Semilattices of stratified extensions (Q6582350)

\textit{P. A. Grillet} [Semigroup Forum 50, No. 1, 25--36 (1995; Zbl 0822.20059)] defined a semigroup \(S\) with zero to be stratified whenever \(\bigcap_{m > 0} S^m = \{0\}\). A semigroup without zero is called stratified if \(S^0\) is stratified. Let \(S\) be a semigroup. The authors define the base of \(S\) to be the subset \(\mbox{Base}(S) = \bigcap_{m > 0} S^m\) and \(S\) to be a stratified extension of \(\mbox{Base}(S)\) if \(\mbox{Base}(S) \neq \emptyset\). If \(\mbox{Base}(S)\) is a trivial subgroup, then \(S\) is a stratified semigroup. \(S\) is called a finitely stratified extension if there exists \(m \in \mathbb{N}\) such that \(S^m = S^{m+1} = \mbox{Base}(S)\). The smallest such \(m\) is called the height of \(S\) and where necessary \(S\) is said to be a finitely stratified extension with height \(m\). If for every \(s \in S\) there is an \(m \in \mathbb{N}\) such that \(s^m \in \mbox{Base}(S)\) then \(S\) is a nil-stratified extension.\N\NThe authors investigate the structural properties of (finitely, nil-) stratified extensions. They describe the quotients, direct products and semilattices of stratified extensions. A semigroup in which every regular \(\mathcal{H}\)-class contains an idempotent is called a strongly \(2\)-chained semigroup. They prove that a strongly \(2\)-chained group-bound semigroup is a semilattice of certain kinds of Archimedean semigroups. They show that a semigroup \(S\) is a finite strongly \(2\)-chained semigroup if and only if \(S = [Y ; S\alpha]\) is a finite semilattice of finite semigroups \(S_\alpha\) where each \(S_\alpha\) is a finitely stratified extension of a completely simple semigroup. They also illustrate the coincidence of strict extensions of a Clifford semigroup by a semigroup and certain semilattice of extension groups.