Convexity lattices
Generalizing the situation for vector spaces over ordered division rings the authors define the concept of convexity lattices namely of complete, biatomic, algebraic lattices with conditions that generalize Moore's four point theorem and the fact that in ordered sets of three points there is just one between the two others. Convexity lattices are called Peano if for distinct atoms p,q,r,s,t with \(q\leq p\bigvee r\), \(s\leq p\bigvee t\), there is an atom \(w\leq (r\bigvee s)\bigwedge (q\bigvee t)\). It is shown that they are Peano if and only if they satisfy the Pasch condition. In Peano convexity lattices the affine rank of an element can be defined, and some exchange property holds. The purpose of the article is to derive the theorems of Radon, Helly, and Carathéodory in the class of Peano convexity lattices. Especially it is shown that Helly's theorem holds in any atomic lattice which satisfies Radon's theorem.
vector spaces over ordered division rings
scientific article; zbMATH DE number 3902717