Homomorphisms of distributive lattices as restrictions of congruences. III. Rectangular lattices and two convex sublattices

TL;DR

This paper extends the understanding of how congruences of finite lattices relate to those of their convex sublattices, providing a stronger, simpler proof for rectangular lattices.

Contribution

It offers a stronger version of Czédli's result on rectangular lattices and presents a concise, elementary proof of the relationship between lattice congruences and sublattices.

Findings

Proved a stronger form of Czédli's theorem for rectangular lattices.

Provided a short, elementary proof of the main result.

Clarified the connection between lattice congruences and convex sublattices.

Abstract

Let $L$ be a finite lattice and let $I$ be an ideal of $L$. Then the restriction map is a bounded lattice homomorphism of the congruence lattice of~$L$ into the congruence lattice of $I$. In a 2009 paper, the authors proved the converse. In a 2012 paper, G. Cz\'edli proved an analogous result for rectangular lattices. In this paper, we prove a stronger form of Cz\'edli's result and provide a short, elementary, and direct proof.