Mutual algebraicity and cellularity

Samuel Braunfeld; Michael C. Laskowski

doi:10.1007/s00153-021-00804-4

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arXiv · 1911.06303

Mutual algebraicity and cellularity

Samuel Braunfeld·Michael C. Laskowski

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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.

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BibTeX RIS

Samuel Braunfeld, Michael C. Laskowski. 2019-11-14. Mutual algebraicity and cellularity. https://doi.org/10.1007/s00153-021-00804-4

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