On the definition and the representability of quasi-polyadic equality algebras (Q2793902)

\textit{Quasi-polyadic equality algebras of dimension \(\alpha\)} (for an ordinal \(\alpha\)) are Boolean algebras supplemented with cylindrifications over finite subsets of \(\alpha\), substitutions for finite transformations over \(\alpha\) (functions from \(\alpha\) to \(\alpha\) that move only finitely many indices in \(\alpha\)), plus diagonal elements for pairs of indices from \(\alpha\). The author proves that the two axioms in the standard axiomatization that involve both substitutions (over arbitrary finite transformations) and cylindrifications (over arbitrary finite subsets) can be restricted to instances involving only substitutions for replacements (transformations that move a single index) and cylindrifications over a single index.NEWLINENEWLINE\textit{Finitary polyadic equality algebras of dimension \(\alpha\)} are Boolean algebras with cylindrifications only over single indices, substitutions only for replacements and transpositions, and diagonal elements for pairs of indices. The author's \textit{transposition algebras} are defined by weakening one of the axioms in an axiomatization of finitary polyadic equality algebras due to \textit{I. Sain} and \textit{R. J. Thompson} [in: Algebraic logic. Papers of a colloquium, Budapest, Hungary, August 8--14, 1988. Amsterdam etc.: North-Holland; Budapest: János Bolyai Mathematical Society. 539--571 (1991; Zbl 0751.03033)]. Commutativity of cylindrifications fails in transposition algebras. Sain and Thompson [loc. cit.] proved that quasi-polyadic equality algebras are definitionally equivalent to finitary polyadic equality algebras. To get a corresponding results for transposition algebras, the author introduces \textit{non-commutative quasi-polyadic equality algebras} by restricting cylindrifications in quasi-polyadic equality algebras to single indices, and correspondingly simplifying the Sain-Thompson axiom set. The class of non-commutative quasi-polyadic equality algebras is shown to be definitionally equivalent to the class of transposition algebras. The proof uses Jónsson's defining relations for full semigroups of finite transformations.NEWLINENEWLINEFor any \(\alpha\)-sequence \(p\) of elements of a base set \(U\), the \textit{weak space determined by \(U\) and \(p\)} is the set of \(\alpha\)-sequences of elements of \(U\) that differ from \(p\) in only finitely many places. A \textit{generalized weak space} is a disjoint union of weak spaces and forms, under the natural set-theoretic definitions, a \textit{generalized weak quasi-polyadic relativized set algebra}. Finally, as an application, the author shows, by means of a similar result for transposition algebras, that every non-commutative quasi-polyadic equality algebra is isomorphic to a generalized weak quasi-polyadic relativized set algebra.