\(\zeta\)-primitive \(S\)-sets (Q1590068)

Let \(S\) be a semigroup with zero and let \(M\) be a centered (right) \(S\)-set. Denote by \(\Gamma(M)\) the set of all \(S\)-subsets of \(M\). A map \(\zeta\colon\Gamma(M)\to\Gamma(M)\) is said to be a conjugate map on \(\Gamma(M)\) if for any \(L\in\Gamma(M)\), \(\zeta(L)=\bigcup\{\zeta(uS^1)\mid u\in L\}\) and \(\zeta^2(L)\subseteq L\). An \(S\)-subset \(L\) of \(M\) is said to be a \(\zeta\)-subset of \(M\) if \(\zeta(L)\subseteq L\). \(L\in\Gamma(M)\) is said to be \(\zeta\)-core-free if \(\bigcup\{aS^1\mid a\in L\text{ with }\zeta(aS^1)\subseteq L\}=\{\Theta\}\). An \(S\)-set \(M\) is said to be \(\zeta\)-primitive if it has a \(\zeta\)-core-free maximal \(S\)-subset. For \(L\in\Gamma(M)\), its \(\zeta\)-centralizer \(C_\zeta(L)\) in \(M\) is defined by \(C_\zeta(L)=\bigcup\{K\mid K\in\Gamma(M)\text{ such that }\{K\cap\zeta(L)\}\cup\{\zeta(K)\cap L\}=\{\Theta\}\}\). The article describes \(\zeta\)-primitive \(S\)-sets. For example, it is proved that an \(S\)-set \(M\) is \(\zeta\)-primitive and \(\zeta\) is not nilpotent on \(M\) if and only if there exists a minimal \(\zeta\)-subset \(N\) of \(M\) with \(C_\zeta(N)=\{\Theta\}\) and a maximal \(S\)-subset \(K\) of \(M\) with \(K\cup N=M\).