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

## TL;DR

This paper characterizes finite distributive lattices that can be fully represented by principal congruences of finite algebras, showing they are planar with specific structural properties and can be realized by small finite algebras.

## Contribution

It establishes necessary and sufficient conditions for finite distributive lattices to be fully (A1)-representable by principal congruences, linking lattice structure to algebraic representation.

## Key findings

- Fully (A1)-representable lattices are planar.
- Such lattices have at most one join-reducible coatom.
- Every chain-representable inclusion can be realized by principal congruences of a small finite algebra.

## Abstract

Let $Q$ be a subset of a finite distributive lattice $D$. An algebra $A$ 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 every subset $Q$ containing $0$, $1$, and all join-irreducible elements of $D$, then $D$ is said to be 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 fully chain-representable in the sense of a recent paper of G. Gr\"atzer. Combining the results of this paper with another paper by the present author, it follows that every fully (A1)-representable finite distributive lattice is "fully representable" even by principal congruences of finite lattices. Finally, we prove that every chain-representable inclusion $Q\subseteq D$ can be represented by the principal congruences of a finite (and quite small) algebra.