Skip to main content
SpringerLink
Log in

On Birkhoff’s Common Abstraction Problem

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

In his milestone textbook Lattice Theory, Garrett Birkhoff challenged his readers to develop a “common abstraction” that includes Boolean algebras and latticeordered groups as special cases. In this paper, after reviewing the past attempts to solve the problem, we provide our own answer by selecting as common generalization of \({\mathcal{B} \mathcal{A}}\) and \({\mathcal{L} \mathcal{G}}\) their join \({\mathcal{B} \mathcal{A} \vee \mathcal{L} \mathcal{G}}\) in the lattice of subvarieties of \({\mathcal{F} \mathcal{L}}\) (the variety of FL-algebras); we argue that such a solution is optimal under several respects and we give an explicit equational basis for \({\mathcal{B} \mathcal{A} \vee \mathcal{L} \mathcal{G}}\) relative to \({\mathcal{F} \mathcal{L}}\). Finally, we prove a Holland-type representation theorem for a variety of FL-algebras containing \({\mathcal{B} \mathcal{A} \vee \mathcal{L} \mathcal{G}}\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Anderson M., C.C. Edwards: A representation theorem for distributive ℓ- monoids, Canad. Math. Bull. 27, 238–240 (1984)

    Article  Google Scholar 

  2. Birkhoff G.: Lattice Theory, Rev. Edition. AMS Publications, New York (1948)

    Google Scholar 

  3. Blount K., C. Tsinakis: The structure of residuated lattices. International Journal of Algebra and Computation 13(4), 437–461 (2002)

    Article  Google Scholar 

  4. Casari, E., Comparative logics and Abelian ℓ-groups, in R. Ferro et al. (eds.), Logic Colloquium ’88, North Holland, 161–190, 1989.

  5. Cignoli, R., I. M. L. D’Ottaviano, and D. Mundici, Algebraic Foundations of Many-Valued Reasoning, Kluwer, Dordrecht, 1999.

  6. Constantinescu C.: Some properties of spaces of measures, Atti Sem. Mat. Fis. Univ. Modena 35, 1–286 (1989)

    Google Scholar 

  7. Darnel M.R.: Theory of Lattice-Ordered Groups. Dekker, New York (1995)

    Google Scholar 

  8. Dvurečenskij A., Ventriglia F.: On two versions of the Loomis-Sikorski theorem for algebraic structures. Soft Computing 12, 1027–1034 (2008)

    Article  Google Scholar 

  9. Ellis D.: Notes on the foundation of lattice theory. Archive for Mathematics 4, 257–260 (1953)

    Article  Google Scholar 

  10. Galatos N., Jipsen P., Kowalski T., Ono H.: Residuated Lattices: An Algebraic Glimpse on Substructural Logics. Elsevier, Amsterdam (2007)

    Google Scholar 

  11. Galatos N., Tsinakis C.: Generalized MV algebras. Journal of Algebra 283, 254–291 (2005)

    Article  Google Scholar 

  12. Georgescu G., Iorgulescu A.: Pseudo-MV algebras. Multiple Valued Logic 6, 95–135 (2001)

    Google Scholar 

  13. Grätzer G.: Two Mal’cev-type theorems in universal algebra. Journal of Combinatorial Theory 8, 334–342 (1970)

    Article  Google Scholar 

  14. Grätzer G., Lakser H., Płonka J.: Joins and direct products of equational classes. Canadian Mathematical Bulletin 12, 741–744 (1969)

    Article  Google Scholar 

  15. Graziano M.G.: Uniformities of Fr´echet-Nikodym type in Vitali spaces. Semigroup Forum 61, 91–115 (2000)

    Article  Google Scholar 

  16. Gumm H.P., Ursini A.: Ideals in universal algebra. Algebra Universalis 19, 45–54 (1984)

    Article  Google Scholar 

  17. Holland W.C.: The lattice-ordered group of automorphisms of an ordered set, Michigan Math. J. 10, 399–408 (1963)

    Google Scholar 

  18. Jipsen, P., and C. Tsinakis, A survey of residuated lattices, in J. Martinez (ed.), Ordered Algebraic Structures, Kluwer, Dordrecht, 19–56, 2002.

  19. J´onsson B., Tsinakis C.: Products of classes of residuated structures. Studia Logica 77, 267–292 (2004)

    Article  Google Scholar 

  20. Kovář, T., A General Theory of Dually Residuated Lattice-Ordered Monoids, Ph.D. Thesis, Palacky University, Olomouc, 1996.

  21. Kowalski T., Paoli F.: Joins and subdirect products of varieties. Algebra Universalis 65(4), 371–391 (2011)

    Article  Google Scholar 

  22. Kühr J.: Ideals of noncommutative DRL monoids. Czechoslovak Mathematical Journal 55, 97–111 (2005)

    Article  Google Scholar 

  23. Ledda A., Konig M., Paoli F., Giuntini R.: MV algebras and quantum computation. Studia Logica 82(2), 245–270 (2006)

    Article  Google Scholar 

  24. Merlier T.: Sur les demi-groups r´eticul´es et les o-demi-groupes. Semigroup Forum 2, 64–70 (1971)

    Article  Google Scholar 

  25. Metcalfe, G., F. Paoli, and C. Tsinakis, Ordered algebras and logic, in H. Hosni and F. Montagna, Probability, Uncertainty, Rationality, Pisa, Edizioni della Normale, 2010, pp. 1–85.

  26. Nakano T.: Rings and partly ordered systems. Mathematische Zeitschrift 99, 355–376 (1967)

    Article  Google Scholar 

  27. Paoli F.: *-autonomous lattices and fuzzy sets. Soft Computing 10, 607–617 (2006)

    Article  Google Scholar 

  28. Rama Rao V.V.: On a common abstraction of Boolean rings and lattice-ordered groups I. Monatshefte für Mathematik 73, 411–421 (1969)

    Article  Google Scholar 

  29. Schmidt K.D.: A common abstraction of Boolean rings and lattice ordered groups. Compositio Mathematica 54, 51–62 (1985)

    Google Scholar 

  30. Schmidt, K. D., Minimal clans: a class of ordered partial semigroups including Boolean rings and partially ordered groups, in Semigroups: Theory and Applications, Springer, Berlin, 1988, pp. 300–341.

  31. Schmidt, K. D., Decomposition and extension of abstract measures on Riesz spaces, Rendiconti dell’Istituto Matematico dell’Università di Trieste, Suppl. Vol. 29, 1998, pp. 135–213.

  32. Swamy K.L.N.: Dually residuated lattice-ordered semigroups. Mathematische Annalen 159, 105–114 (1965)

    Article  Google Scholar 

  33. Swamy K.L.N.: Dually residuated lattice-ordered semigroups II. Mathematische Annalen 160, 64–71 (1965)

    Article  Google Scholar 

  34. Swamy K.L.N.: Dually residuated lattice-ordered semigroups III. Mathematische Annalen 167, 71–74 (1966)

    Article  Google Scholar 

  35. Wyler, O., Clans, Compositio Mathematica 17:172–189, 1965-66.

Download references

Author information

Authors and Affiliations

  1. Department of Pedagogy, Psychology, Philosophy, University of Cagliari, Via Is Mirrionis 1, 09123, Cagliari, Italy

    F. Paoli

  2. Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN, 37240, USA

    C. Tsinakis

Authors
  1. F. Paoli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Tsinakis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to F. Paoli.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Paoli, F., Tsinakis, C. On Birkhoff’s Common Abstraction Problem. Stud Logica 100, 1079–1105 (2012). https://doi.org/10.1007/s11225-012-9452-5

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11225-012-9452-5

Keywords

Search

Navigation