2021/05/21 by Gábor Czédli, Czédli, Gábor, Али Молхаси +1
Computer Science · #06C10 #Advanced Algebra and Logic #FOS: Mathematics #Rings and Algebras (math.RA) #Rough Sets and Fuzzy Logic
paper · pdf · doi:10.48550/arxiv.2105.10604
openalex publication_date 2021/05/21 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
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. Grätzer 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.