Slim patch lattices as absolute retracts and maximal lattices

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 | >...

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Bibliographic Details
Published in:Algebra universalis 2024-08, Vol.85 (3), Article 28
Main Author: Czédli, Gábor
Format: Article
Language:English
Subjects:
Algebra
Homomorphisms
Lattices
Mathematics
Mathematics and Statistics
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Summary: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.
ISSN:0002-5240
1420-8911
DOI:10.1007/s00012-024-00861-9