Around independence and domination in metric abstract elementary classes: assuming uniqueness of limit models

Abstract

We study notions of independence appropriate for a stability theory of metric abstract elementary classes (for short, MAECs). We build on previous notions used in the discrete case, and adapt definitions to the metric case. In particular, we study notions that behave well under superstability-like assumptions. Also, under uniqueness of limit models, we study domination, orthogonality and parallelism of Galois types in MAECs. 212A. Villaveces and P. Zambrano: Around independence and domination in MAEC which witnesses that M is a limit model over M 0 such that a M * M N * . Also, in this section we study notions of orthogonality and parallelism in superstable MAECs, inspired in [23]. In this study, we drop some strong conditions given in [23] and simplify some of the proofs given there. Also, we prove some properties which were not studied in [23]. Some basic definitions and resultsDefinition 2.1 The density character of a topological space is the smallest cardinality of a dense subset of the space. If X is a topological space, we denote its density character by dc(X ). If A is a subset of a topological space X , we define dc(A) := dc(A).We consider a natural adaptation of the notion of Abstract Elementary Class (cf. [3,11]), but work in a context of Continuous Logic that generalizes the "First Order Continuous" setting of [7] by removing the assumption of uniform continuity. 1 We follow the definitions given by Hirvonen and Hyttinen (cf. [15]).Definition 2.2 Let K be a class of L-structures (in the context of Continuous Logic) and ≺ K be a binary relation defined in K. We say that (K, ≺ K ) is a Metric Abstract Elementary Class (shortly MAEC) if 1. K and ≺ K are closed under isomorphism; 2. ≺ K is a partial order in K;1 Uniform continuity guarantees logical compactness in their formalization, but we drop compactness in AEC-like settings. 2 Notice that several of these classes fall under case (1)-however, in general, natural classes of Banach spaces are not axiomatizable in the context of [7]. Math. Log. Quart. 60, No. 3 (2014) / www.mlq-journal.org 213 3. for every m-ary relation symbolLet K be an MAEC. We say that K satisfies the Amalgamation Property (abbreviated AP) if and only if forLet K be an MAEC. We say that K satisfies Joint Embedding Property (abbreviated JEP) if and only if for everyNotice that if K has a prime model, then AP implies JEP. If K is an MAEC which satisfies AP and JEP and has large enough models, then we can construct a large enough model M (which we call a Monster Model) which is homogeneous, i.e., every isomorphism between two K-substructures of M can be extended to an automorphism of M, and also universal, i.e., every model with density character < dc(M) can be K-embedded into M.Under the existence of a monster model M, for all a ∈ M and N ≺ K M, we define ga-tp(a/N ) (the Galois type of a over N ) as the orbit of a under Aut(M/N )We denote the space of Galois types over a model M ∈ K by ga-S(M).Throughout this paper, we assume the existence of a homogenous and universal monster model as...

Limit models in metric abstract elementary classes: the categorical case

“…(Notice however that this distance between orbits does not imply that arbitrary realizations of the types are close.) The proofs of these forms of stationary are in [, Lemma 3.6 & Propositions & 3.15].…”

“…Smooth independence is the most robust property for our purposes, but we will use both εnon-splitting independence and smooth independence in our Assumption 4.1. The reader is urged to refer to [18] for the proofs of the results quoted here.…”

“…Anti-reflexivity) Let M ≺ K N where M is a (μ, ϑ)-limit model witnessed by M := {M i : i < ϑ}. If a | MM N and a ∈ N , then a ∈ M. As mentioned above, the proofs of the propositions in this section are in our paper[18].…”

Model theory of a Hilbert space expanded with an unbounded closed selfadjoint operator

“…In order to deal with the case of analytic structures, Hyttinen and Hirvonen defined metric abstract elementary classes in as a generalization of Shelah's AECs to classes of metric structures (MAECs). After this, Villaveces and Zambrano studied notions of independence and superstability for metric abstract elementary clases (MAECs) .…”

Downward categoricity from a successor inside a good frame

“…The proof of Theorem 6.14 develops orthogonality calculus in this setup (versions of some of our results on orthogonality have been independently derived by Villaveces and Zambrano [VZ14]). We were heavily inspired from Shelah's development of orthogonality calculus in successful good λ-frames [She09a, Section III.6], and use it to define a notion of unidimensionality similar to what is defined in [She09a, Section III.2].…”