An application of the decomposition theorem (New developments of independence notions in model theory)
Pablo Cubides Kovacsics · Institutional Repositories DataBase (IRDB) · 2010
In this paper we present an application of the decomposition theorem for abstract elementary classes.We give a list of conditions implying that any two models of cardinality $\lambda$ which are $L_{\infty,\lambda}$ -equivalent are isomorphic (for a large enough $\lambda$ ).A similar version was proved by Shelah for first order theories in [5].In [2], Rami Grossberg and Olivier Lessmann proposed a number of axioms in order to lift and generalize the decomposition theorem, first proved by Shelah in [5], to abstract elementary classes (hereafter AEC).In this paper we present an application of this abstract version of the theorem.We show that if an AEC satisfies a similar setting to the one proposed in [2] then any two models of cardinality $\lambda$ which are $L_{\infty,\lambda}$ -equivalent are isomorphic (for a large enough $\lambda$ ).At least two main differences between [2] and the approach here outlined are important to mention.Firstly, the choice of axioms is slightly different.Secondly, an additional condition is added to the definition of decomposition.Although these differences will not change the application here discussed, they were needed to reach a detailed and gapless proof of the abstract version of the decomposition theorem.For more details about this see [3].The notation will be standard.We work in an AEC $(\mathcal{K}, \prec)$ with the amalgamation property and arbitrary large models.This enables us to fix a $\overline{\kappa}$ -universal and strongly $\overline{\kappa}-$ model-homogeneous (hence $\overline{\kappa}$ -Galois saturated) model $C\in \mathcal{K}$ for big enough cardinal $\overline{\kappa}$ .Every set and structure is assumed to be respectively a subset and a substructure of $C$ of cardinality less than $\overline{\kappa}$ .Types, which are called in this abstract framework Galois types, are denoted by $gt(a/M)$ (the galois type of $a$ over $M$ ) and correspond simply to orbits in Aut $(C)$ , that is, $gt(a/M)$ is the set of all $b\in C$ such that there is $f\in$ Aut $(C)$ such that $f[M=id$ and $f(a)=b$.We first present the axiomatic setting and start with an axiom that defines an independence relation as a relation between triplets of subsets of C. We denote this relation by A $\downarrow B$ $c$ .