Equivalences, \(\alpha\)-semigroups and \(\alpha\)-congruences
Let \(X\) be a set with more than one point. An \(\alpha\)-semigroup, \(T(X)\), is any semigroup of selfmaps of \(X\) which contains the identity map and all constant maps. An equivalence relation \(E\) on \(X\) is a \(T\)- equivalence relation if \((f(x), f (y)) \in E\) for all \((x, y) \in E\). The complete lattice of all \(T\)-equivalences on \(X\) is denoted by \(T\text{eq}(X)\) and the complete lattice of all equivalence relations on \(X\) is denoted by \({\mathcal E}(X)\). The \(T\)-equivalences turned out to be useful in the search for the smallest proper congruence on \(T(X)\). Specifically, it was shown by \textit{K. H. Hofmann} and \textit{K. D. Magill} jun. [Glasg. Math. J. 30, 301-313 (1988; Zbl 0681.54018)] that \(T(X)\) has a smallest proper congruence if and only if there is a smallest proper \(T\)-equivalence and moreover, one can describe the smallest proper congruence in terms of that \(T\)-equivalence. The work in this paper was motivated by the following question. Given a complete sublattice \(L\) of \({\mathcal E}(X)\) with \(\Delta(X),X \times X\in L\), where \(\Delta(X)\) is the diagonal of \(X\), does there exist an \(\alpha\)-semigroup \(T(X)\) such that \(T\text{eq}(X) = L\)? The question is left unanswered but to get some idea of the type of result obtained, choose \(E \in {\mathcal E}(X)\) and define \(T_ E(X) = \{f \in {\mathcal T}_ X : (f(a), f(b))\in E\;\forall(a, b) \in E\}\) where \({\mathcal T}_ X\) is the semigroup of all selfmaps of \(X\). The author proves, for example, that if \(E,F \in{\mathcal E}(X)\) are noncomparable and if \(E \cap F \neq \Delta(X)\) and \(E \vee F \neq X\times X\), then \(T\text{eq}(X) = \{\Delta(X), (E \cap F), E, F, (E \vee F), X\times X\}\) where \(T(X) = T_ E(X) \cap T_ F(X)\). He goes on to conjecture that if \(L\) is an arbitrary sublattice of \({\mathcal E}(X)\) containing both \(\Delta(X)\) and \(X \times X\), then \(T(X) = \bigcap\{T_ E(X) : E \in L\}\) is the greatest \(\alpha\)-semigroup with \(T\text{eq}(X) = L\). The author also investigates \(\alpha\)-congruences. A congruence \(\sigma\) on a semigroup \(S\) is referred to as an \(\alpha\)-congruence if \((az, bz) \in \sigma\) for all left zeros \(z \in S\) implies \((a, b) \in \sigma\). Among other things, he shows that if \(E, F \in {\mathcal E}(X)\) are noncomparable and if \(E \cap F \neq \Delta(X)\) and \(E \vee F \neq X \times X\), then there are exactly six \(\alpha\)-congruences on \(T_ E(X) \cap T_ F(X)\).
scientific article; zbMATH DE number 611114