\(RGC_n\)-commutative \(\Delta\)-semigroups (Q1393043)

A semigroup \(S\) is called a \(\Delta\)-semigroup if the lattice of congruences of \(S\) forms a chain with respect to inclusion. A semigroup \(S\) is called \(\mathcal R\)-commutative if for every \((a,b)\in S\times S\), there is \(u\in S^1\) such that \(ab=bau\). For a positive integer \(n\) a semigroup is called \(\mathcal{GC}_n\)-commutative if it satisfies the identity \(a^nxa^i=a^ixa^n\) for every integer \(i(\geq 2)\). A semigroup which is \(\mathcal R\)-commutative and \(\mathcal{GC}_n\)-commutative is called an \(\mathcal{RGC}_n\)-semigroup. In this paper the author determines the structure of \(\mathcal{RGC}_n\)-commutative \(\Delta\)-semigroups. Let \(G\) be a nontrivial subgroup of a quasicyclic \(p\)-group (\(p\) is a prime), \(N\) a commutative nil semigroup whose ideals are chain ordered with respect to inclusion, and \(R\) a right zero semigroup of order 2. The following is the main theorem. A semigroup \(S\) is an \(\mathcal{RGC}_n\)-commutative \(\Delta\)-semigroup if and only if \(S\) is isomorphic to one of the following: (1) \(G\) (2) \(G^0\) (3) \(N\) (4) \(N^1\) (5) \(R\) (6) \(R^0\). The reviewer read the results and the proof with much interest.