Abstract
In this paper, we introduce and study the GD′-operations, which are a hyper class of the known D′-operations. GD′-operations are in fact D′-operations, that are generated not only from the same fuzzy negation. Similar with D′-operations, they are not always fuzzy implications. Nevertheless, some sufficient, but not necessary conditions for a GD′-operation to be a fuzzy implication, will be proved. A study for the satisfaction, or the violation of the basic properties of fuzzy implications, such as the left neutrality property, the exchange principle, the identity principle and the ordering property will also be made. This study also completes the study of the basic properties of D′-implications. At the end, surprisingly an unexpected new result for the connection of the QL-operations and D-operations will be presented.
Highlights
The transition from classical to fuzzy logic achieved especially from generalizations of classical tautologies
Different models, or construction methods are always necessary, since they produce new fuzzy implication functions that can be adequate in specific applications
There are many definitions in fuzzy logic that are generalizations of classical tautologies. Such a generalization are D -implications which are the generalization of the classical tautology (1)
Summary
The transition from classical to fuzzy logic achieved especially from generalizations of classical tautologies. Many definitions of fuzzy connectives, classes of fuzzy implications and properties of them are such generalizations [1,2,3,4,5,6,7,8,9]. Many applications have been constructed by using fuzzy implications [9,10,11]. Fuzzy implications are applied in many different areas, such as approximate reasoning, decisionmaking theories, control theories and expert systems, image processing, fuzzy mathematical morphology, robotics, and others. Mas et al in [9] (page 1107) remarked that, the necessity of different classes of fuzzy implications is because they are used to representing imprecise knowledge. Different models, or construction methods are always necessary, since they produce new fuzzy implication functions that can be adequate in specific applications
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