Lattices of retracts of direct products of two finite chains and notes on retracts of lattices

Abstract

Ordered by set inclusion, the retracts of a lattice L together with the empty set form a bounded poset \(Ret (L)\). By a grid we mean the direct product of two non-singleton finite chains. We prove that if G is a grid, then \(Ret (G)\) is a lattice. We determine the number of elements of \(Ret (G)\). Some easy properties of retracts, retractions, and their kernels called retraction congruences of (mainly distributive) lattices are found. Also, we present several examples, including a 12-element modular lattice M such that \(Ret (M)\) is not a lattice.