검색
Article Search

JMB Journal of Microbiolog and Biotechnology

OPEN ACCESS eISSN 0454-8124
pISSN 1225-6951
QR Code

Article

Kyungpook Mathematical Journal 2020; 60(1): 1-20

Published online March 31, 2020

Copyright © Kyungpook Mathematical Journal.

On the Representations of Finite Distributive Lattices

Mark Siggers

Department of Mathematics, Kyungpook National University, Daegu 41566, Republic of Korea
e-mail : mhsiggers@knu.ac.kr

Received: January 24, 2017; Revised: January 16, 2020; Accepted: January 16, 2020

Abstract

A simple but elegant result of Rival states that every sublattice L of a finite distributive lattice ℘ can be constructed from ℘ by removing a particular family ℐL of its irreducible intervals.

Applying this in the case that ℘ is a product of a finite set of chains, we get a one-to-one correspondence between the sublattices of ℘ and the preorders spanned by a canonical sublattice of ℘.

We then show that L is a tight sublattice of the product of chains ℘ if and only if is asymmetric. This yields a one-to-one correspondence between the tight sublattices of ℘ and the posets spanned by its poset J(℘) of non-zero join-irreducible elements.

With this we recover and extend, among other classical results, the correspondence derived from results of Birkhoff and Dilworth, between the tight embeddings of a finite distributive lattice L into products of chains, and the chain decompositions of its poset J(L) of non-zero join-irreducible elements.

Keywords: finite distributive lattice, representation, embedding, product of chains.