Mutual algebraicity and cellularity

TL;DR

This paper characterizes when countable structures are cellular based on mutual algebraicity and introduces MA-presentations to simplify analysis, revealing structural decompositions and extensions.

Contribution

It establishes a precise equivalence between cellularity, $$-categoricity, and mutual algebraicity, and introduces MA-presentations for structural analysis.

Findings

Countable cellular structures are exactly those that are $$-categorical and mutually algebraic.

Mutually algebraic non-cellular structures can be extended to include infinitely many MA-connected components.

MA-presentations enable improved quantifier elimination and structural decomposition.

Abstract

We prove two results intended to streamline proofs about cellularity that pass through mutual algebraicity. First, we show that a countable structure $M$ is cellular if and only if $M$ is $\omega$-categorical and mutually algebraic. Second, if a countable structure $M$ in a finite relational language is mutually algebraic non-cellular, we show it admits an elementary extension adding infinitely many infinite MA-connected components. Towards these results, we introduce MA-presentations of a mutually algebraic structure, in which every atomic formula is mutually algebraic. This allows for an improved quantifier elimination and a decomposition of the structure into independent pieces. We also show this decomposition is largely independent of the MA-presentation chosen.