Interpreting Groups and Fields in Some Nonelementary Classes

Abstract

ABSTRACT. This paper is concerned with extensions of geometric stability theory to some nonelementary classes. We prove the following theorem:

Quasiminimal structures, groups and Zariski-like geometries

“…In 4.1, we present a notion of plane curve that works in our setting, in 4.2 we reformulate our group configuration theorem in terms of indiscernible arrays (Lemma 4.17), and finally in 4.3 we adopt Hrushovski's and Zilber's proof to our setting and show that a suitable plane curve can be used to find the group configuration. The work is continued in a forthcoming paper by the second author [20], where it is shown that an analogue for Zilber's trichotomy holds in a Zariski-like structure: a non locally modular Zariski-like structure interprets either an algebraically closed field or a non-classical group (see [16]). …”

“…A result like this would be in line with the existing studies of geometries in non-elementary cases. However, since the existence of a non-classical group (see [16] and [17] for locally modular cases) is still open, to prove something like this seems very difficult, and if it turns out that there are nonclassical groups, the playground is completely open. Another open question is whether B. Zilber's pseudo-exponentiation and quantum algebras (not at a root of unity) satisfy our axioms for Zariski-like geometries.…”

Locally modular geometries in homogeneous structures

“…. , 0, c 1 +· · ·+c n −a+d) ∈ S. As above, it follows c 1 (ii)⇒(iii): By [6] (also in [1], these groups have been studied), there is a group (G, +) such that (a) Every member of G is a permutation of V .…”

“…The problem we had in [2] was that we were not able to show that strongly minimal groups in M eq are abelian, see [5] and [6] for more on this problem in the general case. Thus at that time it looked as if the geometries could be wild.…”

“…All the hard work for the main theorem of this paper has already been done: the needed group constructions have been done in [3] and [6], and the theory of M eq and 1-basedness has been developed in [2]. So in this paper we just put these things together.…”