On t-semigroups (Q1891655)

We discuss the notion of a t-semigroup (i.e., a semigroup \(S\) in which the relation ``is an ideal of'' is transitive among the ideals of \(S\)). In connection with the congruence extension property (CEP) and the ideal extension property (IEP), we determine that each of these properties for a semigroup \(S\) implies that \(S\) is a t-semigroup. For a cyclic semigroup \(S\), we establish the equivalence of \(S\) being a t-semigroup, \(S\) having CEP, and \(S\) having IEP. Examples are provided for illustrative purposes.