A solution to the finitizability problem for quantifier logics with equality

TL;DR

This paper introduces a finite axiomatization for certain algebraic structures modeling first-order logic with equality, providing a solution to the finitizability problem by using relativized semantics and specific semigroup conditions.

Contribution

It establishes a finite schema axiomatization for classes of algebras related to first-order logic with equality, using rich semigroups and guarded semantics, and proves their representability and amalgamation properties.

Findings

CPEA_T is axiomatizable by a finite schema.

Atomic algebras in CPEA_T are completely representable under certain conditions.

CPEA_T has the super amalgamation property.

Abstract

We consider countable so-called rich subsemigroups of (\omega\omega,\circ); each such semigroup $T$ gives a variety CPEA_T that is axiomatizable by a finite schema of equations taken in a countable subsignature of that of \omega-dimensional cylindric-polyadic algebras with equality where substitutions are restricted to maps in T. It is shown that for any such T, A\in CPEA_T iff A is representable as a concrete set algebra of \omega-ary relations. The operations in the signature are set-theoretically interpreted like in polyadic equality set algebras, but such operations are relativized to a union of cartesian spaces that are not necessarily disjoint. This is a form of guarding semantics. We show that CPEA_T is canonical and atom-canonical. Imposing an extra condition on T, we prove that atomic algebras in CPEA_T are completely representable and that CPEA_T has the super amalgamation property. If T is rich and {\it finitely represented}, it is shown that CPEA_T is term definitionally equivalent to a finitely axiomatizable Sahlqvist variety. Such semigroups exist. This can be regarded as a solution to the central finitizability problem in algebraic logic for first order logic {\it with equality} if we do not insist on full fledged commutativity of quantifiers. The finite dimensional case is approached from the view point of guarded and clique guarded (relativized) semantics of fragments of first order logic using finitely many variables. Both positive and negative results are presented.