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

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 ( N 1 ∗ , O ( 1 ) , N 1 , G ( 1 ) ) - or ( N 2 ∗ , O ( 2 ) , N 2 , G ( 2 ) ) -implications constructed from the tuples ( N 1 ∗ , O ( 1 ) , N 1 , G ( 1 ) ) or ( N 2 ∗ , O ( 2 ) , N 2 , G ( 2 ) ) satisfy a certain property P, we now investigate whether circledcircG,N-composition of ( N 1 ∗ , O ( 1 ) , N 1 , G ( 1 ) ) - and ( N 2 ∗ , O ( 2 ) , N 2 , G ( 2 ) ) -implications satisfies the same property or not. If not, then we attempt to characterise those implications ( N 1 ∗ , O ( 1 ) , N 1 , G ( 1 ) ) -, ( N 2 ∗ , O ( 2 ) , N 2 , G ( 2 ) ) -implications satisfying the property P such that circledcircG,N-composition of ( M 1 ∗ , O ( 1 ) , M 1 , G ( 1 ) ) - and ( M 2 ∗ , O ( 2 ) , M 2 , 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.