Semantical conditions for the definability of functions and relations

TL;DR

This paper establishes semantic conditions for defining functions and relations in first-order logic, with applications to universal algebra and classical theorems like Pixley's and Baker-Pixley's.

Contribution

It provides semantical criteria equivalent to the existence of formulas with specific syntactic forms for definability, extending to functions and applying to algebraic structures.

Findings

Semantical conditions for relation definability established

Analogous results for function definability presented

Generalizations of Pixley's and Baker-Pixley's theorems obtained

Abstract

Let $\mathcal{L}\subseteq \mathcal{L}^{\prime }$ be first order languages, let $R\in \mathcal{L}^{\prime }-\mathcal{L}$ be a relation symbol, and let $% \mathcal{K}$ be a class of $\mathcal{L}^{\prime }$-structures. In this paper we present semantical conditions equivalent to the existence of an $\mathcal{L}$-formula $\varphi \left( \vec{x}\right) $ such that $\mathcal{K}\vDash \varphi (\vec{x})\leftrightarrow R(\vec{x})$, and $\varphi $ has a specific syntactical form (e.g., quantifier free, positive and quantifier free, existential horn, etc.). For each of these definability results for relations we also present an analogous version for the definability of functions. Several applications to natural definability questions in universal algebra have been included; most notably definability of principal congruences. The paper concludes with a look at term-interpolation in classes of structures with the same techniques used for definability. Here we obtain generalizations of two classical term-interpolation results: Pixley's theorem for quasiprimal algebras, and the Baker-Pixley Theorem for finite algebras with a majority term.