\(F\)-abundant semigroups (Q2731111)

A semigroup \(S\) is called abundant if every \({\mathcal L}^*\)-class and \({\mathcal R}^*\)-class of \(S\) contains at least one idempotent. For each \(a\in S\), \(a^+\) (resp. \(a^*\)) denotes an element of \(R^*_a\cap E(S)\) (resp. \(L^*_a\cap E(S)\)). Let \(\sigma\) be a minimum cancellative congruence on an abundant semigroup \(S\). If the \(\sigma\)-class \(\sigma_a\) of the element \(a\) in \(S\) contains a greatest element \(m_a\) under the natural partial order, then this element is uniquely determined. An abundant semigroup is called \(F\)-abundant if each \(\sigma\)-class of \(S\) has a greatest element with respect to the natural partial order. An \(F\)-abundant semigroup \(S\) is called strong if for all \(a\in S\), \(Em_a^+=m_a^+E\) and \(Ea^*=a^*E\).NEWLINENEWLINENEWLINEFollowing \textit{R. McFadden} and \textit{L. O'Carroll}'s work on \(F\)-inverse semigroups [Proc. Lond. Math. Soc., III. Ser. 22, 652-666 (1971; Zbl 0219.20042)] and \textit{C. C. Edward}'s work on \(F\)-regular semigroups [Semigroup Forum 19, 331-345 (1980; Zbl 0432.20052)], the author in this paper obtains some properties of \(F\)-abundant semigroups and establishes the structure of strongly \(F\)-abundant semigroups. In addition, the author also proves that all IC quasi-adequate semigroups are of type \(W\), which answers an open problem raised by \textit{A. El-Qallali} and \textit{J. B. Fountain} [Proc. R. Soc. Edinb., Sect. A 91, 79-90 (1981; Zbl 0501.20043)].