Fully chain-representable distributive lattices

Research paper by Gábor Czédli

Indexed on: 30 May '17Published on: 30 May '17Published in: arXiv - Mathematics - Rings and Algebras


Following a recent pioneering paper of G.\ Gr\"atzer on representations of certain subsets of distributive lattices as the sets of principal congruences in congruence lattices of finite lattices, we call a finite distributive lattice $D$ fully chain-representable if for every subset $Q$ of $D$ containing $0$, $1$, and the set $J(D)$ of nonzero join-irreducible elements of $D$ there exists a finite chain $C$ with the following properties: (1) the edges of $C$ are colored with the elements of $J(D)$, (2) for every interval $[x,y]$ of $C$, the join of colors occurring in $[x,y]$ belongs to $Q$, and (3) each element of $Q$ is represented by such a join. We prove that a finite distributive lattice $D$ is fully chain-representable if and only if it is planar and it has at most one join-reducible coatom.