The lattice of congruence lattices of algebras on a finite set
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Abstract
The congruence lattices of all algebras defined on a fixed finite set A ordered by inclusion form a finite atomistic lattice \(\mathcal {E}\). We describe the atoms and coatoms. Each meet-irreducible element of \(\mathcal {E}\) being determined by a single unary mapping on A, we characterize completely those which are determined by a permutation or by an acyclic mapping on the set A. Using these characterisations we deduce several properties of the lattice \(\mathcal {E}\); in particular, we prove that \(\mathcal {E}\) is tolerance-simple whenever \(|A|\ge 4\).
Keywords
Congruence lattice Unary operation Monounary algebra Join-irreducible element Meet-irreducible element Tolerance simpleMathematics Subject Classification
Primary 08A30 Secondary 06B15 08A60 06A15 08A35 08A99 20M20References
- 1.Grätzer, G., Schmidt, E.T.: Characterizations of congruence lattices of abstract algebras. Acta Sci. Math. (Szeged) 24, 34–59 (1963)MathSciNetMATHGoogle Scholar
- 2.Jakubíková-Studenovská, D.: Lattice of quasiorders of monounary algebras. Miskolc Math. Notes 10, 41–48 (2009)MathSciNetMATHGoogle Scholar
- 3.Jakubíková-Studenovská, D., Pócs, J.: Monounary Algebras. P.J. Šafárik Univ, Košice (2009)MATHGoogle Scholar
- 4.Jakubíková-Studenovská, D., Pöschel, R., Radeleczki, S.: The lattice of compatible quasiorders of acyclic monounary algebras. Order 28, 481–497 (2011)MathSciNetCrossRefMATHGoogle Scholar
- 5.Jakubíková-Studenovská, D., Pöschel, R., Radeleczki, S.: Irreducible quasiorders of monounary algebras. J. Aust. Math. Soc. 93, 259–276 (2013)MathSciNetCrossRefMATHGoogle Scholar
- 6.Jakubíková-Studenovská, D., Pöschel, R., Radeleczki, S.: The lattice of quasiorder lattices of algebras on a finite set. Algebra Universalis 75, 197–220 (2016)MathSciNetCrossRefMATHGoogle Scholar
- 7.Janowitz, M.: Tolerances, interval orders, and semiorders. Czechoslov. Math. J. 44(119), 21–38 (1994)MathSciNetMATHGoogle Scholar
- 8.Janowitz, M.F., Radeleczki, S.: Aggregation on a finite lattice. Order 33, 371–388 (2016)MathSciNetCrossRefMATHGoogle Scholar
- 9.Kindermann, M.: Über die Äquivalenz von Ordnungspolynomvollständigkeit und Toleranzeinfachheit endlicher Verbände. In: Contributions to general algebra (Proc. Klagenfurt Conf., Klagenfurt, 1978), pp. 145–149. Heyn, Klagenfurt (1979)Google Scholar
- 10.Pöschel, R.: Galois connections for operations and relations. In: Denecke, K., Erné, M., Wismath, S. (eds.) Galois connections and applications. Mathematics and its Applications, vol. 565, pp. 231–258. Kluwer, Dordrecht (2004)Google Scholar
- 11.Pöschel, R., Kalužnin, L.: Funktionen- und Relationenalgebren. Deutscher Verlag der Wissenschaften, Berlin (1979). Birkhäuser Verlag Basel. Math. Reihe 67, (1979)Google Scholar
- 12.Pöschel, R., Radeleczki, S.: Endomorphisms of quasiorders and related lattices. In: Dorfer, G., Eigenthaler, G., Kautschitsch, H., More, W., Müller, W. (eds.) Contributions to General Algebra 18 (Proceedings of the Klagenfurt Conference 2007 (AAA73+CYA22), Febr. 2007), pp. 113–128. Verlag Heyn GmbH and Co KG (2008)Google Scholar
- 13.Radeleczki, S., Schweigert, D.: Notes on locally order-polynomially complete lattices. Algebra Universalis 53, 397–399 (2005)MathSciNetCrossRefMATHGoogle Scholar
- 14.Werner, H.: Which partition lattices are congruence lattices? In: Lattice theory (Proc. Colloq., Szeged, 1974), Colloq. Math. Soc. János Bolyai, vol. 14, pp. 433–453. North-Holland, Amsterdam (1976)Google Scholar
- 15.Zádori, L.: Generation of finite partition lattices. In: Lectures in universal algebra (Szeged, 1983), Colloq. Math. Soc. János Bolyai, vol. 43, pp. 573–586. North-Holland, Amsterdam (1986)Google Scholar
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