On the Concepts of Convexity and Weak O-Minimality for Partially Ordered Structures

In this paper, we consider a generalization of the concept of weak o-minimality to partially ordered sets. However, the concept of weak o-minimality is based on the concept of a convex set, the direct transfer of which to partial orders, as it will be shown in the work, is not, in our opinion, the most successful, since then in the class of weakly o-minimal partially ordered structures, it is possible to define any mathematical structure. Moreover, as it will be shown, this can be done using such a simple operation as the intersection of intervals. The article is devoted to the search for various generalizations of the concept of “convex set” to partial orders. Since convex sets on a line also have other properties, such as the ability to represent them as a union or intersection of intervals, convex sets are connected, all these properties can be used as the basis for the definition of a “convex set” for partially ordered structures. Thus, the representation of a convex set as a union of nested intervals (half-intervals, segments) gives us the concept of an “internally convex set,” and the intersection of intervals gives us the concept of an “externally convex set”. In the article, we will build examples that show the non-equivalence of the introduced concepts.