A characterization of interval-valued residuated lattices

International Journal of Approximate Reasoning

Volumn 49, Issue 2, 2008, Pages 478-487

  Van Gasse, B ^a   Cornelis, C ^a   Deschrijver, G ^a   Kerre, E E ^a  

^a GHENT UNIVERSITY   (Belgium)

Author keywords

Interval valued fuzzy set theory; Residuated lattices; Triangle algebras

Indexed keywords

EQUATIONAL REPRESENTATION; INTERVAL-VALUED; INTERVAL-VALUED FUZZY SET THEORY; LEFT-CONTINUOUS T-NORMS; OPEN PROBLEMS; RESIDUATED LATTICES; TRIANGLE ALGEBRAS; UNIT INTERVAL;

ALGEBRA;

EID: 52949104052     PISSN: 0888613X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.ijar.2008.04.006     Document Type: Article

References (26)

1. Alcalde C., Burusco A., and Fuentes-González R. A constructive method for the definition of interval-valued fuzzy implication operators. Fuzzy Sets and Systems 153 (2005) 211-227
2. Bustince H. Indicator of inclusion grade for interval-valued fuzzy sets. Application to approximate reasoning based on interval-valued fuzzy sets. International Journal of Approximate Reasoning 23 (2000) 137-209
3. Chang C.C. Algebraic analysis of many valued logics. Transactions of the American Mathematical Society 88 2 (1958) 467-490
4. Chang C.C. A new proof of the completeness of the Lukasiewicz axioms. Transactions of the American Mathematical Society 93 1 (1959) 74-80
5. Cornelis C., Deschrijver G., and Kerre E.E. Implication in intuitionistic fuzzy and interval-valued fuzzy set theory: construction, classification, application. International Journal of Approximate Reasoning 35 1 (2003) 55-95
6. Cornelis C., Deschrijver G., and Kerre E.E. Advances and challenges in interval-valued fuzzy logic. Fuzzy Sets and Systems 157 5 (2006) 622-627
7. Deschrijver G., and Kerre E.E. Classes of intuitionistic fuzzy t-norms satisfying the residuation principle. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 11 6 (2003) 691-709
8. Deschrijver G., Cornelis C., and Kerre E.E. On the representation of intuitionistic fuzzy t-norms and t-conorms. IEEE Transactions on Fuzzy Systems 12 1 (2003) 45-61
9. Deschrijver G. A representation of t-norms in interval-valued L-fuzzy set theory. Fuzzy Sets and Systems 159 13 (2008) 1597-1618
10. Dilworth R.P., and Ward M. Residuated lattices. Transactions of the American Mathematical Society 45 (1939) 335-354
11. Esteva F., Garcia-Calvés P., and Godo L. Enriched interval bilattices and partial many-valued logics: an approach to deal with graded truth and imprecision international journal of uncertainty. Fuzziness and Knowledge-Based Systems 2 1 (1994) 37-54
12. Esteva F., and Godo L. Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets and Systems 124 (2001) 271-288
13. Gehrke M., Walker C., and Walker E. Some comments on interval-valued fuzzy sets. International Journal of Intelligent Systems 11 (1996) 751-759
14. Goguen J.A. L-fuzzy sets. Journal of Mathematical Analysis and Applications 18 (1967) 145-174
15. Hájek P. Metamathematics of Fuzzy Logic Trends in Logic-Studia Logica Library (1998), Kluwer Academic Publishers
16. Höhle U. Commutative, residuated L-monoids. In: Höhle U., and Klement E.P. (Eds). Non-classical Logics and their Applications to Fuzzy Subsets: A Handbook of the Mathematical Foundations of Fuzzy Set Theory (1995), Kluwer Academic Publishers 53-106
17. Huntington E.V. Sets of independent postulates for the algebra of logic. Transactions of the American Mathematical Society 5 (1904) 288-309
18. Jenei S. A more efficient method for defining fuzzy connectives. Fuzzy Sets and Systems 90 (1997) 25-35
19. Klement E.P., Mesiar R., and Pap E. Triangular norms. Trends in Logic-Studia Logica Library (2000), Kluwer Academic Publishers
20. Rasiowa H., and Sikorski R. The mathematics of metamathematics. Polska Akad Nauk (1963)
21. Schweizer B., and Sklar A. Statistical matric spaces. Pacific Journal of Mathematics 10 (1960) 313-334
22. Turunen E. Mathematics behind fuzzy logic. Physica-Verlag (1999)
23. Van Gasse B., Cornelis C., Deschrijver G., and Kerre E.E. On the properties of a generalized class of t-norms in interval-valued fuzzy logics. New Mathematics and Natural Computation 2 1 (2006) 29-42
24. B. Van Gasse, C. Cornelis, G. Deschrijver, E.E. Kerre, Triangle algebras: towards an axiomatization of interval-valued residuated lattices, in: Lecture Notes in Computer Science, Proceedings of the Fifth International Conference on Rough Sets and Current Trends in Computing (RSCTC 2006), 2006, pp. 117-126.
25. Van Gasse B., Cornelis C., Deschrijver G., and Kerre E.E. Triangle algebras: a formal logic approach to interval-valued residuated lattices. Fuzzy Sets and Systems 159 (2008) 1042-1060
26. Zadeh L.A. Fuzzy sets. Information and Control 8 (1965) 338-353