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The \(\circledast\)-composition of fuzzy implications: closures with respect to properties, powers and families. (English) Zbl 1373.03037

Summary: Recently, N. R. Vemuri and B. Jayaram [ibid. 247, 51–67 (2014; Zbl 1334.03028)] proposed a novel method of generating fuzzy implications from a given pair of fuzzy implications. Viewing this as a binary operation \(\circledast\) on the set \(\mathbb{I}\) of fuzzy implications they obtained, for the first time, a monoid structure \((\mathbb{I}, \circledast)\) on the set \(\mathbb{I}\). Some algebraic aspects of \((\mathbb{I}, \circledast)\) had already been explored and hitherto unknown representation results for the Yager’s families of fuzzy implications were obtained in [the authors, ibid. 247, 51–67 (2014; Zbl 1334.03028)]. However, the properties of fuzzy implications generated or obtained using the \(\circledast\)-composition have not been explored. In this work, the preservation of the basic properties like neutrality, ordering and exchange principles, the functional equations that the obtained fuzzy implications satisfy, the powers with reference to \(\circledast\) and their convergence, and the closures of some families of fuzzy implications with reference to the operation \(\circledast\), specifically the families of \((S, N)\)-, \(R\)-, \(f\)- and \(g\)-implications, are studied. This study shows that the \(\circledast\)-composition carries over many of the desirable properties of the original fuzzy implications to the generated fuzzy implications and further, due to the associativity of the \(\circledast\)-composition one can obtain, often, infinitely many new fuzzy implications from a single fuzzy implication through self-composition with reference to the \(\circledast\)-composition.

MSC:

03B52 Fuzzy logic; logic of vagueness
68T37 Reasoning under uncertainty in the context of artificial intelligence

Citations:

Zbl 1334.03028
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References:

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