Semigroups whose regular \({\mathcal J}\)-classes form finite lattices (Q1891646)

A \(\mathcal J\)-class of a semigroup \(S\) is defined to be regular if it contains at least one regular element. The family of all regular \(\mathcal J\)-classes of a semigroup \(S\) is denoted by \({\mathcal J}_R(S)\) and is partially ordered in the usual manner. That is, \(J_a\leq J_b\) if \(S^1aS^1\subseteq S^1bS^1\). The symbol \(S(X)\) denotes the semigroup of all continuous selfmaps of the topological space \(X\) where the binary operation is composition. In this paper, we characterize, among the dendrites, those spaces \(X\) for which \({\mathcal J}_R(S(X))\) forms a finite lattice. Specifically, what we do is to exhibit one particular space and eight infinite classes of spaces and we show that, among the dendrites, the subcontinua of these spaces are exactly those spaces with the property that the partially ordered family of all regular \(\mathcal J\)-classes of their semigroups are finite lattices. All this is made precise in Theorem 3.6, the last result of the paper.