One-dimensional asymptotic classes of finite structures

Abstract

Abstract. A collection C of finite L-structures is a 1-dimensional asymptotic class if for every m ∈ N and every formula ϕ(x,ȳ), whereȳ = (y 1 , . . . , y m ):(i) There is a positive constant C and a finite set E ⊂ R >0 such that forOne-dimensional asymptotic classes are introduced and studied here. These classes come equipped with a notion of dimension that is intended to provide for the study of classes of finite structures a concept that is central in the development of model theory for infinite structures. Connections with the model theory of infinite structures are also drawn.

Variations Sur Un Thème De Aldama Et Shelah

“…Counterexample 2.5 (A centraliser that is not nice). In an infinite extraspecial 3-group K, which is supersimple of rank 1 (see [18]) and whose conjugacy classes are all finite, choose (a n ) n 1 such that the chain of centralisers…”

Some connections between finite and infinite model theory

“…More recently, a direction of research initiated by Macpherson and Steinhorn [28] and continued by Elwes [13,14] studies classes of nite structures in which denable sets have a uniform asymptotic behaviour, as the cardinalities of the universes increase. The complete theory T of a non-principal ultraproduct of such a class of nite structures (called aǹ asymptotic class') is simple with nite SU-rank and there is a notion of measure on the denable subsets of models of T , but T is not necessarily smoothly approximable.…”

“…Before continuing we note that there is a line of research [28,13,14,15], not discussed here, which studies the connection between classes of nite structures in which denable sets have uniform behaviour, asymptotically, and (innite) simple structures with nite SUrank and with a measure on the denable subsets, but which are not necessarily smoothly approximable. A question not answered here is whether the approach in this article has anything in common with the work about`asymptotic classes' and`measurable structures'…”

“…(7) The random graph ts within the framework presented in sections 5 and 6 as well as within the framework of`asymptotic classes' and`measurable structures' [28,13,14,15]. …”

Measurable groups of low dimension

“…For instance, even the possible structure of a group of rank 1 is not yet fully understood. In this paper we prove some elementary results about low-dimensional supersimple groups and group-actions under various extra hypotheses, most notably that of measurability: the assumption that the system of definable sets admits a finitary counting measure, as developed in [8,9,3,4].…”