Retracts of rectangular distributive lattices and some related observations

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.