Interval decomposition lattices are balanced

TL;DR

This paper proves that the lattice of interval decompositions in various discrete structures is balanced, using a general axiomatic approach to intervals and join-irreducible partitions.

Contribution

It provides a novel axiomatic characterization of intervals and demonstrates that the lattice of all interval decompositions is balanced.

Findings

Lattice of interval decompositions is balanced.

Characterization of join-irreducible partitions into intervals.

General axiomatic framework for intervals.

Abstract

Intervals in binary or n-ary relations or other discrete structures generalize the concept of interval in a linearly ordered set. Join-irreducible partitions into intervals are characterized in the lattice of all interval decompositions of a set, in a general sense of intervals defined axiomatically. This characterization is used to show that the lattice of interval decompositions is balanced.