On RDGCn-commutative permutable semigroups

In this paper, we describe the RDGCn-commutative permutable semigroups and RDGCn-commutative ∆-semigroups. A semigroup S is called permutable, if for each pair of congruences ρ, σ on S, ρ ◦ σ = σ ◦ ρ. S is said to be a ∆-semigroup if the lattice of all congruences of S is a chain with respect to inclusion. Nagy, Schein, Tamura and Trotter etc. investigated the ∆-semigroups in special classes of semigroups([5], [6], [9]–[12]) and Cherubini, Varisco, Hamilton and the first author etc. investigated the permutable semigroups in special classes of semigroups ([1], [3], [4]). The aim of this note is to describe RDGCn-commutative permutable semigroups and RDGCn-commutative ∆-semigroups. A semigroup S is said to be right duo if, every right ideal of S is a two side ideal. We say that S is generalized conditionally commutative if S satisfies the identity xyx = xyx for every positive integer i ([8]). As a generalization of generalized conditionally commutative semigroups, Nagy [6] introduced the following definition: Definition. For a positive integer n, a semigroup will be called generalized conditionally n-commutative (or GCn-commutative) if it satisfies the identity xyx = xyx for every positive integer i ≥ 2. A semigroup is said to be RDGCn − commutative if it is both right duo and GCn-commutative. Clearly, a band is a GCn-commutative semigroup for every positive integer n. It is easy to see that a group is GCn-commutative if and only if it is commutative, and that GC1-commutative semigroups are just generalized conditionally commutative semigroups. Mathematics subject classification number: 20M10.