Neutrosophic triplet group

Abstract

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 group of three elements that satisfy certain properties with some binary operation. These neutrosophic triplets highly depends on the defined binary operation. Further, in this paper, we utilized these neutrosophic triplets to introduce the innovative notion of neutrosophic triplet group which is 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 neutral of A denoted as neut(a) and anti of A denoted 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 neutron-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 classical Molaei’s generalized group as well as the possible application areas of the neutrosophic triplet groups.

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Acknowledgments

There is no conflict of interest in the manuscript. We are very thankful to Prof. Muhammad Zafarullah from the USA, for his valuable comments and suggestion which improve this paper.

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Correspondence to Florentin Smarandache.

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Smarandache, F., Ali, M. Neutrosophic triplet group. Neural Comput & Applic 29, 595–601 (2018). https://doi.org/10.1007/s00521-016-2535-x

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Keywords

  • Groups
  • Homomorphism
  • Neutrosophic triplet
  • Neutrosophic triplet group
  • Neutro-homomorphism