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Residuated Lattices

  • Lavinia Corina Ciungu
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

Residuation is a fundamental concept of ordered structures and the residuated lattices, obtained by adding a residuated monoid operation to lattices, have been applied in several branches of mathematics, including -groups, ideal lattices of rings and multiple-valued logics. Since the late 1930s, the commutative residuated lattices have been studied by Krull, Dilworth and Ward.

Non-commutative residuated lattices, sometimes called pseudo-residuated lattices, biresiduated lattices or generalized residuated lattices, are the algebraic counterparts of substructural logics, i.e. logics which lack at least one of the three structural rules, namely contraction, weakening and exchange. Particular cases of residuated lattices are the full Lambek algebras (FL-algebras), integral residuated lattices and bounded integral residuated lattices (FL w -algebras).

In this chapter we investigate the properties of a residuated lattice and the lattice of filters of a residuated lattice, we study the Boolean center of an FL w -algebra and we define and study the directly indecomposable FL w -algebras. We prove that any linearly ordered FL w -algebra is directly indecomposable and any subdirectly irreducible FL w -algebra is directly indecomposable. Finally, the FL w -algebras of fractions relative to a meet-closed system is introduced and investigated.

References

  1. 3.
    Bahls, P., Cole, J., Galatos, N., Jipsen, P., Tsinakis, C.: Cancellative residuated lattices. Algebra Univers. 50, 83–106 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 4.
    Balbes, R., Dwinger, P.: Distributive Lattices. University of Missouri Press, Columbia (1974) zbMATHGoogle Scholar
  3. 9.
    Birkhoff, G.: Lattice Theory. Am. Math. Soc., Providence (1967) zbMATHGoogle Scholar
  4. 11.
    Blount, K., Tsinakis, C.: The structure of residuated lattices. Int. J. Algebra Comput. 13, 437–461 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 19.
    Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra. Springer, New York (1981) zbMATHCrossRefGoogle Scholar
  6. 24.
    Buşneag, D., Piciu, D.: Localization of MV-algebras and ℓu-groups. Algebra Univers. 50, 359–380 (2003) zbMATHCrossRefGoogle Scholar
  7. 25.
    Buşneag, D., Piciu, D.: Boolean BL-algebra of fractions. An. Univ. Craiova, Ser. Mat. Inform. 31, 1–19 (2004) MathSciNetzbMATHGoogle Scholar
  8. 26.
    Buşneag, D., Piciu, D.: MV-algebra of fractions and maximal MV-algebra of quotients. J. Mult.-Valued Log. Soft Comput. 10, 363–383 (2004) zbMATHGoogle Scholar
  9. 27.
    Buşneag, D., Piciu, D.: Boolean MV-algebra of fractions. Math. Rep. (Bucur.) 7, 265–280 (2005) MathSciNetGoogle Scholar
  10. 28.
    Buşneag, D., Piciu, D.: Localization of pseudo-BL algebra. Rev. Roum. Math. Pures Appl. L, 495–513 (2005) Google Scholar
  11. 29.
    Buşneag, D., Piciu, D.: BL-algebra of fractions and maximal BL-algebra of quotients. Soft Comput. 9, 544–555 (2005) zbMATHCrossRefGoogle Scholar
  12. 30.
    Buşneag, D., Piciu, D.: On the lattice of filters of a pseudo-BL algebra. J. Mult.-Valued Log. Soft Comput. 12, 217–248 (2006) zbMATHGoogle Scholar
  13. 31.
    Buşneag, D., Piciu, D.: Residuated lattice of fractions relative to a meet-closed system. Bull. Math. Soc. Sci. Math. Roum. 49, 13–24 (2006) Google Scholar
  14. 32.
    Buşneag, D., Piciu, D.: Pseudo-BL algebra of fractions and maximal pseudo-BL algebra of quotients. Southeast Asian Bull. Math. 31, 639–665 (2007) MathSciNetzbMATHGoogle Scholar
  15. 33.
    Buşneag, D., Piciu, D.: Localization of pseudo-MV algebras and -groups with strong unit. Int. Rev. Fuzzy Math. 2, 63–95 (2007) MathSciNetzbMATHGoogle Scholar
  16. 35.
    Buşneag, D., Piciu, D., Jeflea, A.: Archimedean residuated lattices. An. ştiinţ. Univ. “Al.I. Cuza” Iaşi, Mat. LVI, 227–252 (2010) Google Scholar
  17. 93.
    Dilworth, R.P.: Non-commutative residuated lattices. Trans. Am. Math. Soc. 46, 426–444 (1939) MathSciNetGoogle Scholar
  18. 105.
    Dvurečenskij, A., Hyčko, M.: Algebras on subintervals of BL-algebras, pseudo-BL algebras and bounded residuated R-monoids. Math. Slovaca 56, 125–144 (2006) MathSciNetzbMATHGoogle Scholar
  19. 110.
    Dvurečenskij, A., Rachůnek, J.: Probabilistic averaging in bounded non-commutative R-monoids. Semigroup Forum 72, 190–206 (2006) MathSciNetGoogle Scholar
  20. 121.
    Flondor, P., Sularia, M.: On a class of residuated semilattice monoids. Fuzzy Sets Syst. 138, 149–176 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 124.
    Galatos, N.: Varieties of residuated lattices. Ph.D. thesis, Vanderbilt University, Nashville (2003) Google Scholar
  22. 125.
    Galatos, N.: Minimal varieties of residuated lattices. Algebra Univers. 52, 215–239 (2005) MathSciNetCrossRefGoogle Scholar
  23. 126.
    Galatos, N., Jipsen, P.: A survey of generalized basic logic algebras. In: Cintula, P., Hanikova, Z., Svejdar, V. (eds.) Witnessed Years: Essays in Honour of Petr Hajek, pp. 305–331. College Publications, London (2009) Google Scholar
  24. 129.
    Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Elsevier, Amsterdam (2007) Google Scholar
  25. 145.
    Georgescu, G., Popescu, A.: Similarity convergence in residuated structures. Log. J. IGPL 13, 389–413 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 152.
    Georgescu, G., Leuştean, L., Mureşan, C.: Maximal residuated lattices with lifting Boolean center. Algebra Univers. 63, 83–99 (2010) zbMATHCrossRefGoogle Scholar
  27. 156.
    Grätzer, G.: Lattice Theory. First Concepts and Distributive Lattices. A Series of Books in Mathematics. Freeman, San Francisco (1972) Google Scholar
  28. 165.
    Höhle, U.: Commutative residuated monoids. In: Höhle, U., Klement, P. (eds.) Non-classical Logics and Their Applications to Fuzzy Subsets, pp. 53–106. Kluwer Academic, Dordrecht (1995) CrossRefGoogle Scholar
  29. 200.
    Jipsen, P., Tsinakis, C.: A survey of residuated lattices. In: Martinez, J. (ed.) Ordered Algebraic Structures, pp. 19–56. Kluwer Academic, Dordrecht (2002) CrossRefGoogle Scholar
  30. 201.
    Kowalski, T., Ono, H.: Residuated lattices: an algebraic glimpse at logics without contraction. Monograph (2001) Google Scholar
  31. 204.
    Krull, W.: Axiomatische Begründung der allgemeinen Idealtheorie. Sitzungsber. Phys.-Med. Soz. Erlangen 56, 47–63 (1924) Google Scholar
  32. 235.
    Ono, H.: Completions of algebras and completeness of modal and substructural logics. In: Advances in Modal Logic, vol. 4, pp. 335–353. King’s College Publications, London (2003) Google Scholar
  33. 236.
    Ono, H., Komori, Y.: Logics without contraction rule. J. Symb. Log. 50, 169–201 (1985) MathSciNetzbMATHCrossRefGoogle Scholar
  34. 237.
    Pavelka, J.: On fuzzy logic II. Enriched residuated lattices and semantics of propositional calculi. Z. Math. Log. Grundl. Math. 25, 119–134 (1979) MathSciNetzbMATHCrossRefGoogle Scholar
  35. 252.
    Rudeanu, S.: Localizations and fractions in algebra of logic. J. Mult.-Valued Log. Soft Comput. 16, 467–504 (2010) MathSciNetzbMATHGoogle Scholar
  36. 261.
    Ward, M.: Residuated distributive lattices. Duke Math. J. 6, 641–651 (1940) MathSciNetCrossRefGoogle Scholar
  37. 262.
    Ward, M., Dilworth, R.P.: Residuated lattices. Trans. Am. Math. Soc. 45, 335–354 (1939) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Lavinia Corina Ciungu
    • 1
  1. 1.Department of MathematicsUniversity of IowaIowa CityUSA

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