Bare canonicity of representable cylindric and polyadic algebras

TL;DR

This paper demonstrates that for finite dimensions at least 3, the axiomatizations of various algebraic structures related to logic inherently include infinitely many non-canonical formulas and are non-elementary, using random graph-based algebras.

Contribution

It proves the non-canonicity and non-elementarity of axiomatizations for certain algebraic logic varieties in finite dimensions, employing novel algebraic constructions from random graphs.

Findings

Axiomatizations contain infinitely many non-canonical formulas

The classes are non-elementary

Uses algebras derived from random graphs

Abstract

We show that for finite n at least 3, every first-order axiomatisation of the varieties of representable n-dimensional cylindric algebras, diagonal-free cylindric algebras, polyadic algebras, and polyadic equality algebras contains an infinite number of non-canonical formulas. We also show that the class of structures for each of these varieties is non-elementary. The proofs employ algebras derived from random graphs.