Retracts of rectangular distributive lattices and some related observations

TL;DR

This paper investigates the structure of retracts within rectangular distributive lattices, establishing that their collection forms a lattice, and explores properties and examples including a modular lattice with non-lattice retracts.

Contribution

It proves that the set of retracts of a rectangular distributive lattice forms a lattice and provides descriptions and counts of these retracts, along with related properties and examples.

Findings

Ret($G$) forms a lattice for rectangular distributive lattices

The paper describes and counts the retracts of such lattices

Provides examples including a modular lattice with non-lattice retracts

Abstract

By a rectangular distributive lattice we mean the direct product of two non-singleton finite chains. We prove that the retracts (ordered by set inclusion and together with the empty set) of a rectangular distributive lattice $G$ form a lattice, which we denote by Ret($G$). Also, we describe and count the retracts of $G$. Some easy properties of retracts, retractions, and retraction kernels of (mainly distributive) lattices are observed and several examples are presented, including a 12-element modular lattice $M$ such that Ret($M$) is not a lattice.