N-norm and N-conorm in Neutrosophic Logic and Set, and the Neutrosophic Topologies

In this paper we present the N-norms/N-conorms in neutrosophic logic and set as extensions of T-norms/T-conorms in fuzzy logic and set. Also, as an extension of the Intuitionistic Fuzzy Topology we present the Neutrosophic Topologies. 1. Definition of the Neutrosophic Logic/Set: Let T, I, F be real standard or non-standard subsets of ]0, 1[, with sup T = t_sup, inf T = t_inf, sup I = i_sup, inf I = i_inf, sup F = f_sup, inf F = f_inf, and n_sup = t_sup+i_sup+f_sup, n_inf = t_inf+i_inf+f_inf. Let U be a universe of discourse, and M a set included in U. An element x from U is noted with respect to the set M as x(T, I, F) and belongs to M in the following way: it is t% true in the set, i% indeterminate (unknown if it is or not) in the set, and f% false, where t varies in T, i varies in I, f varies in F. Statically T, I, F are subsets, but dynamically T, I, F are functions/operators depending on many known or unknown parameters. 2. In a similar way define the Neutrosophic Logic: A logic in which each proposition x is T% true, I% indeterminate, and F% false, and we write it x(T,I,F), where T, I, F are defined above. 3. As a generalization of T-norm and T-conorm from the Fuzzy Logic and Set, we now introduce the N-norms and N-conorms for the Neutrosophic Logic and Set. We define a partial relation order on the neutrosophic set/logic in the following way: x(T1, I1, F1) ≤ y(T2, I2, F2) iff (if and only if) T1 ≤ T2, I1 ≥ I2, F1 ≥ F2 for crisp components. And, in general, for subunitary set components: x(T1, I1, F1) ≤ y(T2, I2, F2) iff inf T1 ≤ inf T2, sup T1 ≤ sup T2, inf I1 ≥ inf I2, sup I1 ≥ sup I2, inf F1 ≥ inf F2, sup F1 ≥ sup F2.