Order

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The Homomorphism Lattice Induced by a Finite Algebra

  • Brian A. Davey
  • Charles T. Gray
  • Jane G. Pitkethly
Article

Abstract

Each finite algebra A induces a lattice LA via the quasi-order → on the finite members of the variety generated by A, where BC if there exists a homomorphism from B to C. In this paper, we introduce the question: ‘Which lattices arise as the homomorphism lattice LA induced by a finite algebra A?’ Our main result is that each finite distributive lattice arises as LQ, for some quasi-primal algebra Q. We also obtain representations of some other classes of lattices as homomorphism lattices, including all finite partition lattices, all finite subspace lattices and all lattices of the form L1, where L is an interval in the subgroup lattice of a finite group.

Keywords

Homomorphism order Finitely generated variety Quasi-primal algebra Distributive lattice Covering forest 

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References

  1. 1.
    Adams, M.E., Koubek, V., Sichler, J.: Homomorphisms and endomorphisms of distributive lattices. Houston J. Math. 11, 129–145 (1985)MathSciNetMATHGoogle Scholar
  2. 2.
    Adams, M.E., Sichler, J.: Cover set lattices. Canad. J. Math. 32, 1177–1205 (1980)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Behncke, H., Leptin, H.: Classification of C -algebras with a finite dual. J. Functional Analysis 16, 241–257 (1974)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Birkhoff, G., Frink, O.: Representations of lattices by sets. Trans. Amer. Math. Soc. 64, 299–316 (1948)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press, New York (2002)CrossRefMATHGoogle Scholar
  6. 6.
    Davey, B.A., Werner, H.: Distributivity of coproducts over products. Algebra Universalis 12, 387–394 (1981)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Davey, B.A., Werner, H.: Dualities and Equivalences for Varieties of Algebras. In: Contributions to Lattice Theory (Szeged, 1980). Colloq. Math. Soc. János Bolyai, vol. 33, pp. 101–275. North-Holland, Amsterdam (1983)Google Scholar
  8. 8.
    Dean, R.A.: Component subsets of the free lattice on n generators. Proc. Amer. Math. Soc. 7, 220–226 (1956)MathSciNetMATHGoogle Scholar
  9. 9.
    Dörfler, W., Harary, F., Malle, G.: Covers of digraphs. Math. Slovaca 30, 269–280 (1980)MathSciNetMATHGoogle Scholar
  10. 10.
    Foniok, J., Nešetřil, J., Tardif, C.: Generalised dualities and maximal finite antichains in the homomorphism order of relational structures. European J. Combin. 29, 881–899 (2008)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gierz, G., Romanowska, A.: Duality for distributive bisemilattices. J. Austral. Math. Soc. Ser. A 51, 247–275 (1991)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Goralčík, P., Koubek, V., Sichler, J.: Universal varieties of (0,1)-lattices. Canad. J. Math. 42, 470–490 (1990)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Grätzer, G.: Universal Algebra. Van Nostrand, New York (1968)MATHGoogle Scholar
  14. 14.
    Grätzer, G., Lakser, H., Płonka, J.: Joins and direct products of equational classes. Canad. Math. Bull. 12, 741–744 (1969)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Hedrlín, Z.: On universal partly ordered sets and classes. J. Algebra 11, 503–509 (1969)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Hedrlín, Z., Pultr, A.: On full embeddings of categories of algebras. Illinois J. Math. 10, 392–406 (1966)MathSciNetMATHGoogle Scholar
  17. 17.
    Hell, P., Nešetřil, J.: Graphs and Homomorphisms. Oxford University Press, Oxford (2004)CrossRefMATHGoogle Scholar
  18. 18.
    Hoffman, M.E.: An analogue of covering space theory for ranked posets. Electron. J. Combin. 8(1), R32 (2001)MathSciNetMATHGoogle Scholar
  19. 19.
    Hubička, J., Nešetřil, J.: Universal partial order represented by means of oriented trees and other simple graphs. European J. Combin. 26, 765–778 (2005)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Kalman, J.A.: Subdirect decomposition of distributive quasilattices. Fund. Math. 71, 161–163 (1971)MathSciNetMATHGoogle Scholar
  21. 21.
    Keimel, K., Werner, H.: Stone duality for varieties generated by quasi-primal algebras. In: Recent Advances in the Representation Theory of Rings and C -algebras by Continuous Sections (Sem., Tulane Univ., New Orleans, La., 1973). Mem. Amer. Math. Soc., no. 148, pp. 59–85. American Mathematical Society, Providence (1974)Google Scholar
  22. 22.
    Kwuida, L., Lehtonen, E.: On the homomorphism order of labeled posets. Order 28, 251–265 (2011)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Nešetřil, J., Pultr, A., Tardif, C.: Gaps and dualities in Heyting categories. Comment. Math. Univ. Carolin. 48, 9–23 (2007)MathSciNetMATHGoogle Scholar
  24. 24.
    Nešetřil, J., Tardif, C.: Duality theorems for finite structures (characterising gaps and good characterisations). J. Combin. Theory Ser. B 80, 80–97 (2000)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Pálfy, P.P., Pudlák, P.: Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups. Algebra Universalis 80, 22–27 (1980)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Pixley, A.F.: Functionally complete algebras generating distributive and permutable classes. Math. Z. 114, 361–372 (1970)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Pixley, A.F.: The ternary discriminator function in universal algebra. Math. Ann. 191, 167–180 (1971)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Płonka, J.: On distributive quasi-lattices. Fund. Math. 60, 191–200 (1967)MathSciNetMATHGoogle Scholar
  29. 29.
    Pudlák, P., Tůma, J.: Every finite lattice can be embedded in a finite partition lattice. Algebra Universalis 10, 74–95 (1980)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Pultr, A.: Concerning universal categories. Comment. Math. Univ. Carolinae 5, 227–239 (1964)MathSciNetMATHGoogle Scholar
  31. 31.
    Pultr, A., Trnková, V.: Combinatorial, Algebraic and Topological Representations of Groups, Semigroups and Categories. North-Holland, Amsterdam (1980)MATHGoogle Scholar
  32. 32.
    Stallings, J.R.: Topology of finite graphs. Invent. Math. 71, 551–565 (1983)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Welzl, E.: Color-families are dense. Theoret. Comput. Sci. 17, 29–41 (1982)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Werner, H.: Discriminator-Algebras. Studien zur Algebra und ihre Anwendungen, vol. 6. Akademie-Verlag, Berlin (1978)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsLa Trobe UniversityVictoriaAustralia

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