The Rees and Steinfeld theorems for semigroups with involution (Q1281626)

A *-ideal (*-bi-ideal) of a semigroup with involution is an ideal (bi-ideal) which is closed under the involution. An involution semigroup with zero is called completely 0-*-simple if it has no non-trivial *-ideals and contains at least one 0-minimal *-bi-ideal. The authors prove analogues of the Rees theorem and of O. Steinfeld's characterization of 0-direct unions of completely 0-simple semigroups in this setting. The difference from the case without involution arises from the fact that if a semigroup \(S\) with involution has an ideal \(K\) such that \(S\) is the 0-direct union of \(K\) and \(K^*\) then \(S\) is 0-*-simple but not 0-simple.