A representation theorem for weak automorphisms of a universal algebra
Let \({\mathcal A}=(A,{\mathbb{A}})\) be a universal algebra, where A is the underlying set of the algebra and \({\mathbb{A}}\) is the clone of all polynomial operations. Let \({\mathbb{A}}_ n\) denote the set of all n-ary operations of the clone and \({}^ nf\) denote the nth cartesian power of a mapping f. A weak automorphism of \({\mathcal A}\) is a bijection \(\phi\) : \(A\to A\) such that for every n \((n=0,1,2,...)\) and every \(f\in {\mathbb{A}}\) the compositions \(\phi^{-1}f(^ n\phi)\) and \(\phi f(^ n(\phi^{- 1}))\) also belong to \({\mathbb{A}}_ n\). Let W(\({\mathcal A})\) denote the group of weak automorphisms which satisfy \(\phi^{-1}f(^ n\phi)=f\), for every \(f\in {\mathbb{A}}_ n\). \(W_ n({\mathcal A})\) is a normal subgroup of W(\({\mathcal A})\). The sequence W(\({\mathcal A}),W_ 0({\mathcal A}),W_ 1({\mathcal A}),..\). is a descending chain of normal subgroups. Main results. Let \(G,G_ 0,G_ 1,...,G_ n,..\). be a descending chain of normal subgroups of G. There exists an algebra \({\mathcal A}\), such that the chain \(...\triangleleft G_ n\triangleleft...\triangleleft G_ 1\triangleleft G_ 0\triangleleft G\) is isomorphic to the chain \(...\triangleleft W_ n({\mathcal A})\triangleleft...\triangleleft W_ 1({\mathcal A})\triangleleft W_ 0({\mathcal A})\triangleleft W({\mathcal A})\). Consequently \(N=\cap^{\infty}_{n=0}G_ n\) is isomorphic to A(\({\mathcal A})=\cap^{\infty}_{n=0}W_ n({\mathcal A})\).
scientific article; zbMATH DE number 3906562