Skip to main content
SpringerLink
Log in

Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part II: Category Theoretic Approaches

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

For classical sets one has with the cumulative hierarchy of sets, with axiomatizations like the system ZF, and with the category SET of all sets and mappings standard approaches toward global universes of all sets.

We discuss here the corresponding situation for fuzzy set theory. Our emphasis will be on various approaches toward (more or less naively formed) universes of fuzzy sets as well as on axiomatizations, and on categories of fuzzy sets.

What we give is a (critical) survey of quite a lot of such approaches which have been offered in the last approximately 35 years.

Part I was devoted to model based and to axiomatic approaches; the present Part II is devoted to category theoretic approaches.

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. Barr, M., and C. Wells, Toposes, Triples and Theories, Grundl. Math. Wiss. 278. Springer-Verlag, New York, 1985.

    Google Scholar 

  2. Bell, J. L., ‘Some aspects of the category of subobjects of constant objects in a topos’, J. Pure Appl. Algebra 24:245–259, 1982.

    Article  Google Scholar 

  3. Bell, J. L., Toposes and Local Set Theories, Oxford Logic Guides 14. Oxford UniversityPress (Clarendon Press), Oxford, 1988.

    Google Scholar 

  4. Borceux, F., Handbook of Categorical Algebra, Encycl. Math. and its Appl., vols. 50–52, Cambridge Univ. Press, Cambridge, 1994.

    Google Scholar 

  5. Carrega, J. C., ‘The categories Set-H and Fuz-H’, Fuzzy Sets Syst. 9:327–332, 1983.

    Article  Google Scholar 

  6. Cerruti, U., and U. Höhle, ‘Categorical foundations of fuzzy set theory with applications to algebra and topology’, in A. di Nola and A. G. S. Ventre, (eds.), The Mathematics of Fuzzy Systems, Interdisciplinary Systems Res. 88, TUV Rheinland, Köln (Cologne), 1986, pp. 51–86.

    Google Scholar 

  7. Coulon, J., J.-L. Coulon, and U. Höhle, ‘Classification of extremal subobjects of algebras over SM-SET’, in Applications of Category Theory to Fuzzy Subsets (Linz, 1989). Kluwer Acad. Publ., Dordrecht, 1992, pp. 9–31.

    Google Scholar 

  8. DiNola, A., G. Georgescu and A. Iorgulescu, ‘Pseudo-BL-algebras I, II’, Multiple-Valued Logic 8:673–714 and 717–750, 2002.

    Google Scholar 

  9. Eytan, M., ‘Fuzzy sets: A topos-logical point of view’, Fuzzy Sets Syst. 5:47–67, 1981.

    Article  Google Scholar 

  10. Fourman, M. P., ‘The logic of topoi’, in J. Barwise (ed.), Handbook of Mathematical Logic, North-Holland Publ. Comp., Amsterdam, 1977, pp. 1053–1090.

    Google Scholar 

  11. Fourman, M. P., and D. S. Scott, ‘Sheaves and logic’, in M. P. Fourman, C. J. Mulvey, and D. S. Scott (eds.), Applications of Sheaves, Lect. Notes Math. 753. Springer-Verlag, Berlin, 1979, pp. 302–401.

    Google Scholar 

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

    Article  Google Scholar 

  13. Goguen, J. A., ‘Concept respresentation in natural and artificial languages: axioms, extensions and applications for fuzzy sets’, Int. J. Man-Machine Stud. 6:513–561, 1974.

    Article  Google Scholar 

  14. Gottwald, S., A Treatise on Many-valued Logics, Studies in Logic and Computation 9, Research Studies Press, Baldock, 2001.

    Google Scholar 

  15. Gottwald, S., ‘Universes of fuzzy sets and axiomatizations of fuzzy set theory. Part I: Model-based and axiomatic approaches’, Studia Logica 82:211–244, 2006.

    Article  Google Scholar 

  16. Gylys, R. P., ‘Quantal sets and sheaves over quantales’. Liet. Matem. Rink. 34:9–31, 1994.

    Google Scholar 

  17. Hájek, P., Metamathematics of Fuzzy Logic, Trends in Logic 4, Kluwer Acad. Publ., Dordrecht, 1998.

    Google Scholar 

  18. Hájek, P., ‘Fuzzy logics with non-commutative conjunctions’, J. Logic and Computation 13:469–479, 2003.

    Article  Google Scholar 

  19. Higgs, D., A Category Approach to Boolean-valued Set Theory. Preprint, Univ. of Waterloo, 1973.

  20. Higgs, D., ‘Injectivity in the topos of complete Heyting algebra valued sets’, Canadian J. Math. 36:550–568, 1984.

    Google Scholar 

  21. Höhle, U., ‘M-valued sets and sheaves over integral commutative CL-monoids’, in S. E. Rodabaugh et al. (eds.), Applications of Category Theory to Fuzzy Subsets, TheoryDecis. Libr., Ser. B 14. Kluwer Acad. Publ., Dordrecht, 1992, pp. 34–72.

    Google Scholar 

  22. Höhle, U., ‘Commutative, residuated l-monoids’. in U. Höhle and E. P. Klement (eds.), Non-Classical Logics and Their Applications to Fuzzy Subsets, Theory Decis. Libr., Ser. B. 32. Kluwer Acad. Publ., Dordrecht, 1995, pp. 53–106.

    Google Scholar 

  23. Höhle, U., ‘Presheaves over GL-monoids’. in U. Höhle and E. P. Klement (eds.), Non-Classical Logics and Their Applications to Fuzzy Subsets, Theory Decis. Libr., Ser. B 32. Kluwer Acad. Publ., Dordrecht, 1995, pp. 127–157.

    Google Scholar 

  24. Höhle, U., ‘GL-quantales: Q-valuedsets andtheir singletons’, Studia Logica 61:123–148, 1998.

    Article  Google Scholar 

  25. Höhle, U., ‘Classification of subsheaves over GL-algebras’, in S. R. Buss, P. Hájek, and P. Pudlák, (eds.), Logic Colloquium ’98, Lect. Notes Logic 13, pp. 238–261. A K Peters, Ltd., Natick'MA, 2000.

    Google Scholar 

  26. Höhle, U., ‘Many-valued equalities and their representations’, in E. P. Klement and R. Mesiar (eds.), Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms. Elsevier, Dordrecht, 2005, pp. 301–319.

    Google Scholar 

  27. Höhle, U., ‘Sheaves on Quantales’ in: Proceedings Linz Seminar on Fuzzy Set Theory 2005. (in preparation)

  28. Höhle, U., and L. N. Stout, ‘Foundations of fuzzy sets’, Fuzzy Sets Syst. 40:257–296, 1991.

    Article  Google Scholar 

  29. Jenei, S., and F. Montagna, ‘A proof of standard completeness for non-commutative monoidal t-norm logic’, Neural Network World 13:481–489, 2003.

    Google Scholar 

  30. Lawvere, F. W., ‘An elementary theory of the category of sets’, Proc.Nat.Acad. Sci. USA 52:1506–1511, 1964.

    Article  Google Scholar 

  31. Mac Lane, S., and I. Moerdijk, Sheaves in Geometry and Logic, Springer, New York, 1992.

    Google Scholar 

  32. Manes, E. G., Algebraic Theories, Springer, Berlin, 1976.

    Google Scholar 

  33. Mulvey, C. J., and M. Nawaz, ‘Quantales: Quantal sets’, in U. Höhle and E. P. Klement (eds.), Non-Classical Logics and Their Applications to Fuzzy Subsets, Theory Decis. Libr., Ser. B. 32, Kluwer Acad. Publ., Dordrecht, 1995, pp. 159–217.

    Google Scholar 

  34. Pitts, A. M., ‘Fuzzy sets do not form a topos’ Fuzzy Sets Syst. 8:101–104, 1982.

    Article  Google Scholar 

  35. Ponasse, D., ‘Some remarks on the category Fuz(H) of M. Eytan’, Fuzzy Sets Syst. 9:199–204, 1983.

    Article  Google Scholar 

  36. Pultr, A., ‘Closed categories of L-fuzzy sets’, in Vorträge aus dem Problemseminar Automaten-und Algorithmentheorie (Weißig 1975). Techn. Univ. Dresden, Sektion Mathematik, Dresden, 1976, pp. 60–68.

    Google Scholar 

  37. Pultr, A., ‘Fuzziness and fuzzy equality’, in H. J. Skala, S. Termini, and E. Trillas (eds.), Aspects of Vagueness, Theory and Decision Libr. 39. Reidel, Dordrecht, 1984, pp. 119–135.

    Google Scholar 

  38. Rosenthal, K. I., Quantales and Their Applications, Pittman Res. Notes in Math. 234, Longman, Burnt Mill, Harlow, 1990.

    Google Scholar 

  39. Scott, D. S., ‘Continuous lattices’, in F. W. Lawvere (ed.), Toposes, Algebraic Geometry and Logic, Lect. Notes Math. 274. Springer-Verlag, Berlin, 1971, pp. 97–136.

    Google Scholar 

  40. Scott, D. S., ‘Identity and existence in intuitionistic logic’, in M. P. Fourman, C. J. Mulvey, and D. S. Scott (eds.), Applications of Sheaves, Lect. Notes Math. 753. Springer-Verlag, Berlin, 1979, pp. 660–696.

    Google Scholar 

  41. Shimoda, M., ‘Categorical aspects of Heyting-valued models for intuitionistic set theory’, Comment. Math. Univ. Sancti Pauli 30:17–35, 1981.

    Google Scholar 

  42. Stout, L. N., ‘Topoi and categories of fuzzy sets’, Fuzzy Sets Syst. 12:169–184, 1984.

    Article  Google Scholar 

  43. Stout, L. N., ‘A survey of fuzzy set and topos theory’, Fuzzy Sets Syst. 42:3–14, 1991.

    Article  Google Scholar 

  44. Stout, L. N., ‘Categories of fuzzy sets with values in a quantale or projectale’, in U. Höhle and E. P. Klement (eds.), Non-Classical Logics and Their Applications to Fuzzy Subsets, Theory Decis. Libr., Ser. B. 32. Kluwer Acad. Publ., Dordrecht, 1995, pp. 219–234.

    Google Scholar 

  45. Takeuti, G., and S. Titani, ‘Intuitionistic fuzzy logic and intuitionistic fuzzy set theory’, J. Symb. Logic 49:851–866, 1984.

    Article  Google Scholar 

  46. Takeuti, G., and S. Titani, ‘Global intuitionistic fuzzy set theory’, in The Mathematics of Fuzzy Systems, Interdisciplinary Syst. Res. 88. TUV Rheinland, Köln (Cologne), 1986, pp. 291–301.

    Google Scholar 

  47. Takeuti, G., and S. Titani, ‘Fuzzy logic and fuzzy set theory’, Arch. Math. Logic, 32:1–32, 1992.

    Article  Google Scholar 

  48. Wyler, O., Lecture notes on topoi and quasitopoi, World Scientific, Singapore, 1991.

    Google Scholar 

  49. Wyler, O., ‘Fuzzy logic and categories of fuzzy sets’, in U. Höhle and E. P. Klement (eds.), Non-Classical Logics and Their Applications to Fuzzy Subsets, Theory Decis. Libr., Ser. B. 32. Kluwer Acad. Publ., Dordrecht, 1995, pp. 235–268.

    Google Scholar 

  50. Zadeh, L. A., ‘Fuzzy sets’, Information and Control 8:338–353, 1965.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Logic and Philosophy of Science, Leipzig University, Beethovenstr. 15, 04107, Leipzig, Germany

    Siegfried Gottwald

Authors
  1. Siegfried Gottwald
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Siegfried Gottwald.

Additional information

This paper is a version of the invited talk given by the author at the conference Trends in Logic III, dedicated to the memory of A. MOSTOWSKI, H. RASIOWA and C. RAUSZER, and held in Warsaw and Ruciane-Nida from 23rd to 25th September 2005.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gottwald, S. Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part II: Category Theoretic Approaches. Stud Logica 84, 23–50 (2006). https://doi.org/10.1007/s11225-006-9001-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11225-006-9001-1

Keywords

Search

Navigation