On a class of left type-\(A\) monoids
For a semigroup \(S\), let \({\mathcal R}^*\) be the equivalence relation on \(S\) defined by: \(a{\mathcal R}^*b\) if and only if \(a\) and \(b\) are \(\mathcal R\)-related in some oversemigroup of \(S\). A monoid \(S\) is called a left type-\(A\) monoid if (i) for every \(a\in S\), the \({\mathcal R}^*\)-class of \(a\) contains a unique idempotent \(a^+\), (ii) the set of idempotents of \(S\) forms a semilattice, and, (iii) \(ae=(ae)^+a\) for all \(a\in S\) and every idempotent \(e\) of \(S\). Let \(S\) be a left type-\(A\) monoid, \(\tau\) the congruence relation on \(S\) generated by \({\mathcal R}^*\). It is shown that there exists a congruence relation \(\rho\) on \(S\) such that \(\rho\) induces the least right cancellative congruence on each of the \(\tau\)-classes. The left type-\(A\) monoid is called weakly proper if \(\rho\cap{\mathcal R}^*\) is the equality on \(S\). The authors prove several results which serve as a motivation for the study of this class of monoids and they give a structure theorem for these weakly proper left type-\(A\) monoids.
scientific article; zbMATH DE number 1423175