Slim patch lattices as absolute retracts and maximal lattices

Abstract

We prove that slim patch lattices are exactly the absolute retracts with more than two elements for the category of slim semimodular lattices with length-preserving lattice embeddings as morphisms. Also, slim patch lattices are the same as the maximal objects L in this category such that \(|L|>2.\) Furthermore, slim patch lattices are characterized as the algebraically closed lattices L in this category such that \(|L|>2.\) Finally, we prove that if we consider \(\{0,1\}\)-preserving lattice homomorphisms rather than length-preserving ones, then the absolute retracts for the class of slim semimodular lattices are the at most 4-element boolean lattices.