2012/08/31 by Gábor Czédli
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Combinatorial Mathematics #Complete lattice #Finite set #Lattice (music) #Regular polygon #Rough Sets and Fuzzy Logic #Set (abstract data type) #math.RA #msc:05E99 #msc:06C10 #msc:52C99
paper · pdf · doi:10.1007/s00012-014-0282-3
published as Algebra Universalis 71, 385-404 (2014) · 20 pages, no figure
arxiv created 2012/10/12 · openalex publication_date 2014/04/09 · openalex created_date 2016/06/24 · arxiv updated 2021/02/18 · openalex updated_date 2026/08/05
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.