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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Second author partially supported by NSF Grant DMS-1855789.
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Braunfeld, S., Laskowski, M.C. Mutual algebraicity and cellularity. Arch. Math. Logic 61, 841–857 (2022). https://doi.org/10.1007/s00153-021-00804-4
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DOI: https://doi.org/10.1007/s00153-021-00804-4