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Algebra universalis

, Volume 68, Issue 1–2, pp 17–37 | Cite as

Isotone maps on lattices

  • G. M. Bergman
  • G. Grätzer
Article

Abstract

Let \({\mathcal{L} = (Li | i \in I)}\) be a family of lattices in a nontrivial lattice variety V, and let \({\varphi_{i} : L_{i} \rightarrow M}\), for \({i \in I}\), be isotone maps (not assumed to be lattice homomorphisms) to a common lattice M (not assumed to lie in V). We show that the maps \({\varphi_{i}}\) can be extended to an isotone map \({\varphi : L \rightarrow M}\), where \({L = {\rm Free}_{V} \mathcal{L}}\) is the free product of the L i in V. This was known for V = L, the variety of all lattices. The above free product L can be viewed as the free lattice in V on the partial lattice P formed by the disjoint union of the L i . The analog of the above result does not, however, hold for the free lattice L on an arbitrary partial lattice P. We show that the only codomain lattices M for which that more general statement holds are the complete lattices. On the other hand, we prove the analog of our main result for a class of partial lattices P that are not-quite-disjoint unions of lattices. We also obtain some results similar to our main one, but with the relationship lattices : orders replaced either by semilattices : orders or by lattices : semilattices. Some open questions are noted.

2010 Mathematics Subject Classification

Primary: 06B25 Secondary: 06B20 06B23 

Keywords and phrases

free product of lattices varieties prevarieties and quasivarieties of lattices isotone map free lattice on a partial lattice semilattice 

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References

  1. 1.
    Banaschewski B., Bruns G.: Categorical characterization of the MacNeille completion. Arch. Math. (Basel) 18, 369–377 (1967)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bruns G., Lakser H.: Injective hulls of semilattices. Canad. Math. Bull. 13, 115–118 (1970)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Day A.: Injectives in non-distributive equational classes of lattices are trivial. Arch. Math. (Basel) 21, 113–115 (1970)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Freese, R.: Ordinal sums of projectives in varieties of lattices. Preprint, 5 pp. (2005) http://www.math.hawaii.edu/~ralph/proj.pdf
  5. 5.
    Freese, R., Ježek, J., Nation, J.B.: Free lattices. Mathematical Surveys and Monographs, vol. 42. American Mathematical Society, Providence (1995)Google Scholar
  6. 6.
    Freese R., Nation J.B.: Projective lattices. Pacific J. Math. 75, 93–106 (1978)MathSciNetMATHGoogle Scholar
  7. 7.
    Grätzer G.: A property of transferable lattices. Proc. Amer. Math. Soc. 43, 269–271 (1974)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Grätzer G.: General Lattice Theory, 2nd edn. Birkhäuser, Basel (1998, 2007)Google Scholar
  9. 9.
    Grätzer G.: Lattice Theory: Foundation. Birkhäuser, Basel (2011)MATHCrossRefGoogle Scholar
  10. 10.
    Grätzer G., Kelly D.: Products of lattice varieties. Algebra Universalis 21, 33–45 (1985)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Grätzer G., Lakser H., Platt C.R.: Free products of lattices. Fund. Math. 69, 233–240 (1970)MathSciNetMATHGoogle Scholar
  12. 12.
    Horn A., Kimura N.: The category of semilattices. Algebra Universalis 1, 26–38 (1971)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Mal’cev, A.I.: Multiplication of classes of algebraic systems. Sibirsk. Mat. Ž. 8, 346–365 (1967) (Russian)Google Scholar
  14. 14.
    Nelson E.: An elementary proof that there are no nontrivial injective lattices. Algebra Universalis 10, 264–265 (1980)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Sorkin, Yu.I.: Free unions of lattices. Mat. Sbornik, N.S. 30 (72), 677–694 (1952) (Russian)Google Scholar
  16. 16.
    Zhao D., Zhao B.: The categories of m-semilattices. Northeast Math. J. 14(4), 419–430 (1998)MathSciNetGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.University of CaliforniaBerkeleyUSA
  2. 2.Department of MathematicsUniversity of ManitobaWinnipegCanada

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