A new class of fuzzy implications derived from non associative structures and its characterizations

Article type: Research Article

Authors: Park, Choonkil^a | Rehman, Noor^{b; *} | Ali, Abbas^c | Alahmadi, Reham A.^d | Khalaf, Mohammed M.^e | Hila, Kostaq^f

Affiliations: [a] Department of Mathematics, Research Institute of Natural Sciences, Hanyang University, Seoul, Republic of Korea | [b] Department of Mathematics & Statistics, Bacha Khan University, Charsadda, Khyber Pakhtunkhwa, Pakistan | [c] Department of Mathematics & Statistics, Riphah International University, Islamabad, Pakistan | [d] Basic Sciences Department, College of Science and Theoretical Studies, Saudi Electronic University, Riyadh, Kingdom of Saudi Arabia | [e] Department of Mathematics, Higher Institute of Engineering and Technology, King Marriott, Egypt | [f] Department of Mathematical Engineering, Polytechnic University of Tirana, Tirana, Albania

Correspondence: [*] Corresponding author. Noor Rehman, Department of Mathematics & Statistics, Bacha Khan University, Charsadda, Khyber Pakhtunkhwa, Pakistan. E-mail: [email protected].

Abstract: In clasical logic, it is possible to combine the uniary negation operator ¬ with any other binary operator in order to generate the other binary operators. In this paper, we introduce the concept of (N∗, O, N, G)-implication derived from non associative structures, overlap function O, grouping function G and two different fuzzy negations N∗ and N are used for the generalization of the implication p → q ≡ ¬ [p ∧ ¬ (¬ p ∨ q)] . We show that (N∗, O, N, G)-implication are fuzzy implication without any restricted conditions. Further, we also study that some properties of (N∗, O, N, G)-implication that are necessary for the development of this paper. The key contribution of this paper is to introduced the concept of circledcircG,N-compositions on (N∗, O, N, G)-implications. If (N1∗,O(1),N1,G(1)) - or (N2∗,O(2),N2,G(2)) -implications constructed from the tuples (N1∗,O(1),N1,G(1)) or (N2∗,O(2),N2,G(2)) satisfy a certain property P, we now investigate whether circledcircG,N-composition of (N1∗,O(1),N1,G(1)) - and (N2∗,O(2),N2,G(2)) -implications satisfies the same property or not. If not, then we attempt to characterise those implications (N1∗,O(1),N1,G(1)) -, (N2∗,O(2),N2,G(2)) -implications satisfying the property P such that circledcircG,N-composition of (M1∗,O(1),M1,G(1)) - and (M2∗,O(2),M2,G(2)) -implications also satisfies the same property. Further, we introduced sup-circledcircO-composition of (N∗, O, N, G)-implications constructed from tuples (N∗, O, N, G) . Subsequently, we show that under which condition sup-circledcircO-composition of (N∗, O, N, G)-implications are fuzzy implication. We also study the intersections between families of fuzzy implications, including RO-implications (residual implication), (G, N)-implications, QL-implications, D-implications and (N∗, O, N, G)-implications.

Keywords: Overlape function, grouping function, fuzzy implication, fuzzy negation

DOI: 10.3233/JIFS-222878

Journal: Journal of Intelligent & Fuzzy Systems, vol. 45, no. 3, pp. 4949-4977, 2023