a, b Δ S) if and only if an+1 = ba n and b n+1= ab n for some positive integer n is the least wealdy separative congruence.