Mutually algebraic structures and expansions by predicates

TL;DR

This paper introduces mutually algebraic structures and theories, characterizes them through model-theoretic properties, and explores their reducts and elementary extensions, advancing understanding of their structural and logical features.

Contribution

It defines mutually algebraic structures and theories, proves their equivalences, and analyzes their reducts and extensions, providing new insights into their model-theoretic properties.

Findings

Mutually algebraic theories are equivalent to weakly minimal and trivial theories.

Every structure has a maximal mutually algebraic reduct.

Elementary extensions of mutually algebraic structures have a specific structure theorem.

Abstract

We introduce the notions of a mutually algebraic structures and theories and prove many equivalents. A theory $T$ is mutually algebraic if and only if it is weakly minimal and trivial if and only if no model $M$ of $T$ has an expansion $(M,A)$ by a unary predicate with the finite cover property. We show that every structure has a maximal mutually algebraic reduct, and give a strong structure theorem for the class of elementary extensions of a fixed mutually algebraic structure.