Skip to main content
SpringerLink
Log in

Algebras with compatible quantale-enriched structures

  • Research Paper
  • Published:
Science China Information Sciences Aims and scope Submit manuscript

Abstract

Based on quantale-enriched category, we consider algebras with compatible quantale-enriched structures, which can be viewed as fuzzification of ordered algebraic structures. We mainly study groupoids and semigroups with compatible quantale-enriched structures from this viewpoint. Some basic concepts such as ideals, homomorphisms, residuated quantale-enriched groupoids are developed and some examples of them are given. Our approach gives a complement to the approach initiated by Rosenfeld to study fuzzy abstract algebra, and these two approaches are combined in the present paper to study fuzzy aspects of abstract algebra structures.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. Lawvere F W. Metric spaces, generalized logic, and closed categories. Rendiconti del Seminario Matematico e Fisico di Milano 1973, 43: 135–166

    Article  MathSciNet  Google Scholar 

  2. Wagner K R. Solving recursive domain equations with enriched categories. Ph.D. Thesis, Pittsburgh: School of Computer Science, Carnegie-Mellon University, 1994

    Google Scholar 

  3. Wagner K R. Liminf convergence in Ω-categories. Theoret Comp Sci, 1997, 184: 61–104

    Article  MATH  Google Scholar 

  4. Lai H, Zhang D. Complete and directed complete Ω-categories. Theoret Comp Sci, 2007, 388: 1–25

    Article  MathSciNet  MATH  Google Scholar 

  5. Zhang D. An enriched category approach to many valued Topology. Fuzzy Sets Syst, 2007, 158: 349–366

    Article  MATH  Google Scholar 

  6. Fan L. A new approach to quantitative domain theory. Electr Notes Theoret Comp Sci, 2001, 45: 77–87

    Article  Google Scholar 

  7. Yao W. Quantitative domains via fuzzy sets (I): Continuity of fuzzy directed complete posets. Fuzzy Sets Syst, 2010, 161: 973–987

    Article  MATH  Google Scholar 

  8. Lai H, Zhang D. Many-valued complete distributivity. arXiv:math/0603590v2, May 2006

    Google Scholar 

  9. Axler S, Ribet K A. Lattices and Ordered Algebraic Structures. London: Springer, 2005

    Google Scholar 

  10. Rosenfeld A. Fuzzy groups. J Math Anal Appl, 1971, 35: 512–517

    Article  MathSciNet  MATH  Google Scholar 

  11. Bělohlávek R, Vychodil V. Algebras with fuzzy Equalities. Fuzzy Sets Syst, 2006, 157: 161–201

    Article  MATH  Google Scholar 

  12. Rosenthal K I. Quantales and Their Applications. New York: Longman Scientific & Technical, 1990

    MATH  Google Scholar 

  13. Adámek J, Herrlich H, Strecker G E. Abstract and Concrete Categories. New York: Wiley, 1990

    MATH  Google Scholar 

  14. Kelly G M. Basic concepts of enriched category theory. London Mathematical Society Lecture Notes Series, Vol.64. Cambridge: Cambridge University Press, 1982

    MATH  Google Scholar 

  15. Bělohlávek R. Fuzzy Relational Systems: Foundations and Principles. New York: Kluwer Academic Publishers, 2002

    Book  Google Scholar 

  16. Johnstone P T. Stone Spaces. Cambridge: Cambridge University Press, 1982

    MATH  Google Scholar 

  17. Goguen J A. L-fuzzy sets. J Math Anal Appl, 1967, 18: 145–174

    Article  MathSciNet  MATH  Google Scholar 

  18. Xie X Y. An Introduction to Ordered Semigroup Theory. Beijing: Science Press, 2001. 269

    Google Scholar 

  19. Kehayopulu N. On prime, weakly prime ideals in ordered semigroups. Semigroup Forum, 1992, 44: 341–346

    Article  MathSciNet  MATH  Google Scholar 

  20. Kehayopulu N, Tsingelis M. Fuzzy sets in ordered Groupoids. Semigroup Forum, 2002, 65: 128–132

    Article  MathSciNet  MATH  Google Scholar 

  21. Zadeh L A. The concept of a linguistic variable and its applications in approximate reasoning. Info Sci, 1975, 8: 199–251, 301–357; 1975, 9: 43–80

    Article  MathSciNet  MATH  Google Scholar 

  22. Rodabaugh S E. Powerset operator foundations for poslat fuzzy set theories and topologies, Chapter 2. In: Höhle U, Rodabaugh S E, eds. Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory. Boston, Dordrecht, London: Kluwer Academic Publishers, 1999. 91–116

    Chapter  Google Scholar 

  23. Yao W. An approach to fuzzy frames via fuzzy posets. Fuzzy Sets Syst, 2011, 166: 75–89

    Article  MATH  Google Scholar 

  24. Liu W J. Fuzzy invariant subgroups and fuzzy ideals. Fuzzy Sets Syst, 1982, 8: 133–139

    Article  MATH  Google Scholar 

  25. Liu W J. Operations on fuzzy ideals. Fuzzy Sets Syst, 1983, 11: 37–41

    Google Scholar 

  26. Wang K Y. Some researches on fuzzy domains and fuzzy quantales. Ph.D. Thesis, College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, 2012

    Google Scholar 

  27. Burris S, Sankappanavar H P. A Course in Universal Algebra. New York: Springer, 1981

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, 710062, China

    Bin Zhao & Min Liu

Authors
  1. Bin Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Min Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bin Zhao.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhao, B., Liu, M. Algebras with compatible quantale-enriched structures. Sci. China Inf. Sci. 57, 1–10 (2014). https://doi.org/10.1007/s11432-014-5141-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11432-014-5141-5

Keywords

Search

Navigation