\(RC\)-commutative \(\Delta\)-semigroups (Q1186840)

A semigroup \(S\) is called \(R\)-commutative if for every \(a,b\in S\) we have \(ab\in baS^ 1\). A semigroup \(S\) is said to be conditionally commutative if for every \(a,b\in S\), the assumption \(ab=ba\) implies \(axb=bxa\) for all \(x\in S\). A semigroup will be called an \(RC\)-commutative semigroup if it is an \(R\)-commutative and conditionally commutative semigroup. A semigroup \(S\) will be called a \(\Delta\)-semigroup if the lattice of all congruences of \(S\) is a chain with respect to inclusion. The results of this paper can be summarized as follows (Theorem 20): A semigroup is an \(RC\)-commutative \(\Delta\)-semigroup if and only if it is isomorphic to either \(G\) or \(G^ 0\) or \(R\) or \(R^ 0\) or \(N\) or \(N^ 1\), where \(G\) is a non-trivial subgroup of a quasicyclic \(p\)-group, \(p\) is a prime, \(R\) is a right zero semigroup of order 2 and \(N\) is a commutative nil semigroup whose ideals form a chain with respect to inclusion.