Orthogonal decomposition of definable groups

TL;DR

This paper introduces a new concept of cohesiveness in model theory to analyze the structure of definable groups, showing they can be decomposed into cohesive orthogonal subsets, especially in o-minimal structures.

Contribution

It defines cohesiveness as a dual to orthogonality, proves that definable groups decompose into cohesive orthogonal parts, and characterizes groups in o-minimal structures and their quotients.

Findings

Definable groups decompose into products of cohesive orthogonal subsets.

Groups with dimension one or definably simple are cohesive.

Abelian groups in finite unions of o-minimal structures are quotients of products of locally definable groups.

Abstract

Orthogonality in model theory captures the idea of absence of non-trivial interactions between definable sets. We introduce a somewhat opposite notion of cohesiveness, capturing the idea of interaction among all parts of a given definable set. A cohesive set is indecomposable, in the sense that if it is internal to the product of two orthogonal sets, then it is internal to one of the two. We prove that a definable group in an o-minimal structure is a product of cohesive orthogonal subsets. If the group has dimension one, or it is definably simple, then it is itself cohesive. Finally, we show that an abelian group definable in the disjoint union of finitely many o-minimal structures is a quotient, by a discrete normal subgroup, of a direct product of locally definable groups in the single structures.