Triangular family of intervals

Intervals are one of the most basic concepts arises from betweenness relations. A particular family of intervals is shown below. This sort of family can be formed even by metric betweenness on a path graph or order betweenness on a total order. This set system is an atomistic Moore family with a triangular structure. The number of points at some covering distance from the maximum is the covering distance plus one.



The triangular structure of the intervals has three edges. These correspond to the the join irreducibles and the meet irreducibles of the lattice. The edges on the side are the meet irreducibles and the bottom edge is the join irreducibles. The meet irreducibles are the rays and the join irreducibles are the singleton intervals. Every interval can be expressed as an intersection of rays or as a union of singleton intervals. The inner elements of the triangle. The triangular structure seems to be defining character of the set of intervals.