The Universal Theory of First Order Algebras and Various Reducts

TL;DR

This paper develops a universal axiomatization for first order algebras and their reducts, providing a modular framework that encompasses various logical fragments and their algebraic structures.

Contribution

It introduces a universal set of axioms characterizing first order algebras and their reducts, extending to positive existential and quantifier-free cases.

Findings

Universal axioms characterize first order algebras.

Modular approach applies to various algebraic fragments.

Framework relates to theories and can incorporate function symbols.

Abstract

First order formulas in a relational signature can be considered as operations on the relations of an underlying set, giving rise to multisorted algebras we call first order algebras. We present universal axioms so that an algebra satisfies the axioms iff it embeds into a first order algebra. Importantly, our argument is modular and also works for, e.g., the positive existential algebras (where we restrict attention to the positive existential formulas) and the quantifier-free algebras. We also explain the relationship to theories, and indicate how to add in function symbols.