On semilattices of quasi-rectangular groups (Q1307046)

We say that a semigroup \(S\) is quasi-regular if for every \(a\in S\) there exist \(x\in S\) and a positive integer \(n\) with \(a^n=a^nxa^n\). If, moreover, the mapping \(f_e\) such that \(f_e(x)=exe\) for all \(x\in S\) is an endomorphism of \(S\) for every idempotent \(e\in S\) then \(S\) is called a \(C^*\)-quasi-regular semigroup. A quasi-regular semigroup such that its idempotents form a subsemigroup that is a rectangular band is called a quasi-rectangular group. A necessary and sufficient condition for a \(C^*\)-quasi-regular semigroup is given to be a strong semilattice of quasi-rectangular groups. Some consequences of this result are derived and examples are given.