On the Definition of Atanassov's Intuitionistic Fuzzy Subrings and Ideals

Saifur Rahman ; Helen K. Saikia ; B. Davvaz

Bulletin of the Malaysian Mathematical Society, Tome 36 (2013) no. 2

Résumé

On the basis of the concept of grades of a fuzzy point to belongingness ($\in$) or quasi-coincident ($q$) or belongingness and quasi-coincident ($\in \wedge q$) or belongingness or quasi-coincident ($\in \vee q$) in an intuitionistic fuzzy set of a ring, the notion of a ($\alpha, \beta$)-intuitionistic fuzzy subring and ideal is introduced by applying the Lukasiewicz 3-valued implication operator. Using the notion of fuzzy cut set of an intuitionistic fuzzy set, the support and $\alpha$-level set of an intuitionistic fuzzy set are defined and it is established that, for $ \alpha \neq \in \wedge q$, the support of a ($\alpha, \beta$)-intuitionistic fuzzy ideal of a ring is an ideal of the ring. It is also established that the level sets of an intuitionistic fuzzy ideal with thresholds ($s, t$) of a ring is an ideal of the ring. We investigate that an intuitionistic fuzzy set $A$ of a ring is a ($\in, \in$) (or ($\in, \in \vee q$ ) or ($\in \wedge q, \in $) )-intuitionistic fuzzy ideal of the ring if and only if $A$ is an intuitionistic fuzzy ideal with thresholds ($0,1$) (or ($0,0.5$) or ($0.5, 1$)) of the ring respectively. We also establish that $A$ is a ($\in, \in$) (or ($\in, \in \vee q$ ) or ($\in \wedge q, \in $) )-intuitionistic fuzzy ideal of the ring if and only if for any $a \in (0,1] $ (or $ a \in (0,0.5 ]$ or $a \in (0.5,1]$ ), $A_a$ is a fuzzy ideal of the ring. Finally, we investigate that an intuitionistic fuzzy set of a ring is an intuitionistic fuzzy ideal with thresholds ($s,t$) of the ring if and only if for any $a \in ( s,t]$, the cut set $A_a$ is a fuzzy ideal of $R$.

Détail

Classification : 08A72, 16D25