Note on the description of join-distributive lattices by permutations

Abstract

Let L be a join-distributive lattice with length n and width(Ji L) ≤ 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.

The Congruences of a Finite Lattice

“…There are really only two general positive results: 1. The ideal lattice of a distributive lattice with zero is the congruence lattice of a lattice-see E. T. Schmidt [177] (also P. Pudlák [168]).…”

“…We have quite a bit of flexibility to construct a planar diagram for an ordered set, but for a lattice, we are much more constrained because L has a zero, which must be the lowest element and a unit, which must be the highest element-contrast this with Figure 1. 1. All lattices with five or fewer elements are planar; all but the five chains are shown in the first two rows of Figure 1.…”

“…The lattices L 1 = (L 1 , ∨, ∧) and L 2 = (L 2 , ∨, ∧) are isomorphic as algebras (in symbols, L 1 ∼ = L 2 ), and the map ϕ : L 1 → L 2 is an isomorphism iff ϕ is one-to-one and onto and ϕ(a ∨ b) = ϕa ∨ ϕb, (1) ϕ(a ∧ b) = ϕa ∧ ϕb (2) for a, b ∈ L 1 . A map, in general, and a homomorphism, in particular, is called injective if it is onto, surjective if it is one-to-one, and bijective if it is one-to-one and onto.…”

“…Lemma 1. 1. A reflexive binary relation α on a lattice L is a congruence relation iff the following three properties are satisfied for x, y, z, t ∈ L: (ii) x ≤ y ≤ z, x ≡ y (mod α), and y ≡ z (mod α) imply that x ≡ z (mod α).…”

Modular and semimodular lattices

“…For more details, see also On the other hand, it is worth mentioning that several new results have appeared recently about some geometric aspects of semimodular lattices; only to refer to planar semimodular lattices, see Grätzer and Edward Knapp [44,45,46] and Czédli and Schmidt [25,26], or to semimodular lattices that can be "represented" in higher dimensional spaces, see the web site of Schmidt (http://www.math.bme.hu/ schmidt/) for more details. As for convex geometries, there has been some recent improvement as well, see, e.g., the papers of Adaricheva, Gorbunov, Tumanov and Czédli [2,1,19].…”