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.
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References
Adaricheva, K., Bolat, M.: Representation of convex geometries by circles on the plane. Discrete Math 342, 726–746 (2019)
Adaricheva, K., Czédli, G.: Note on the description of join-distributive lattices by permutations. Algebra Universalis 72, 155–162 (2014)
Czédli, G.: Finite convex geometries of circles. Discret. Math. 330, 61–75 (2014)
Czédli, G.: Slim patch lattices as absolute retracts and maximal lattices. arXiv:2105.12868 (2021)
Czédli, G., Dékány, T., Gyenizse, G., Kulin, J.: The number of slim rectangular lattices. Algebra Universalis 75, 33–50 (2016)
Czédli, G., Grätzer, G. Grätzer, G., Wehrung, F (eds.): Planar Semimodular Lattices and Their Diagrams. Chapter. Basel, Birkhäuser (2014)
Czédli, G., Grätzer, G.: A new property of congruence lattices of slim, planar, semimodular lattices. Categories Gen. Algebraic Struct. with Appl. 16(1), 1–28 (2022). available online at https://cgasa.sbu.ac.ir/article_101508.html
Czédli, G., Grätzer, G., Lakser, H.: Congruence structure of planar semimodular lattices: the general swing lemma. Algebra Universalis 79:40 18 pp (2018)
Czédli, G., Kurusa, Á.: A convex combinatorial property of compact sets in the plane and its roots in lattice theory. Categ. Gen. Algebr. Struct. Appl., (2022) 11, 57–92 (2019). http://cgasa.sbu.ac.ir/article_82639.htmlhttp://cgasa.sbu.ac.ir/article_82639.html
Czédli, G., Makay, G.: Swing lattice game and a direct proof of the swing lemma for planar semimodular lattices. Acta. Sci Math. (Szeged) 83, 13–29 (2017)
Czédli, G., Schmidt, E.T.: The Jordan-Hölder theorem with uniqueness for groups and semimodular lattices. Alg. Universalis 66, 69–79 (2011)
Czédli, G., Schmidt, E.T.: Slim semimodular lattices. I. A visual approach. Order 29, 481–497 (2012)
Dilworth, R.P.: A decomposition theorem for partially ordered sets. Ann. Math. 51(2), 161–166 (1950)
Dushnik, B, Miller, E.W.: Partially ordered sets. Amer. J. Math. 63, 600–610 (1941)
Grätzer, G.: Lattice Theory: Foundation. Basel, Birkhäuser (2011)
Grätzer, G.: On a result of Gábor Czédli concerning congruence lattices of planar semimodular lattices. Acta. Sci. Math. (Szeged) 81, 25–32 (2015)
Grätzer, G.: Congruences in slim, planar, semimodular lattices: The Swing Lemma. Acta. Sci. Math. (Szeged) 81, 381–397 (2015)
Grätzer, G.: Congruences of fork extensions of slim, planar, semimodular lattices. Algebra Universalis 76, 139–154 (2016)
Grätzer, G.: Notes on planar semimodular lattices. VIII. Congruence lattices of SPS lattices. Algebra Universalis 81(15), 3 (2020)
Grätzer, G., Knapp, E.: Notes on planar semimodular lattices. I. Construction. Acta. Sci. Math.(Szeged) 73, 445–462 (2007)
Grätzer, G., Knapp, E.: Notes on planar semimodular lattices. III. Congruences of rectangular lattices. Acta. Sci. Math. (Szeged) 75, 29–48 (2009)
Grätzer, G., Nation, J.B.: A new look at the Jordan-Hölder theorem for semimodular lattices. Algebra Universalis 64, 309–311 (2010)
Kelly, D., Rival, I.: Planar lattices. Canadian J. Math. 27, 636–665 (1975)
Milner, E.C., Pouzet, M.: A note on the dimension of a poset. Order 7, 101–102 (1990)
Molkhasi, A.: On strongly algebraically closed lattices. Zh. Sib. Fed. Univ. Mat. Fiz. 9, 202–208 (2016)
Molkhasi, A.: Strongly algebraically closed lattices in ℓ-groups and semilattices. Zh. Sib. Fed. Univ. Mat. Fiz. 11, 258–263 (2018)
Molkhasi, A.: On strongly algebraically closed orthomodular lattices. Southeast Asian Bull. Math. 42, 83–88 (2018)
Molkhasi, A.: Refinable and strongly algebraically closed lattices. Southeast Asian Bull. Math. 44, 673–680 (2020)
Rabinovitch, I., Rival, I.: The rank of a distributive lattice. Discrete Math. 25, 275–279 (1979)
Ranitović, M.G., Tepavčević, A.: On Planarity of Join-between Lattices. J. Mult.-Valued Log. Soft Comput. 38, 183–196 (2022)
Ranitović, M.G., Tepavčević, A.: Representation of slim lattice by poset. Filomat 35(3), 919–925 (2021). https://www.researchgate.net/publication/350546656
Reinhold, B.: Absolute retracts in group theory. Bull. Amer. Math. Soc. 52, 501–506 (1946)
Schmid, J.: Algebraically and existentially closed distributive lattices. Z. Math. Logik Grundlagen Math. 25, 525–530 (1979)
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This research of the first author was supported by the National Research, Development and Innovation Fund of Hungary under funding scheme K134851.
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Czédli, G., Molkhasi, A. Absolute Retracts for Finite Distributive Lattices and Slim Semimodular Lattices. Order 40, 127–148 (2023). https://doi.org/10.1007/s11083-021-09592-1
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DOI: https://doi.org/10.1007/s11083-021-09592-1
Keywords
- Absolute retract
- Slim semimodular lattice
- Algebraically closed lattice
- Strongly algebraically closed lattice
- Distributive lattice