O-asymptotic classes of finite structures, pseudofinite dimension and forking (2014)
Abstract: My research aims to study the of ultraproducts of finite structures and the study of forking, pseudofinite dimensions and other model-theoretic properties, specifically in pseudofinite structures and classes of finite linearly ordered structures. The main results obtained during my Ph.D can be separated in two main topics: Pscudofinitc dimcnsions and forking, and 0-asymptotic classcs of finitc structurcs. Studying classes of finite structures (e.g 1-dimensional asymptotic classcxs) one can ask whether the notions of pseudofinite dimensions of Hrushovski and Wagner provide information about independence relations and other model-theoretic properties in their ultraproducts. In this setting, I proved that an instance of dividing in an ultraproduct of finite structures can be realized as a decrease in the pseudofinite dimension; thus implying, as a corollary, a generalization of a well-known result in 1-dimensional asymptotic classes; namely, that every infinite ultraproduct of models in such a class is supersimple of U-rank 1. In the study of classes of finite linearly ordered structures, I stated the definition of O-asymptotic classcs as a way to meld ideas from 1-dimensional asymptotic classes and 0-minimality. The main examples of these classes are the class of finite linear orders and the class of cyclic grolllxs Z/(2N + I)Z with the natural order inherited by the order in the integers when we take the representative-s - N < - (N-1) < ? <-1<0<1< ? < N ? 1 < N. Results obtained Include: a cell-decomposition result for 0-asymptotic classes melding ideas from the combinatorial cell decomposition for 1-dimensional asymptotic clas.scxs, and the cell decomposition theorem in O-minimal structures; and a classification of the ultraproducts of 0-asymptotic classes: if every ultraproduct of a class C is o-minimal, then C is an O-asymptotic class; every infinite ultraproduct of structures in an 0-asymptotic class is superrosy of U-thorn-rank 1 and NTP2 of inp-rank l. I also present a preliminary collection of results towards isolate conditions under which dense 0-minimal structures can be obtained as quotients of ultraproducts of 0-asymptotic classes