Submission date: 14 October 2022

Abstract:

We study a variation of Chatzidakis and Pillay's expansions of a theory by a generic predicate, requiring in addition that the predicate satisfy a universal condition. We show that for a theory T, these expansions (model companions) always exist in any structure definable in T if and only if T is nfcp, and that if T eliminates ∃^∞ this expansion exists whenever the universal condition is given by an algebraic ternary relation. When T is NSOP_1, we show that, in any relational expansion of T with free-amalgmation properties, Conant-independence is inherited from Kim-independence in T, and that the expansion is either NSOP_1 or both TP_2 and strictly NSOP_4. In the specific setting of an algebraic ternary relation where T is additionally weakly minimal (so the NSOP_1 case is in fact simple), we apply Hrushovski's group configuration theorem to characterize the non-simple case as the case where the algebraic ternary relation is essentially the graph of a rank-one group operation.

Mathematics Subject Classification: 03C45, 03C10

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Full text arXiv 2210.07524: pdf, ps.