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Implications over intuitionistic fuzzy sets: Difference between revisions
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=== | === List of implications === | ||
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== References == | == References == | ||
Revision as of 09:45, 11 March 2021
For the various definitions of implication of over intuitionistic fuzzy sets, the functions sg(x) and sg(x) have been used:
[math]\displaystyle{ \text{sg}(x) = \begin{cases} 1 \text{ if } x \gt 0 \\ 0 \text{ if } x \leq 0 \end{cases}, }[/math] [math]\displaystyle{ \overline{\text{sg}}(x) = \begin{cases} 0 \text{ if } x \gt 0 \\ 1 \text{ if } x \leq 0 \end{cases}. }[/math]
List of implications
| No. | Ref. | Year | Implication:
{<x, Implication MEMBERSHIP expression, Implication NON-MEMBERSHIP expression >|x ∈ E} |
|---|
| No. | Ref. | Year | Implication MEMBERSHIP expression |
Implication NON-MEMBERSHIP expression |
|---|---|---|---|---|
| →1 | max(νA(x),min(μA(x),μB(x))) | min(μA(x),νB(x)) | ||
| →2 | sg(μA(x)-μB(x)) | νB(x).sg(μA(x)-μB(x)) | ||
| →3 | 1-(1-μ(x)).sg(μA(x)-μB(x)) | νB.sg(μA(x)-μB(x)) | ||
| →4 | max(νA(x),μB(x)) | min(μA(x),νB(x)) | ||
| →5 | min(1,νA(x)+μB(x)) | max(0,μA(x)+νB(x)-1) | ||
| →6 | νA(x)+μA(x)μB(x) | μA(x)νB(x) | ||
| →7 | min(max(νA(x),μB(x)),max(μA(x),νA(x)), max(μB(x),νB(x))) | max(min(μA(x),νB(x)), min(μA(x),νA(x)),min(μB(x),νB(x))) | ||
| →8 | 1-(1-min(νA(x),μB(x))).sg(μA(x)-μB(x)) | max(μA(x),νB(x)).sg(μA(x)-μB(x)),sg(νB(x)-νA(x)) | ||
| →9 | νA(x)+μA(x)2μB(x) | μA(x)νA(x)+μA(x)2νB(x) | ||
| →10 | μA(x).sg(1-μA(x))+sg(1-μA(x)).(sg(1-μB(x))+νA(x).sg(1-μB(x))) | νB.sg(1-μA(x))+μA(x).sg(1-μA(x)).sg(1-μB(x)) | ||
| →11 | 1-(1-μB(x)).sg(μA(x)-μB(x)) | νB(x).sg(μA(x)-μB(x)).sg(νB(x)-νA(x)) | ||
| →12 | max(νA(x),μB(x)) | 1-max(νA(x),μB(x)) | ||
| →13 | νA(x)+μB(x)-νA(x).μB(x) | μA(x).νB(x) | ||
| →14 | 1-(1-μB(x)).sg(μA(x)-μB(x))-νB(x).sg(μA(x)-μB(x)).sg(νB(x)-νA(x)) | νB(x).sg(νB(x)-νA(x)) | ||
| →15 | 1-sg(μA(x)-μB(x)).sg(νB(x)-νA(x)) | sg(sg(μA(x)-μB(x))+sg(νB(x)-νA(x))) | ||
| →16 | max(sg(μA(x)),μB(x)) | min(sg(μA(x)),νB(x)) | ||
| →17 | max(νA(x),μB(x)) | min(μA(x).νA(x)+μA(x)2,νB(x)) | ||
| →18 | max(νA(x),μB(x)) | min(1-νA(x),νB(x)) | ||
| →19 | max(1-sg(sg(μA(x))+sg(1-νA(x))),μB(x)) | min(sg(1-νA(x)),νB(x)) | ||
| →20 | max(sg(μA(x)),sg(μA(x)))) | min(sg(μA(x)),sg(μB(x))) | ||
| →21 | max(νA(x),μB(x).(μB(x)+νB(x))) | min(μA(x).(μA(x)+νA(x)),νB(x).(μB(x)2+νB(x)+μB(x).νB(x))) | ||
| →22 | max(νA(x),1-νB(x)) | min(1-νA(x),νB(x)) | ||
| →23 | 1-min(sg(1-νA(x)),sg(1-νB(x))) | min(sg(1-νA(x)),sg(1-νB(x))) | ||
| →24 | sg(μA(x)-μB(x)).sg(νB(x)-νA(x)) | sg(μA(x)-μB(x)).sg(νB(x)-νA(x)) | ||
| →25 | max(νA(x),sg(μA(x)).sg(1-νA(x)),μB(x).sg(νB(x)).sg(1-μB(x))) | min(μA(x),νB(x)) | ||
| →26 | max(sg(1-νA(x)),μB(x)) | min(sg(μA(x)),νB(x)) | ||
| →27 | max(sg(1-νA(x)),sg(μB(x))) | min(sg(μA(x)),sg(1-νB(x))) | ||
| →28 | max(sg(1-νA(x)),μB(x)) | min(μA(x),νB(x)) | ||
| →29 | max(sg(1-νA(x)),sg(1-μB(x))) | min(μA(x),sg(1-νB(x))) | ||
| →30 | max(1-μA(x),min(μA(x),1-νB(x))) | min(μA(x),νB(x)) | ||
| →31 | sg(μA(x)+νB(x)-1) | νB(x).sg(μA(x)+νB(x)-1) | ||
| →32 | 1-νB(x).sg(μA(x)+νB(x)-1) | νB(x).sg(μA(x)+νB(x)-1) | ||
| →33 | 1-min(μA(x),νB(x)) | min(μA(x),νB(x)) | ||
| →34 | min(1,2-μA(x)-μB(x)) | max(0,μA(x)+νB(x)-1) | ||
| →35 | 1-μA(x).νB(x) | μA(x).νB(x) | ||
| →36 | min(1-min(μA(x),νB(x)),max(μA(x),1-μA(x)),max(1-νB(x),νB(x))) | max(min(μA(x),νB(x)),min(μA(x),1-μA(x)),min(1-νB(x),νB(x))) | ||
| →37 | 1-max(μA(x),νB(x)).sg(μA(x)+νB(x)-1) | max(μA(x),νB(x)).sg(μA(x)+νB(x)-1) | ||
| →38 | 1-μA(x)+(μA(x)2.(1-νB(x))) | μA(x)(1-μA(x))+μA(x)2.νB(x) | ||
| →39 | (1-νB(x)).sg(1-μA(x))+sg(1-μA(x)).(sg(νB(x))+(1-μA(x)).sg(νB(x))) | νB(x).sg(1-μA(x))+μA(x).sg(1-μA(x)).sg(νB(x)) | ||
| →40 | 1-sg(μA(x)+νB(x)-1) | 1-sg(μA(x)+νB(x)-1) | ||
| →41 | max(sg(μA(x)),1-νB(x)) | min(sg(μA(x)),νB(x)) | ||
| →42 | max(sg(μA(x)),sg(1-νB(x))) | min(sg(μA(x)),sg(1-νB(x))) | ||
| →43 | max(sg(μA(x)),1-νB(x)) | min(sg(μA(x)),νB(x)) | ||
| →44 | max(sg(μA(x)),1-νB(x)) | min(μA(x),νB(x)) | ||
| →45 | max(sg(μA(x)),sg(νB(x))) | min(μA(x),sg(1-νB(x))) | ||
| →46 | max(νA(x),min(1-νA(x),μB(x))) | 1-max(νA(x),μB(x)) | ||
| →47 | sg(1-νA(x)-μB(x)) | (1-μB(x)).sg(1-νA(x)-μB(x)) | ||
| →48 | 1-(1-μB(x)).sg(1-νA(x)-μB(x)) | (1-μB(x)).sg(1-νA(x)-μB(x)) | ||
| →49 | min(1,νA(x)+μB(x)) | max(0,1-νA(x)-μB(x)) | ||
| →50 | νA(x)+μB(x)-νA(x).μB(x) | 1-νA(x)-μB(x)+νA(x).μB(x) | ||
| →51 | min(max(νA(x),μB(x)),max(1-νA(x),νA(x)),max(μB(x),1-μB(x))) | max(1-max(νA(x),μB(x)),min(1-νA(x),νA(x)),min(μB(x),1-μB(x))) | ||
| →52 | 1-(1-min(νA(x),μB(x))).sg(1-νA(x)-μB(x)) | 1-min(νA(x),μB(x)).sg(1-νA(x)-μB(x)) | ||
| →53 | νA(x)+(1-νA(x))2.μB(x) | (1-νA(x)).νA(x)+(1-νA(x))2.(1-μB(x)) | ||
| →54 | μB(x)sg(νA(x))+sg(νA(x)).(sg(1-μB(x))+νA(x).sg(1-μB(x))) | (1-μB(x)).sg(νA(x))+(1-νA(x)).sg(νA(x)).sg(1-μB(x)) | ||
| →55 | 1-sg(1-νA(x)-μB(x)) | 1-sg(1-νA(x)-μB(x)) | ||
| →56 | max(sg(1-νA(x)),μB(x)) | min(sg(1-νA(x)),1-μB(x)) | ||
| →57 | max(sg(1-νA(x)),sg(μB(x))) | min(sg(1-νA(x)),sg(μB(x))) | ||
| →58 | max(sg(1-νA(x)),sg(1-μB(x))) | 1-max(νA(x),μB(x)) | ||
| →59 | max(sg(1-νA(x)),μB(x)) | 1-max(νA(x),μB(x)) | ||
| →60 | max(sg(1-νA(x)),sg(1-μB(x))) | min(1-νA(x),sg(μB(x))) | ||
| →61 | max(μB(x),min(νB(x),νA(x))) | min(νB(x),μA(x)) | ||
| →62 | sg(νB(x)-νA(x)) | μA(x).sg(νB(x)-νA(x)) | ||
| →63 | 1-(1-νA(x)).sg(νB(x)-νA(x)) | μA(x).sg(νB(x)-νA(x)) | ||
| →64 | μB(x)+νB(x).νA(x) | νB(x).μA(x) | ||
| →65 | 1-(1-min(μB(x),νA(x))).sg(νB(x)-νA(x)) | max(νB(x),μA(x)).sg(νB(x)-νA(x)).sg(μA(x)-μB(x)) | ||
| →66 | μB(x)+νB(x)2νA(x) | νB(x).μB(x)+νB(x)2μA(x) | ||
| →67 | νA(x).sg(1-νB(x))+sg(1-νB(x)).(sg(1-νA(x))+μB(x).sg(1-νA(x))) | μA(x).sg(1-νB(x))+νB(x).sg(1-νB(x)).sg(1-νA(x)) | ||
| →68 | 1-(1-νA(x)).sg(νB(x)-νA(x)) | μA(x).sg(νB(x)-νA(x)).sg(μA(x)-μB(x)) | ||
| →69 | 1-(1-νA(x)).sg(νB(x)-νA(x))-μA(x).sg(νB(x)-νA(x)).sg(μA(x)-μB(x)) | μA(x).sg(μA(x)-μB(x)) | ||
| →70 | max(sg((νB(x)),νA(x)) | min(sg(νB(x)),μA(x)) | ||
| →71 | max(μB(x),νA(x)) | min(νB(x).μB(x)+νB(x)2,μA(x)) | ||
| →72 | max(μB(x),νA(x)) | min(1-μB(x),μA(x)) | ||
| →73 | max(1-max(sg(νB(x)),sg(1-μB(x))),νA(x)) | min(sg(1-μB(x)),μA(x)) | ||
| →74 | max(sg(νB(x)),sg(νA(x))) | min(sg(νB(x)),sg(νA(x))) | ||
| →75 | max(μB(x),νA(x).(νA(x)+μA(x))) | min(νB(x).(νB(x)+μB(x)),μA(x).(νA(x)2+μA(x))+νA(x).μA(x)) | ||
| →76 | max(μB(x),1-μA(x)) | min(1-μB(x),μA(x)) | ||
| →77 | 1-min(sg(1-μB(x)),sg(1-μA(x))) | min(sg(1-μB(x)),sg(1-μA(x))) | ||
| →78 | max(sg(1-μB(x)),νA(x)) | min(sg(νB(x)),μA(x)) | ||
| →80 | max(sg(1-μB(x)),νA(x)) | min(νB(x),μA(x)) | ||
| →81 | max(sg(1-μB(x)),sg(1-νA(x))) | min(νB(x),sg(1-μA(x))) | ||
| →82 | max(1-νB(x),min(νB(x),1-μA(x))) | min(νB(x),μA(x)) | ||
| →83 | sg(νB(x)+μA(x)-1) | μA(x).sg(νB(x)+μA(x)-1) | ||
| →84 | 1-μA(x).sg(νB(x)+μA(x)+1) | μA(x).sg(νB(x)+μA(x)+1) | ||
| →85 | 1-νB(x)+νB(x)2.(1-μA(x)) | νB(x).(1-νB(x))+νB(x)2 | ||
| →86 | (1-μA(x)).sg(1-νB(x))+sg(1-νB(x))sg(μA(x)+min(1-νB(x),sg(μA(x)))) | μA(x).sg(1-νB(x))+νB(x).sg(1-νB(x)).sg(μA(x)) | ||
| →87 | max(sg(νB(x)),1-μA(x)) | min(sg(νB(x)),μA(x)) | ||
| →88 | max(sg(νB(x)),sg(1-μA(x))) | min(sg(νB(x)),sg(1-μA(x))) | ||
| →89 | max(sg(νB(x)),1-μA(x)) | min(νB(x),μA(x)) | ||
| →90 | max(sg(νB(x)),sg(μA(x))) | min(νB(x),sg(1-μA(x))) | ||
| →91 | max(μB(x),min(1-μB(x),νA(x))) | 1-max(μB(x),νA(x)) | ||
| →92 | sg(1-μB(x)-νA(x)) | min(1-νA(x),sg(1-μB(x)-νA(x))) | ||
| →93 | 1-min(1-νA(x),sg(1-μB(x)-νA(x))) | min(1-νA(x),sg(1-μB(x)-νA(x))) | ||
| →94 | μB(x)+(1-μB(x))2.νA(x)) | (1-μB(x)).μB(x)+(1-μB(x))2.(1-νA(x)) | ||
| →95 | min(νA(x),sg(μB(x)))+sg(μB(x)).(sg(1-νA(x))+min(μB(x),sg(1-νA(x)))) | min(1-νA(x),sg(μB(x)))+min(min(1-μB(x),sg(μB(x))),sg(1-νA(x))) | ||
| →96 | max(sg(1-μB(x)),νA(x)) | min(sg(1-μB(x)),1-νA(x) | ||
| →97 | max(sg(1-μB(x)),sg(νA(x))) | min(sg(1-μB(x)),sg(νA(x))) | ||
| →98 | max(sg(1-μB(x)),νA(x)) | 1-max(μB(x),νA(x)) | ||
| →99 | max(sg(1-μB(x)),sg(1-νA(x))) | min(1-μB(x),sg(νA(x))) | ||
| →100 | max(min(νA(x),sg(μA(x))),μB(x)) | min(min(μA(x),sg(νA(x))),νB(x)) | ||
| →101 | max(min(νA(x),sg(μA(x))),min(μB(x),sg(νB(x)))) | min(min(μA(x),sg(νA(x))),min(νB(x),sg(μB(x)))) | ||
| →102 | max(νA(x),min(μB(x),sg(νB(x)))) | min(μA(x),min(νB(x),sg(μB(x)))) | ||
| →103 | max(min(1-μA(x),sg(μA(x))),1-νB(x)) | min(μA(x),sg(1-μA(x)),νB(x)) | ||
| →104 | max(min(1-μA(x),sg(μA(x))),min(1-νB(x),sg(νB(x)))) | min(min(μA(x),sg(1-μA(x))),min(νB(x),sg(1-νB(x)))) | ||
| →105 | max(1-μA(x),min(1-νB(x),sg(νB(x)))) | min(μA(x),min(νB(x),sg(1-νB(x)))) | ||
| →106 | max(min(νA(x),sg(1-νA(x))),μB(x)) | min(min(1-νA(x),sg(νA(x))),1-μB(x)) | ||
| →107 | max(min(νA(x),sg(1-νA(x))),min(μB(x),sg(1-μB(x)))) | min(min(1-νA(x),sg(νA(x))),min(1-μB(x),sg(μB(x)))) | ||
| →108 | max(νA(x),min(μB(x),sg(1-μB(x)))) | min(1-νA(x),min(1-μB(x),sg(μB(x)))) | ||
| →109 | νA(x)+min(sg(1-μA(x)),μB(x)) | μA(x).νA(x)+min(sg(1-μA(x)),νB(x)) | ||
| →110 | max(νA(x),μB(x)) | min(μA(x).νA(x)+sg(1-μA(x)),νB(x)) | ||
| →111 | max(νA(x),μB(x).νB(x)+sg(1-μB(x))) | min(μA(x).νA(x)+sg(1-μA(x)),νB(x).(μB(x).νB(x)+sg(1-μB(x)))+sg(1-νB(x))) | ||
| →112 | νA(x)+μB(x)-νA(x).μB(x) | μA(x).νA(x)+sg(1-μA(x)).νB(x) | ||
| →113 | νA(x)+(μB(x).νB(x)-νA(x).(μB(x).νB(x)+sg(1-μB(x))) | (μA(x).νA(x)+sg(1-μA(x))).(νB(x).(μB(x).νB(x)+sg(1-μB(x)))+sg(1-νB(x))) | ||
| →114 | 1-μA(x)+min(sg(1-μA(x)),1-νB(x)) | μA(x).(1-μA(x))+min(sg(1-μA(x)),νB(x)) | ||
| →115 | 1-min(μA(x),νB(x)) | min(μA(x)(1-μA(x))+sg(1-μA(x)),νB(x)) | ||
| →116 | max(1-μA(x),(1-νB(x)).νB(x)+sg(νB(x))) | min(μA(x).(1-μA(x))+sg(1-μA(x)),νB(x).((1-νB(x)).νB(x)+sg(νB(x)))+sg(1-νB(x))) | ||
| →117 | 1-μA(x)-νB(x)+μA(x).νB(x) | (μA(x).(1-μA(x))+sg(1-μA(x))).νB(x) | ||
| →118 | (1-μA(x)).sg(νB(x))+μA(x).νB(x).(1-νB(x)) | (μA(x)-μA(x)2+sg(1-μA(x))).((1-νB(x)).νB(x)2+sg(1-νB(x)))+sg(1-νB(x))(x)) | ||
| →119 | νA(x)+min(sg(νA(x)),μB(x)) | (1-νA(x)).νA(x)+min(sg(νA(x)),1-μB(x)) | ||
| →120 | max(νA(x),μB(x)) | min((1-νA(x)).νA(x)+sg(νA(x)),1-μB(x)) | ||
| →121 | max(νA(x),μB(x).(1-μB(x))+sg(1-μB(x))) | min((1-νA(x)).νA(x)+sg(νA(x)),(1-μB(x)).(μB(x).(1-μB(x))+sg(1-μB(x)))+sg(μB(x))) | ||
| →122 | νA(x)+μB(x)-νA(x).μB(x) | ((1-νA(x)).νA(x)+sg(νA(x))).(1-μB(x)) | ||
| →123 | νA(x)+μB(x).(1-μB(x)-νA(x).(μB(x).(1-μB(x))+sg(1-μB(x))) | ((1-νA(x)).νA(x)+sg(νA(x))).(((1-μB(x)).(μB(x).(1-μB(x))+sg(1-μB(x))))+sg(μB(x))) | ||
| →124 | μB(x)+min(sg(1-νB(x)),νA(x)) | νB(x).μB(x)+min(sg(1-νB(x)),μA(x)) | ||
| →125 | max(μB(x),νA(x)) | min(νB(x).μB(x)+sg(1-νB(x)),μA(x)) | ||
| →126 | max(μB(x),νA(x).μA(x)+sg(1-νA(x))) | min(νB(x).μB(x)+sg(1-νB(x)),μA(x).(νA(x).μA(x)+sg(1-νA(x)))+sg(1-μA(x))) | ||
| →127 | μB(x)+νA(x)-μB(x).νA(x) | (νB(x).μB(x)+sg(1-νB(x))).μA(x) | ||
| →128 | μB(x)+νA(x).μA(x)-μB(x).(νA(x).μA(x)+sg(1-νA(x))) | (νB(x).μB(x)+sg(1-νB(x))).(μA(x).(νA(x).μA(x)+sg(1-νA(x)))+sg(1-μA(x))) | ||
| →129 | 1-νB(x)+min(sg(1-νB(x)),1-μA(x)) | νB(x).(1-νB(x))+min(sg(1-νB(x)),μA(x)) | ||
| →130 | 1-min(νB(x),μA(x)) | min(νB(x).(1-νB(x))+sg(1-νB(x)),μA(x)) | ||
| →131 | max(1-νB(x),(1-μA(x)).μA(x)+sg(μA(x))) | min(νB(x).(1-νB(x))+sg(1-νB(x)),μA(x).((1-μA(x)).μA(x)+sg(μA(x)))+sg(1-μA(x))) | ||
| →132 | 1-μA(x).νB(x) | (νB(x).(1-νB(x))+sg(1-νB(x))).μA(x) | ||
| →133 | 1-νB(x)+(1-μA(x)).μA(x)-(1-νB(x)).((1-μA(x)).μA(x)+sg(μA(x))) | (νB(x).(1-νB(x))+sg(1-νB(x))).(μA(x).((1-μA(x)).μA(x)+sg(μA(x)))+sg(1-μA(x))) | ||
| →134 | μB(x)+min(sg(μB(x)),νA(x)) | (1-μB(x)).μB(x)+min(sg(μB(x)),1-νA(x)) | ||
| →135 | max(μB(x),νA(x)) | min((1-μB(x)).μB(x)+sg(μB(x)),1-νA(x)) | ||
| →136 | max(μB(x),νA(x).(1-νA(x))+sg(1-νA(x))) | min((1-μB(x)).μB(x)+sg(μB(x)),(1-νA(x)).(νA(x).(1-νA(x))+sg(1-νA(x)))+sg(νA(x))) | ||
| →137 | μB(x)+νA(x)-μB(x).νA(x) | ((1-μB(x)).μB(x)+sg(μB(x))).(1-νA(x)) | ||
| →138 | μB(x)+νA(x).(1-νA(x))-μB(x). |
((1-μB(x)).μB(x)+sg(μB(x))).(1-νA(x).(νA(x).(1-νA(x)) + sg(1-νA(x)) + sg(νA(x)) | ||
| →139 | (νA(x) + μB(x))/2 | (μA(x) + νB(x))/2 | ||
| →140 | (νA(x) + μB(x) + min(νA(x), μB(x)))/3 | (μA(x) + νB(x) + max(μA(x), νB(x)))/3 | ||
| →141 | (νA(x) + μB(x) + max(νA(x), μB(x)))/3 | (μA(x) + νB(x) + min(μA(x), νB(x)))/3 | ||
| →142 | (3 - μA(x) - νB(x) - max(μA(x), νB(x)))/3 | (μA(x) + νB(x) + max(μA(x), νB(x)))/3 | ||
| →143 | (1 - μA(x) + μb(x) + min(1 - μA(x), μB(x)) )/3 | (2 + μA(x) - μB(x) - min(1 - μA(x), μB(x)))/3 | ||
| →144 | (1 + νA(x) - νb(x) + min(νA(x), 1 - νB(x)) )/3 | (2 - νA(x) + νB(x) + min(νA(x), 1 - νB(x)))/3 | ||
| →145 | (νA(x) + μB(x) + min(νA(x), μB(x)))/3 | (3 - νA(x) - μB(x) - min(νA(x), μB(x)))/3 | ||
| →146 | (3 - μA(x) - νB(x) - min(μA(x), νB(x)))/3 | (μA(x) + νB(x) + min(μA(x), νB(x)))/3 | ||
| →147 | (1 - μA(x) + μb(x) + max(1 - μA(x), μB(x)) )/3 | (2 + μA(x) - μb(x) - max(1 - μA(x), μB(x)) )/3
|
References
- On some properties of intuitionistic fuzzy implications, Michał Baczyński, 2003
- Intuitionistic fuzzy implications and Modus Ponens, Krassimir Atanassov, 2005
- A property of intuitionistic fuzzy implications, Yun Shi and Violeta Tasseva, 2005
- On some intuitionistic fuzzy implications, Krassimir Atanassov, 2006
- On a new intuitionistic fuzzy implication of Gaines-Rescher's type, Beloslav Riečan, Krassimir Atanassov, 2007
- A study on some intuitionistic fuzzy implications, Violeta Tasseva, Desislava Peneva, 2007
- On intuitionistic fuzzy subtraction, generated by an implication from Kleene-Dienes type, Lilija Atanassova, 2009
- A new intuitionistic fuzzy implication, Lilija Atanassova, 2009
- Intuitionistic fuzzy implications and axioms for implications, Krassimir Atanassov and Dimitar Dimitrov, 2010
- Four modal forms of intuitionistic fuzzy implication →@ and two related intuitionistic fuzzy negations. Part 1, Lilija Atanassova, 2010
- Some Remarks about L. Atanassova’s Paper “A New Intuitionistic Fuzzy Implication”, Piotr Dworniczak, 2010
- On the basic properties of the negations generated by some parametric intuitionistic fuzzy implications, Piotr Dworniczak, 2011
- On some two-parametric intuitionistic fuzzy implications, Piotr Dworniczak, 2011
- Second Zadeh's intuitionistic fuzzy implication, Krassimir Atanassov, 2011
