The representations of polyadic-like equality algebras

TL;DR

This paper establishes algebraic representation theorems for classes of polyadic-like equality algebras, linking set algebra axiomatizations with algebraic structures and addressing open problems in the field.

Contribution

It provides new representation theorems for classes of polyadic-like equality algebras, extending the understanding of their algebraic structure and axiomatization.

Findings

Boolean set algebras with union of Cartesian products are axiomatizable

Representation theorems for classes CPE, CPES, and mCPE are established

Answers to some open problems in algebraic logic are provided

Abstract

It is stated that Boolean set algebras with unit V, where V is a union of Cartesian products, are axiomatizable. The axiomatization coincides with that of cylindric polyadic equality algebras (class CPE). This is an algebraic representation theorem for the class CPE by relativized polyadic set algebras in the class Gp. Similar representation theorems are claimed for the classes strong cylindric polyadic equality algebras (CPES) and cylindric m-quasi polyadic equality algebras (mCPE). These are polyadic-like equality algebras with infinite substitution operators and single cylindrifications. They can be regarded also as infinite transformation systems equipped with diagonals and cylindrifications. No representation theorem or neat embedding theorem has proven for this class of algebras yet, except for the locally finite case. The theorems occuring in the paper answer some unsolved problems.