Ordered Algebraic Structures pp 19-56

Part of the Developments in Mathematics book series (DEVM, volume 7)

A Survey of Residuated Lattices

  • P. Jipsen
  • C. Tsinakis


Residuation is a fundamental concept of ordered structures and categories. In this survey we consider the consequences of adding a residuated monoid operation to lattices. The resulting residuated lattices have been studied in several branches of mathematics, including the areas of lattice-ordered groups, ideal lattices of rings, linear logic and multi-valued logic. Our exposition aims to cover basic results and current developments, concentrating on the algebraic structure, the lattice of varieties, and decidability.

We end with a list of open problems that we hope will stimulate further research.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AF88]
    M. Anderson and T. Feil, Lattice-Ordered Groups: an introduction. D. Reidel Publishing Company, 1988.MATHCrossRefGoogle Scholar
  2. [BCGJT]
    P. Bahls, J. Cole, N. Galatos, P. Jipsen, C. Tsinakis, Cancellative residuated lattices. preprint 2001.Google Scholar
  3. [Be74]
    J. Berman, Homogeneous lattices and lattice-ordered groups. Colloq. Math. 32 (1974), 13–24.MathSciNetMATHGoogle Scholar
  4. [Bi67]
    G. Birkhoff, Lattice Theory. (3rd ed), Colloquium Publications 25 Amer. Math. Soc., 1967.MATHGoogle Scholar
  5. [BvA1]
    W. J. Blok and C. J. van Alten, The finite embeddability property for residuated lattices, pocrims and BCK-algebras. Preprint.Google Scholar
  6. [BvA2]
    W. J. Blok and C. J. van Alten, The finite embeddability property for partially ordered biresiduated integral groupoids. Preprint.Google Scholar
  7. [BT]
    K. Blount and C. Tsinakis, The structure of Residuated Lattices. Preprint.Google Scholar
  8. [BJ72]
    T. S. Blyth and M. F. Janowitz, Residuation Theory. (1972) Pergamon Press.MATHGoogle Scholar
  9. [BS81]
    S. Burris and H. P. Sankappanavar, A Course in Universal Algebra. Springer Verlag (1981); online at http://www.thoralf . uwaterloo . ca/ MATHCrossRefGoogle Scholar
  10. [Bu85]
    S. Burris, A simple proof of the hereditary undecidability of the theory of lattice-ordered abelian groups. Alg. Univ. 20 (1985), no. 3, 400–401.MathSciNetMATHCrossRefGoogle Scholar
  11. [Ch58]
    C. C. Chang, Algebraic analysis of many valued logics. Trans. AMS 88 (1958), 467–490.MATHCrossRefGoogle Scholar
  12. [Ch59]
    C. C. Chang, A new proof of the completeness of the Lukasiewicz axioms. Trans. AMS 93 (1959), 74–80.MATHGoogle Scholar
  13. [COMOO]
    R. Cignoli, I. D’Ottaviano and D. Mundici, Algebraic foundations of manyvalued reasoning. Trends in Logic—Studia Logica Library 7 (2000); Kluwer Acad. Publ., Dordrecht.Google Scholar
  14. [Co02]
    J. Cole, Examples of Residuated Orders on Free Monoids. In these Proceedings, 205–212.Google Scholar
  15. [Di38]
    R. P. Dilworth, Abstract residuation over lattices. Bull. AMS 44 (1938), 262–268.MathSciNetCrossRefGoogle Scholar
  16. [Di39]
    R. P. Dilworth, Non-commutative residuated lattices. Trans. AMS 46 (1939), 426–444.MathSciNetGoogle Scholar
  17. [Fe92]
    I. M. A. Ferreirim, On varieties and quasivarieties of hoops and their reducts. Ph. D. thesis (1992); University of Illinois at Chicago.Google Scholar
  18. [Fr80]
    R. Freese, Free modular lattices. Trans. AMS 261 (1980) no. 1, 81–91.MathSciNetMATHCrossRefGoogle Scholar
  19. [Fu63]
    L. Fuchs, Partially Ordered Algebraic Systems. (1963) Pergamon Press.MATHGoogle Scholar
  20. [Ga00]
    N. Galatos, Selected topics on residuated lattices. Qualifying paper (2000), Vanderbilt University.Google Scholar
  21. [Ga02]
    N. Galatos, The undecidability of the word problem for distributive residuated lattices. In these Proceedings, 231–243.Google Scholar
  22. [GG83]
    A. M. W. Glass and Y. Gurevich, The word problem for lattice-ordered groups. Trans. AMS 280 (1983) no. 1, 127–138.Google Scholar
  23. [GH89]
    A. M. W. Glass and W. C. Holland (editors), Lattice-Ordered Groups. Kluwer Academic Publishers, 1989, 278–307.MATHCrossRefGoogle Scholar
  24. [GU84]
    H.P. Gumm and A. Ursini, Ideals in universal algebras. Alg. Univ. 19 (1984) no. 1, 45–54.MathSciNetMATHCrossRefGoogle Scholar
  25. [Gu67]
    Y. Gurevich, Hereditary undecidability of a class of lattice-ordered Abelian groups. (Russian) Algebra i Logika Sem. 6 (1967) no. 1, 45–62.Google Scholar
  26. [GL84]
    Y. Gurevich and H. R. Lewis, The word problem for cancellation semigroups with zero. Journal of Symbolic Logic 49 (1984), 184–191.MathSciNetMATHCrossRefGoogle Scholar
  27. [P. Hájek]
    , Metamathematics of Fuzzy Logic. 4 Trends in Logic; (1998) Kluwer Acad. Publ., Dordrecht.Google Scholar
  28. [HRT]
    J. Hart, L. Rafter and C. Tsinakis, The structure of Commutative Residuated lattices. Intern’1 Jour. of Alg. and Comput.; to appear.Google Scholar
  29. [Hi66]
    N. G. Hisamiev, Universal theory of lattice-ordered Abelian groups. (Russian) Algebra i Logika Sem. 5 (1966) no. 3, 71–76.MathSciNetGoogle Scholar
  30. [HM79]
    W. C. Holland and S. H. McCleary, Solvability of the word problem in free lattice-ordered groups. Houston J. Math. 5 (1979) no. 1, 99–105.MathSciNetMATHGoogle Scholar
  31. [KO]
    T. Kowalski and H. Ono, Residuated Lattices. Preprint.Google Scholar
  32. [K000]
    T. Kowalski and H. Ono, Splittings in the variety of residuated lattices. Alg. Univ. bf 44 (2000) no. 3–4, 283–298.Google Scholar
  33. [Kr24]
    W. Krull, Axiomatische Begründung der algemeinen Idealtheorie, Sitzungsberichte der physikalischmedizinischen Societät zu Erlangen 56 (1924), 47–63.Google Scholar
  34. [McK96]
    R. McKenzie, An algebraic version of categorical equivalence for varieties and more general algebraic categories. In Logic and Algebra, A. Ursini, P. Aglianò, Eds.; (1996) Marcel Dekker, 211–244.Google Scholar
  35. [MT44]
    J. C. C. McKinsey and A. Tarski, The algebra of topology. Ann. of Math. (2) 45 (1944), 141–191.MathSciNetMATHCrossRefGoogle Scholar
  36. [Mu86]
    D. Mundici, Interpretation of AF C* -algebras in Lukasiewicz sentential calculus. J. Funct. Anal. 65 (1986) no. 1, 15–63.MathSciNetMATHCrossRefGoogle Scholar
  37. [NPM99]
    V. Novâk, I. Perfilieva, and J. Mockor, Mathematical Principles of Fuzzy Logic. 517 Kluwer Intern’1 Ser. in Engin. and Comp. Sci. (1999), Kluwer Acad. Publ ., Dordrecht.Google Scholar
  38. [OK85]
    H. Ono and M. Komori, Logics without the contraction rule. Journal of Symbolic Logic 50 (1985) 169–201.MathSciNetMATHCrossRefGoogle Scholar
  39. [OT99]
    M. Okada and K. Terui, The finite model property for various fragments of intuitionistic linear logic. Journal of Symbolic Logic, 64 (2) (1999), 790–802.MathSciNetMATHCrossRefGoogle Scholar
  40. [PT89]
    W. B. Powell and C. Tsinakis, Free products in varieties of lattice-ordered groups. In “Lattice-Ordered Groups” ; A. M. W. Glass and W. C. Holland, Eds.; (1989) Kluwer Acad. Publ., 278–307.CrossRefGoogle Scholar
  41. [Ur72]
    A. Ursini, Sulle variet’ di algebre con una buona teoria degli ideali. (Italian) Boll. Un. Mat. Ital. (4) 6 (1972), 90–95.MathSciNetMATHGoogle Scholar
  42. [vA1]
    C. J. van Alten, Representable biresiduated lattices. Jour. of Alg.; to appear.Google Scholar
  43. [vA2]
    C. J. van Alten, The termwise equivalence of the varieties of group cones and generalized cancellative hoops. Preprint.Google Scholar
  44. [Wa37]
    M. Ward, Residuation in structures over which a multiplication is defined. Duke Math. Jour. 3 (1937), 627–636.CrossRefGoogle Scholar
  45. [Wa38]
    M. Ward, Structure Residuation. Annals of Math., 2nd Ser. 39 (3) (1938), 558–568.CrossRefGoogle Scholar
  46. [Wa40]
    M. Ward, Residuated distributive lattices. Duke Math. Jour. 6 (1940), 641–651.CrossRefGoogle Scholar
  47. [WD38]
    M. Ward and R. P. Dilworth, Residuated lattices. Proc. Nat. Acad. of Sci. 24 (1938), 162–164.CrossRefGoogle Scholar
  48. [WD39]
    M. Ward and R. P. Dilworth, Residuated lattices. Trans. AMS 45 (1939), 335–354.MathSciNetCrossRefGoogle Scholar
  49. [We86]
    V. Weispfenning, The complexity of the word problem for abelian L-groups, Theor. Comp. Sci. 48 (1986) no. 1, 127–132.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2002

Authors and Affiliations

  • P. Jipsen
    • 1
  • C. Tsinakis
    • 1
  1. 1.Department of MathematicsVanderbilt UniversityNashvilleUSA

Personalised recommendations