Absolute retracts for finite distributive lattices and slim semimodular lattices

TL;DR

This paper characterizes the absolute retracts for finite distributive, slim semimodular, and bounded dimensional distributive lattices, revealing that they are finite boolean lattices and certain products of chains, extending previous results.

Contribution

It identifies the absolute retracts for these classes of finite lattices, generalizing earlier findings and clarifying their algebraic closure properties.

Findings

Absolute retracts for slim semimodular lattices are singleton lattices.

For finite distributive lattices, absolute retracts are exactly finite boolean lattices.

In bounded dimensional cases, absolute retracts are finite boolean lattices and products of chains.

Abstract

We describe the absolute retracts for the following classes of finite lattices: (1) slim semimodular lattices, (2) finite distributive lattices, and for each positive integer $n$, (3) at most $n$-dimensional finite distributive lattices. Although the singleton lattice is the only absolute retract for the first class, this result has paved the way to some other classes. For the second class, we prove that the absolute retracts are exactly the finite boolean lattices; this generalizes a 1979 result of J. Schmid. For the third class, the absolute retracts are the finite boolean lattices of dimension at most $n$ and the direct products of $n$ nontrivial finite chains. Also, we point out that in each of these classes, the algebraically closed lattices and the strongly algebraically closed lattices are the same as the absolute retracts. Slim (and necessarily planar) semimodular lattices were introduced by G. Gr\"atzer and E. Knapp in 2007, and they have been intensively studied since then. Algebraically closed and strongly algebraically closed lattices have been investigated by J. Schmid and, in several papers, by A. Molkhasi.