Andness Directedness for t-Norms and t-Conorms
Abstract
:1. Introduction
2. Preliminaries
2.1. Fuzzy Operators: t-Norms and t-Conorms
- (commutativity)
- when and (monotonicity)
- (associativity)
- (identity)
- (commutativity);
- when and (monotonicity);
- (associativity);
- (identity).
2.2. Andness and Orness
3. Andness Directedness for t-Norms
- Establish the relationship between the parameter and andness. Compute the andness of for p in . As we cannot consider the full range of p, we will select a finite subset of this range large enough to have a good approximation. We denote this finite subset of the range by . This process results in pairs . As andness is monotonic with respect to p, we can interpolate (e.g., linearly) the relationship between p and the andness easily.Figure 1 displays the andness for the t-norms (left) and t-conorms (right) discussed in Section 2. Therefore, we include , , , , and as t-norms, and , , , , as t-conorms. It is worth noting that in the figure only four lines are visible because the lines for (Dombi’s t-norm) and are indistinguishable. In Figure 1, we consider positive parameters , p, and w for all t-norms and t-conorms. We have discussed before that the parameter for both t-norm and t-conorm and can take any value except zero. For this reason, we represent the andness of these function independently in Figure 2.
- Establish the parameter given the andness. Given the andness level and the pairs check if there is a value . If this is the case, select . Otherwise, find a pair of values and that satisfy . Then, first, compute and, second, compute . For example, given for a given t-norm ⊤, we compute , or, in other words, we approximate the parameter p as the inverse of the andness for the given andness .We will use , , and to denote this process. For example, for , we obtain the following 5 parameters. For , ; for , ; , ; and for , .
3.1. Differences between t-Norms for a Given Andness
3.2. Differences between t-Norms and High Hyperconjunction for a Given Andness
3.3. Analysis
4. Conclusions and Future Directions
Funding
Data Availability Statement
Conflicts of Interest
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t-norm | ||||
andness | 1.000 | 1.250 | 1.500 | 2.000 |
t-conorm | ||||
andness | 0.000 | −0.250 | −0.500 | −1.000 |
Case | |||||
---|---|---|---|---|---|
(0.8, 0.8) | 0.3566074 | 0.5738743 | 0.4089883 | 0.4146452 | 0.5387788 |
(0.5, 0.5) | 0.1217016 | 0 | 0.1120718 | 0.06492248 | 0 |
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Torra, V. Andness Directedness for t-Norms and t-Conorms. Mathematics 2022, 10, 1598. https://doi.org/10.3390/math10091598
Torra V. Andness Directedness for t-Norms and t-Conorms. Mathematics. 2022; 10(9):1598. https://doi.org/10.3390/math10091598
Chicago/Turabian StyleTorra, Vicenç. 2022. "Andness Directedness for t-Norms and t-Conorms" Mathematics 10, no. 9: 1598. https://doi.org/10.3390/math10091598
APA StyleTorra, V. (2022). Andness Directedness for t-Norms and t-Conorms. Mathematics, 10(9), 1598. https://doi.org/10.3390/math10091598