Notes on the description of join-distributive lattices by permutations

TL;DR

This paper demonstrates the equivalence of two permutation-based descriptions of join-distributive lattices and introduces a new characterization method using trajectories.

Contribution

It establishes the equivalence between combinatorial and lattice-theoretical descriptions of join-distributive lattices and introduces a novel trajectory-based characterization.

Findings

Proves the equivalence of two permutation-based descriptions.

Characterizes join-distributive lattices using trajectories.

Provides a unified understanding of lattice descriptions.

Abstract

Let L be a join-distributive lattice with length n and width(Ji L) \leq k. There are two ways to describe L by k-1 permutations acting on an n-element set: a combinatorial way given by P.H. Edelman and R.E. Jamison in 1985 and a recent lattice theoretical way of the second author. We prove that these two approaches are equivalent. Also, we characterize join-distributive lattices by trajectories.