ON O-MINIMALITY FOR EXPANSIONS OF A DENSE MEET-TREE

Abstract

This paper aims to define the notion of o-minimality for partially ordered sets. Originally, the notion of o-minimality was introduced for linearly ordered sets in the following way: A linearly ordered structure is said to be o-minimal if any definable subset is a finite union of intervals and points.
For partially ordered sets, this definition does not work. One of the main reasons for this is that the complement of an interval need not be a finite union of intervals, as happens in linearly ordered sets. Here we suggest a notion of a generalized interval which makes possible defining o-minimality for such a partial case of partially ordered sets as a dense meet-tree in a classical way: an expansion of a dense meet-tree is said to be o-minimal if any definable subset is a finite union of generalized interval and points. We think that this approach allows us to transfer
the machinery for investigating o-minimality for linearly ordered structures to partially ordered structures.