Absolute Retracts for Finite Distributive Lattices and Slim Semimodular Lattices

Abstract

Let n denote a positive integer. We describe the absolute retracts for the following five categories of finite lattices: (1) slim semimodular lattices, which were introduced by G. Grätzer and E. Knapp in (Acta. Sci. Math. (Szeged), 73 445–462 2007), and they have been intensively studied since then, (2) finite distributive lattices (3) at most n-dimensional finite distributive lattices, (4) at most n-dimensional finite distributive lattices with cover-preserving {0,1}-homomorphisms, and (5) finite distributive lattices with cover-preserving {0,1}-homomorphisms. Although the singleton lattice is the only absolute retract for the first category, this result has paved the way to some other classes. For the second category, we prove that the absolute retracts are exactly the finite boolean lattices; this generalizes a 1979 result of J. Schmid. For the third category and also for the fourth, the absolute retracts are the finite boolean lattices of dimension at most n and the direct products of n nontrivial finite chains. For the fifth category, the absolute retracts are the same as those for the second category. Also, we point out that in each of these classes, the algebraically closed lattices and the strongly algebraically closed lattices (investigated by J. Schmid and, in several papers, by A. Molkhasi) are the same as the absolute retracts.