ACTA issues

On the set of principal congruences in a distributive congruence lattice of an algebra

Gábor Czédli

Acta Sci. Math. (Szeged) 84:3-4(2018), 357-375

Abstract. Let $Q$ be a subset of a finite distributive lattice $D$. An algebra $A$ \emph{represents the inclusion $Q\subseteq D$ by principal congruences} if the congruence lattice of $A$ is isomorphic to $D$ and the ordered set of principal congruences of $A$ corresponds to $Q$ under this isomorphism. If there is such an algebra for \emph{every} subset $Q$ containing $0$, $1$, and all join-irreducible elements of $D$, then $D$ is said to be \emph{fully (A1)-representable}. We prove that every fully (A1)-representable finite distributive lattice is planar and it has at most one join-reducible coatom. Conversely, we prove that every finite planar distributive lattice with at most one join-reducible coatom is \emph{fully chain-representable} in the sense of a recent paper of G. Grätzer. Combining the results of this paper with another result of the present author, it follows that every fully (A1)-representable finite distributive lattice is ``fully representable'' even by principal congruences of \emph{finite lattices}. Finally, we prove that every \emph{chain-representable} inclusion $Q\subseteq D$ can be represented by the principal congruences of a finite (and quite small) algebra.

DOI: 10.14232/actasm-017-538-7

AMS Subject Classification (1991): 06B10

Keyword(s): distributive lattice, principal lattice congruence, congruence lattice, chain-representability

Received June 1, 2017 and in final form February 8, 2018. (Registered under 38/2017.)