Ordered Algebraic Structures pp 19-56 | Cite as

# A Survey of Residuated Lattices

Chapter

## Abstract

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.

## Keywords

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.

## Preview

Unable to display preview. Download preview PDF.

## References

- [AF88]M. Anderson and T. Feil,
*Lattice-Ordered Groups: an introduction*. D. Reidel Publishing Company, 1988.MATHCrossRefGoogle Scholar - [BCGJT]P. Bahls, J. Cole, N. Galatos, P. Jipsen, C. Tsinakis,
*Cancellative residuated lattices*. preprint 2001.Google Scholar - [Be74]J. Berman,
*Homogeneous lattices and lattice-ordered groups*. Colloq. Math.**32**(1974), 13–24.MathSciNetMATHGoogle Scholar - [Bi67]G. Birkhoff,
*Lattice Theory*. (3rd ed), Colloquium Publications**25**Amer. Math. Soc., 1967.MATHGoogle Scholar - [BvA1]W. J. Blok and C. J. van Alten,
*The finite embeddability property for residuated lattices, pocrims and BCK-algebras*. Preprint.Google Scholar - [BvA2]W. J. Blok and C. J. van Alten,
*The finite embeddability property for partially ordered biresiduated integral groupoids*. Preprint.Google Scholar - [BT]K. Blount and C. Tsinakis,
*The structure of Residuated Lattices*. Preprint.Google Scholar - [BJ72]T. S. Blyth and M. F. Janowitz,
*Residuation Theory*. (1972) Pergamon Press.MATHGoogle Scholar - [BS81]S. Burris and H. P. Sankappanavar,
*A Course in Universal Algebra*. Springer Verlag (1981); online at**http://www.thoralf . uwaterloo . ca/**MATHCrossRefGoogle Scholar - [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 - [Ch58]C. C. Chang,
*Algebraic analysis of many valued logics*. Trans. AMS**88**(1958), 467–490.MATHCrossRefGoogle Scholar - [Ch59]C. C. Chang,
*A new proof of the completeness of the Lukasiewicz axioms*. Trans. AMS**93**(1959), 74–80.MATHGoogle Scholar - [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 - [Co02]J. Cole,
*Examples of Residuated Orders on Free Monoids*. In these Proceedings, 205–212.Google Scholar - [Di38]R. P. Dilworth,
*Abstract residuation over lattices*. Bull. AMS**44**(1938), 262–268.MathSciNetCrossRefGoogle Scholar - [Di39]R. P. Dilworth,
*Non-commutative residuated lattices*. Trans. AMS**46**(1939), 426–444.MathSciNetGoogle Scholar - [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 - [Fr80]R. Freese,
*Free modular lattices*. Trans. AMS**261**(1980) no. 1, 81–91.MathSciNetMATHCrossRefGoogle Scholar - [Fu63]L. Fuchs,
*Partially Ordered Algebraic Systems*. (1963) Pergamon Press.MATHGoogle Scholar - [Ga00]N. Galatos,
*Selected topics on residuated lattices*. Qualifying paper (2000), Vanderbilt University.Google Scholar - [Ga02]N. Galatos,
*The undecidability of the word problem for distributive residuated lattices*. In these Proceedings, 231–243.Google Scholar - [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 - [GH89]A. M. W. Glass and W. C. Holland (editors),
*Lattice-Ordered Groups*. Kluwer Academic Publishers, 1989, 278–307.MATHCrossRefGoogle Scholar - [GU84]H.P. Gumm and A. Ursini,
*Ideals in universal algebras*. Alg. Univ.**19**(1984) no. 1, 45–54.MathSciNetMATHCrossRefGoogle Scholar - [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 - [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 - [P. Hájek],
*Metamathematics of Fuzzy Logic*.**4**Trends in Logic; (1998) Kluwer Acad. Publ., Dordrecht.Google Scholar - [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 - [Hi66]N. G. Hisamiev,
*Universal theory of lattice-ordered Abelian groups*. (Russian) Algebra i Logika Sem.**5**(1966) no. 3, 71–76.MathSciNetGoogle Scholar - [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 - [KO]T. Kowalski and H. Ono,
*Residuated Lattices*. Preprint.Google Scholar - [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 - [Kr24]W. Krull,
*Axiomatische Begründung der algemeinen Idealtheorie*, Sitzungsberichte der physikalischmedizinischen Societät zu Erlangen**56**(1924), 47–63.Google Scholar - [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 - [MT44]J. C. C. McKinsey and A. Tarski,
*The algebra of topology*. Ann. of Math. (2)**45**(1944), 141–191.MathSciNetMATHCrossRefGoogle Scholar - [Mu86]D. Mundici,
*Interpretation of AF C* -algebras in Lukasiewicz sentential calculus*. J. Funct. Anal.**65**(1986) no. 1, 15–63.MathSciNetMATHCrossRefGoogle Scholar - [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 - [OK85]H. Ono and M. Komori,
*Logics without the contraction rule*. Journal of Symbolic Logic**50**(1985) 169–201.MathSciNetMATHCrossRefGoogle Scholar - [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 - [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 - [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 - [vA1]C. J. van Alten,
*Representable biresiduated lattices*. Jour. of Alg.; to appear.Google Scholar - [vA2]C. J. van Alten,
*The termwise equivalence of the varieties of group cones and generalized cancellative hoops*. Preprint.Google Scholar - [Wa37]M. Ward,
*Residuation in structures over which a multiplication is defined*. Duke Math. Jour.**3**(1937), 627–636.CrossRefGoogle Scholar - [Wa38]M. Ward,
*Structure Residuation*. Annals of Math., 2nd Ser.**39**(3) (1938), 558–568.CrossRefGoogle Scholar - [Wa40]M. Ward,
*Residuated distributive lattices*. Duke Math. Jour.**6**(1940), 641–651.CrossRefGoogle Scholar - [WD38]M. Ward and R. P. Dilworth,
*Residuated lattices*. Proc. Nat. Acad. of Sci.**24**(1938), 162–164.CrossRefGoogle Scholar - [WD39]M. Ward and R. P. Dilworth,
*Residuated lattices*. Trans. AMS**45**(1939), 335–354.MathSciNetCrossRefGoogle Scholar - [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