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AN AXIOMATIC APPROACH TO FREE AMALGAMATION

Published online by Cambridge University Press:  19 June 2017

GABRIEL CONANT
GABRIEL CONANT*
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF NOTRE DAME NOTRE DAME, IN 46656, USAE-mail: gconant@nd.edu
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Abstract

We use axioms of abstract ternary relations to define the notion of a free amalgamation theory. These form a subclass of first-order theories, without the strict order property, encompassing many prominent examples of countable structures in relational languages, in which the class of algebraically closed substructures is closed under free amalgamation. We show that any free amalgamation theory has elimination of hyperimaginaries and weak elimination of imaginaries. With this result, we use several families of well-known homogeneous structures to give new examples of rosy theories. We then prove that, for free amalgamation theories, simplicity coincides with NTP2 and, assuming modularity, with NSOP3 as well. We also show that any simple free amalgamation theory is 1-based. Finally, we prove a combinatorial characterization of simplicity for Fraïssé limits with free amalgamation, which provides new context for the fact that the generic Kn-free graphs are SOP3, while the higher arity generic $K_n^r$-free r-hypergraphs are simple.

Keywords

Primary 03C45Secondary 03C15free amalgamationhomogeneous structuressimplicityrosinessTP2SOPn
Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 2 , June 2017 , pp. 648 - 671
Copyright
Copyright © The Association for Symbolic Logic 2017 

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