Mutual algebraicity and cellularity

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.

18 pages; to appear in Archive for Mathematical Logic

References in corpus (3)

• Uniformly bounded arrays and mutually algebraic structures
• Worst case expansions of complete theories
• Counting siblings in universal theories

Cited by in corpus (3)

• Worst case expansions of complete theories
• Counting siblings in universal theories
• Theories with few non-algebraic types over models, and their decompositions