Vol. 1, No. 1, 2022

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Worst-case expansions of complete theories

Samuel Braunfeld and Michael C. Laskowski

Vol. 1 (2022), No. 1, 15–30

Given a complete theory T and a subset Y Xk, we precisely determine the worst-case complexity, with respect to further monadic expansions, of an expansion (M,Y ) by Y of a model M of T with universe X. In particular, although by definition monadically stable/NIP theories are robust under arbitrary monadic expansions, we show that monadically NFCP (equivalently, mutually algebraic) theories are the largest class that is robust under anything beyond monadic expansions. We also exhibit a paradigmatic structure for the failure of each of monadic NFCP/stable/NIP and prove each of these paradigms definably embeds into a monadic expansion of a sufficiently saturated model of any theory without the corresponding property.

mutually algebraic, monadic expansions
Mathematical Subject Classification
Primary: 03C45
Received: 23 July 2021
Revised: 28 May 2022
Accepted: 9 June 2022
Published: 24 June 2022
Samuel Braunfeld
Computer Science Institute
Charles University
Czech Republic
Michael C. Laskowski
Department of Mathematics
University of Maryland
College Park, MD
United States