TY - JOUR

T1 - On the representations of finite distributive lattices

AU - Siggers, Mark

N1 - Publisher Copyright:
© Kyungpook Mathematical Journal.

PY - 2020

Y1 - 2020

N2 - A simple but elegant result of Rival states that every sublattice L of a finite distributive lattice P can be constructed from P by removing a particular family IL of its irreducible intervals. Applying this in the case that P is a product of a finite set C of chains, we get a one-to-one correspondence L → DP(L) between the sublattices of P and the preorders spanned by a canonical sublattice C° of P. We then show that L is a tight sublattice of the product of chains P if and only if DP(L) is asymmetric. This yields a one-to-one correspondence between the tight sublattices of P and the posets spanned by its poset J(P) 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.

AB - A simple but elegant result of Rival states that every sublattice L of a finite distributive lattice P can be constructed from P by removing a particular family IL of its irreducible intervals. Applying this in the case that P is a product of a finite set C of chains, we get a one-to-one correspondence L → DP(L) between the sublattices of P and the preorders spanned by a canonical sublattice C° of P. We then show that L is a tight sublattice of the product of chains P if and only if DP(L) is asymmetric. This yields a one-to-one correspondence between the tight sublattices of P and the posets spanned by its poset J(P) 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.

KW - Embedding

KW - Finite distributive lattice

KW - Product of chains

KW - Representation

UR - http://www.scopus.com/inward/record.url?scp=85085927931&partnerID=8YFLogxK

U2 - 10.5666/KMJ.2020.60.1.1

DO - 10.5666/KMJ.2020.60.1.1

M3 - Article

AN - SCOPUS:85085927931

SN - 1225-6951

VL - 60

SP - 1

EP - 20

JO - Kyungpook Mathematical Journal

JF - Kyungpook Mathematical Journal

IS - 1

ER -