Cohen Generic Structures with Functions

2023/10/17 by Nathanael Ackerman, Mohammad Golshani, Ackerman, Nathanael +3 · 1 citation
Mathematics · Computer Science · #Advanced Topology and Set Theory #Advanced Algebra and Logic #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2310.11582

Abstract

Suppose \mathscrL^-⊆ \mathscrL are languages where \mathscrL ∖\mathscrL^- is relational. Additionally, let K be a strong \textrmFraïssé class in \mathscrL. We consider the partial ordering, under substructure, of those elements in K whose reduct to \mathscrL^- are substructures of a fixed \mathscrL^--structure M^-. In this paper, we establish that, under general conditions, this partial order satisfies the |M^-|-chain condition. Furthermore, under these conditions, we demonstrate that any generic for such a partial order satisfies the theory of the \Fraisse limit of K, provided M^- satisfies the theory of\textrmFraïssé limit of its age. We also provide general conditions that guarantee all such generics to be rigid, as well as conditions ensuring that these generics possess large automorphism groups.

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