Uniformly bounded arrays and mutually algebraic structures

TL;DR

This paper characterizes mutually algebraic theories through the concept of uniformly bounded arrays in atomic formulas, providing a new criterion for identifying such theories in models.

Contribution

It introduces a verifiable notion of uniformly bounded arrays and establishes an equivalence with mutual algebraicity for complete and incomplete theories.

Findings

Mutually algebraic theories are characterized by uniformly bounded arrays in atomic formulas.

The paper provides a criterion to verify mutual algebraicity in models.

It bridges the concept of array bounds with the algebraic property of theories.

Abstract

We define an easily verifiable notion of an atomic formula having uniformly bounded arrays in a structure $M$. We prove that if $T$ is a complete $L$-theory, then $T$ is mutually algebraic if and only if there is some model $M$ of $T$ for which every atomic formula has uniformly bounded arrays. Moreover, an incomplete theory $T$ is mutually algebraic if and only if every atomic formula has uniformly bounded arrays in every model $M$ of $T$.