Coordinatization of join-distributive lattices

TL;DR

This paper provides a new coordinatization method for join-distributive lattices, linking them to permutations and extending to related structures like antimatroids and convex geometries.

Contribution

It introduces a permutation-based coordinatization for join-distributive lattices, generalizing previous combinatorial descriptions within lattice theory.

Findings

Lattices can be described by k-1 permutations acting on {1,...,n}.

The coordinatization applies to antimatroids and convex geometries.

Provides a structural framework connecting permutations and lattice elements.

Abstract

Join-distributive lattices are finite, meet-semidistributive, and semimodular lattices. They are the same as Dilworth's lattices in 1940, and many alternative definitions and equivalent concepts have been discovered or rediscovered since then. Let L be a join-distributive lattice of length n and let k denote the width of the set of join-irreducible elements of L. A result of P.H. Edelman and R.E. Jamison, translated from Combinatorics to Lattice Theory, says that L can be described by k-1 permutations acting on the set {1,...,n}. We prove a similar result within Lattice Theory: there exist k-1 permutations acting on {1,...,n} such that the elements of L are coordinatized by k-tuples over {0,...,n}, and the permutations determine which k-tuples are allowed. Since the concept of join-distributive lattices is equivalent to that of antimatroids and convex geometries, our result offers a coordinatization for these combinatorial structures.