Representations of polyadic-like equality algebras

Assuming the usual finite axiom schema of polyadic equality algebras, axiom (P10) is changed to a stronger version. It is proved that infinite dimensional, polyadic equality algebras satisfying the resulting system of axioms are representable. The foregoing stronger axiom is not given with a first order schema. The latter is to be expected knowing the negative results for the Halmos schema axiomatizability of the representable, infinite dimensional, polyadic equality algebras. Furthermore, Halmos’ well-known classical theorem that “locally finite polyadic equality algebras of infinite dimension α are representable” is generalized for locally-m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{m}}$$\end{document} polyadic equality algebras, where m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{m}}$$\end{document} is an arbitrary infinite cardinal and m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{m}}$$\end{document} < α. Also, a neat embedding theorem is proved for the foregoing classes of polyadic-like equality algebras (a neat embedding theorem does not exists for polyadic equality algebras).