Soft Computing

, Volume 21, Issue 18, pp 5245–5264

# Constructive methods for intuitionistic fuzzy implication operators

Foundations

## Abstract

In this paper, two novel constructive methods for fuzzy implication operators are proposed. And they can be used to resolve the questions about how to construct intuitionistic fuzzy implication operators to inference sentence “If x is A, then y is B,” where A and B are intuitionistic fuzzy sets over X and Y, respectively. In the first method, cut sets and representation theorem of intuitionistic fuzzy sets are utilized through triple valued fuzzy implications as well as fuzzy relations to obtain seven intuitionistic fuzzy implication operators. In the second method, the Lebesgue measure of the fuzzy implications with value one and zero is used, and one intuitionistic fuzzy implication operator can be obtained. Finally, properties of those operators are discussed.

### Keywords

Intuitionistic fuzzy implication operators Lebesgue measure Cut sets Fuzzy relation Representation theorem

## Notes

### Acknowledgments

This work was supported by the National Science Foundation of China (No. 61473327).

### Conflict of interest

Yiying Shi, Xuehai Yuan and Yuchun Zhang declare that they have no conflict of interest.

### Ethical approval

This article does not contain any studies with human participants performed by any of the authors.

### References

1. Alcalde C, Burusco A, González RF (2005) A constructive method for the definition of interval-valued fuzzy implication operators. Fuzzy Sets Syst 153(2):211–227
2. Atanassov K, Gargov G (1998) Elements of intuitionistic fuzzy logic. Part I. Fuzzy Sets Syst 95(1):39–52
3. Baczyński M, Jayaram B (2008) (S, N)- and R-implications: a state-of-the-art survey. Fuzzy Sets Syst 159(14):1836–1859
4. Baczyński M, Jayaram B (2010) QL-implications: some properties and intersections. Fuzzy Sets Syst 161(2):158–188
5. Baczyński M (2014) Distributivity of implication operations over t-representable t-norms in interval-valued fuzzy set theory: The case of nilpotent t-norms. Inf Sci 257(1):388–399
6. Bedregal BC, Dimuro GP, Santiago RHN (2010) On interval fuzzy S-implications. Inf Sci 180(8):1373–1389
7. Bedregal BC (2010) On interval fuzzy negations. Fuzzy Sets Syst 161(17):2290–2313
8. Bustince H, Burillo P (1996) Structures on intuitionistic fuzzy relations. Fuzzy Sets Syst 78(3):293–303
9. Bustince H (2000) Construction of intuitionistic fuzzy relations with predetermined properties. Fuzzy Sets Syst 109(3):379–403
10. Bustince H, Barrenechea E, Mohedano V (2004) Intuitionistic fuzzy implication operators-an expression and main properties. Int J Unc Fuzz Knowl Based Syst 12(3):387–406
11. Carbonell M, Torrens J (2010) Continuous R-implications generated from representable aggregation functions. Fuzzy Sets Syst 161(17):2276–2289
12. Cintula P, Metcalfe G (2010) Admissible rules in the implication-negation fragment of intuitionistic logic. Ann Pure Appl Logic 162(2):162–171
13. Cornelis C, Deschrijver G, Kerre EE (2004) Implication in intuitionistic fuzzy and interval-valued fuzzy set theory: construction, classification, application. Int J Approx Reason 35(1):55–95
14. Deschrijver G, Kerre EE (2005) Implicators based on binary aggregation operators in interval-valued fuzzy set theory. Fuzzy Sets Syst 153(2):229–248
15. Dubois D, Prade H (1991) Fuzzy sets in approximate reasoning, Part 1: Inference with possibility distributions. Fuzzy Sets Syst 40(1):143–202
16. Dubois D, Hüllermeier E, Prade H (2003) On the representation of fuzzy rules in terms of crisp rules. Inf Sci 151(5):301–326
17. Fodor JC (1991) On fuzzy implication operators. Fuzzy Sets Syst 42(3):293–300
18. Gorzałczany MB (1989) An interval-valued fuzzy inference method—some basic properties. Fuzzy Sets Syst 31(2):243–251
19. Kóczy L, Hirota K (1993) Interpolative reasoning with insufficient evidence in sparse fuzzy rule bases. Inf Sci 71(1–2):169–201
20. Li DC, Li YM, Xie YJ (2011) Robustness of interval-valued fuzzy inference. Inf Sci 181(20):4754–4764
21. Li DC, Li YM (2012) Algebraic structures of interval-valued fuzzy (S, N) (S, N)-implications. Int J Approx Reason 53(6):892–900
22. Luo CZ, Wang PZ (1990) Representation of compositional relations in fuzzy reasoning. Fuzzy Sets Syst 36(1):77–81
23. Mas M, Monserrat M, Torrens J, Trillas E (2007) A Survey on fuzzy implication functions. IEEE Trans Fuzzy Syst 15(6):1107–1121
24. Pei D (2012) Formalization of implication based fuzzy reasoning method. Int J Approx Reason 53(5):837–846
25. Shi YY, Yuan XH (2015) A possibility-based method for ranking fuzzy numbers and applications to decision making. J Intell Fuzzy Syst 29:337–349
26. Wang G (2000) Triple I method and interval valued fuzzy reasoning. Sci China Ser E 43(3):242–253
27. Wang Z, Xu ZS, Liu SS (2014) Direct clustering analysis based on intuitionistic fuzzy implication. Appl Soft Comput 23(5):1–8Google Scholar
28. Wu WZ, Leung Y, Shao MW (2013) Generalized fuzzy rough approximation operators determined by fuzzy implicators. Int J Approx Reason 54(9):1388–1409
29. Xu Y, Kerre EE, Ruan D, Song Z (2001) Fuzzy reasoning based on the extension principle. Int J Intell Syst 16(4):469–495
30. Yao O (2012) On fuzzy implications determined by aggregation operators. Inf Sci 193(11):153–162
31. Yu XH, Xu ZS (2013) Prioritized intuitionistic fuzzy aggregation operators. Inf Fusion 14(1):108–116
32. Yuan XH, Li HX, Sun KB (2011) The cut sets, decomposition theorems and representation theorems on intuitionistic fuzzy sets and interval valued fuzzy sets. Sci China Inf Sci 54:91–110
33. Zadeh LA (1973) Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans Syst Man Cybern 3(1):28–44
34. Zheng M, Shi Z, Liu Y (2014) Triple I method of approximate reasoning on Atanassov’s intuitionistic fuzzy sets. Int J Approx Reason 55(6):1369–1382
35. Zhou L, Wu WZ, Zhang WX (2009) On characterization of intuitionistic fuzzy rough sets based on intuitionistic fuzzy implicators. Inf Sci 179(7):883–898