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

Ordered by set inclusion, the retracts of a lattice L together with the empty set form a bounded poset R e t (L) . By a grid we mean the direct product of two non-singleton finite chains. We prove that if G is a grid, then R e t (G) is a lattice. We determine the number of elements of R e t (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 R e t (M) is not a lattice. [ABSTRACT FROM AUTHOR]