Semantical conditions for the definability of functions and relations

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 ${\varphi(\vec{x})}$ such that ${\mathcal{K}\vDash \varphi(\vec{x}) \leftrightarrow R(\vec{x})}$, where ${\varphi}$ 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.