A Study of Generalized QL′-Implications
Abstract
:1. Introduction
2. Preliminaries
- (i)
- Strict if it is continuous and strictly decreasing;
- (ii)
- Strong if it is an involution, i.e.,
- (iii)
- Non-filling, if
- (i)
- is a continuous fuzzy negation;
- (ii)
- is a strictly decreasing fuzzy negation.
- (i)
- Idempotent, if
- (ii)
- Positive, if
- (i)
- is strictly decreasing.
- (ii)
- is continuous.
- (iii)
- is strong.
- (i)
- The left neutrality property, if
- (ii)
- The exchange principle, if
- (iii)
- The identity principle, if
- (iv)
- The ordering property, if
- (v)
- The left ordering property, if
- (vi)
- The right ordering property, if
- (i)
- (ii)
3. The Main Results
3.1. GQL-Implications
- (i)
- Proposition 4 gives a necessary but not sufficient condition for a GQL-operation to be a fuzzy implication, when is a non-filling fuzzy negation. Note that is a non-filling fuzzy negation, satisfies (11), but is not a fuzzy implication (see Example 5).
- (ii)
- By Proposition 4, it is obvious that if is a non-filling negation and the pair does not satisfy the law of contradiction (11), i.e., , for some , then the obtained GQL-operation, for any t-conorm S and fuzzy negation , is not a fuzzy implication.
3.2. GQL-Implications and the Left Neutrality Property (20)
3.3. GQL-Implications and the Exchange Principle (21)
- (i)
- Proposition 6 gives us the sufficient condition that if there is an such that , then violates (21).
- (ii)
3.4. GQL-Implications and the Identity Principle (22)
- (i)
- satisfies (22).
- (ii)
- , for any .
3.5. GQL-Implications and the Left Ordering Property (24)
4. Results
5. Discussion
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Grammatikopoulos, D.S.; Papadopoulos, B. A Study of Generalized QL′-Implications. Mathematics 2022, 10, 3742. https://doi.org/10.3390/math10203742
Grammatikopoulos DS, Papadopoulos B. A Study of Generalized QL′-Implications. Mathematics. 2022; 10(20):3742. https://doi.org/10.3390/math10203742
Chicago/Turabian StyleGrammatikopoulos, Dimitrios S., and Basil Papadopoulos. 2022. "A Study of Generalized QL′-Implications" Mathematics 10, no. 20: 3742. https://doi.org/10.3390/math10203742