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 $\left(N_1^{\ast },O^{\left(1\right)},N_1,G^{\left(1\right)}\right)$ - or $\left(N_2^{\ast },O^{\left(2\right)},N_2,G^{\left(2\right)}\right)$ -implications constructed from the tuples $(N_1^{\ast },O^{\left(1\right)},N_1,G^{\left(1\right)})$ or $\left(N_2^{\ast },O^{\left(2\right)},N_2,G^{\left(2\right)}\right)$ satisfy a certain property P, we now investigate whether _{circledcircG,N}-composition of $\left(N_1^{\ast },O^{\left(1\right)},N_1,G^{\left(1\right)}\right)$ - and $\left(N_2^{\ast },O^{\left(2\right)},N_2,G^{\left(2\right)}\right)$ -implications satisfies the same property or not. If not, then we attempt to characterise those implications $\left(N_1^{\ast },O^{\left(1\right)},N_1,G^{\left(1\right)}\right)$ -, $\left(N_2^{\ast },O^{\left(2\right)},N_2,G^{\left(2\right)}\right)$ -implications satisfying the property P such that _{circledcircG,N}-composition of $\left(M_1^{\ast },O^{\left(1\right)},M_1,G^{\left(1\right)}\right)$ - and $\left(M_2^{\ast },O^{\left(2\right)},M_2,G^{\left(2\right)}\right)$ -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 R _O -implications (residual implication), (G, N)-implications, QL-implications, D-implications and (N^∗, O, N, G)-implications.