Neutrosophic Triplet Group (revisited) - Florentin Smarandache; Mumtaz Ali (Neutrosophic Sets and Systems, 2019)
We have introduced for the first time the notion of neutrosophic triplet since 2014, which has the form (x, neut(x), anti(x)) with respect to a given binary well-defined law, where neut(x) is the neutral of x, and anti(x) is the opposite of x. Then we define the neutrosophic triplet group (2016), prove several theorems about it, and give some examples. This paper is an improvement and a development of our 2016 published paper. Groups are the most fundamental and rich algebraic structure with respect to some binary operation in the study of algebra. In this paper, for the first time, we introduced the notion of neutrosophic triplet, which is a collection of three elements that satisfy certain axioms with respect to a binary operation. These neutrosophic triplets highly depend on the defined binary operation. Further, in this paper, we used these neutrosophic triplets to introduce the innovative notion of neutrosophic triplet group, which is a completely different from the classical group in the structural properties. A big advantage of neutrosophic triplet is that it gives a new group (neutrosophic triplet group) structure to those algebraic structures, which are not group with respect to some binary operation in the classical group theory. In neutrosophic triplet group, we apply the fundamental law of Neutrosophy that for an idea A, we have the neutral of A denoted as neut(a) and the opposite of A dented as anti(A) to capture this beautiful picture of neutrosophic triplet group in algebraic structures. We also studied some interesting properties of this newly born structure. We further defined neutro-homomorphisms for neutrosophic triplet groups. A neutro-homomorphism is the generalization of the classical homomorphism with two extra conditions. As a further generalization, we gave rise to a new field or research called Neutrosophic Triplet Structures (such as neutrosophic triplet ring, neutrosophic triplet field, neutrosophic triplet vector space, etc.). In the end, we gave main distinctions and comparison of neutrosophic triplet group with the Molaei’s generalized group as well as the possible application areas of the neutrosophic triplet groups. In this paper we improve our [13] results on neutrosophic triplet groups.