Semilattices with a group of automorphisms
An algebra \({\mathcal S}=\langle S;\wedge ,F \rangle \) with a binary operation \(\wedge \) and a set \(F\) of unary operations is an \(\mathcal F\)-semilattice if \(\mathcal F = \langle F;\cdot ,{}^{-1},\text{id} \rangle \) is a group and \({\mathcal S}\) satisfies the following identities: (a) \(\wedge \) is a semilattice operation, (b) \(\text{id}(x)=x\), (c) \(f(g(x)) = (f\cdot g)(x)\) for all \(f,g\in F\), and (d) \(f(x\wedge y) = f(x)\wedge f(y)\) for all \(f\in F\). In other words, an \(\mathcal F\)-semilattice is a semilattice expanded with a set \(F\) of new operations which forms an automorphism group of the semilattice. The author describes up to isomorphism all simple algebras of \(\mathcal F\)-semilattices in the case when \(\mathcal F\) is commutative. An example of a simple algebra that does not fit in the previous description for non-commutative \(\mathcal F\) is given. This generalizes investigations by \textit{J. Ježek} [Algebra Univers. 15, 162-175 (1982; Zbl 0512.06006)].
scientific article; zbMATH DE number 1407076