A Survey of Residuated Lattices

  • P. Jipsen
  • C. Tsinakis
Part of the Developments in Mathematics book series (DEVM, volume 7)


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.


Word Problem Residuated Lattice Modular Lattice Equational Basis Gentzen System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  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

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