Neutrosophic triplet group

Original Article
  • 81 Downloads

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.

Keywords

Groups Homomorphism Neutrosophic triplet Neutrosophic triplet group Neutro-homomorphism 

References

  1. 1.
    Atanassov TK (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20:87–96MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Dummit DS, Foote RM (2004) Abstract algebra, 3rd edn. John Viley & Sons Inc, New JerseyMATHGoogle Scholar
  3. 3.
    Herstein IN (1975) Topics in algebra. Xerox college publishing, LexingtonMATHGoogle Scholar
  4. 4.
    Kandasamy WBV, Smarandache F (2006) Some neutrosophic algebraic structures and neutrosophic n-algebraic structures. Hexis, Frontigan, p 219MATHGoogle Scholar
  5. 5.
    Kandasamy WBV, Smarandache F (2006) N-algebraic structures and s-n-algebraic structures. Hexis, Phoenix, p 209MATHGoogle Scholar
  6. 6.
    Kandasamy WBV, Smarandache F (2004) Basic neutrosophic algebraic structures and their applications to fuzzy and neutrosophic models. Hexis, Frontigan, p 149MATHGoogle Scholar
  7. 7.
    Molaei MR (1999) Generalized groups. Bul Inst Politehn Ia, si Sect I 45(49):21–24MathSciNetMATHGoogle Scholar
  8. 8.
    Smarandache F (1999) A unifying field in logics. Neutrosophy: neutrosophic probability, set and logic. Rehoboth: American Research PressGoogle Scholar
  9. 9.
    Smarandache F (2006) Neutrosophic set, a generalization of the intuitionistic fuzzy set, In: 2006 IEEE international conference on granular computing, 10–12 May 2006, pp 38–42. doi:10.1109/GRC.2006.1635754
  10. 10.
    Smarandache F, Ali M (2008) Neutrosophic triplet as extension of matter plasma, unmatter plasma, and antimatter plasma. In: 69th annual gaseous electronics conference, Bochum, Germany, Veranstaltungszentrum & Audimax, Ruhr-Universitat, 10–14 Oct 2016, http://meetings.aps.org/Meeting/GEC16/Session/HT6.112
  11. 11.
    Surowski DB (1995) The uniqueness aspect of the fundamental theorem of finite Abelian groups. Amer Math Monthly 102:162–163MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Zadeh AL (1965) Fuzzy sets. Inform Control 8:338–353MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The Natural Computing Applications Forum 2016

Authors and Affiliations

  1. 1.University of New MexicoGallupUSA
  2. 2.Department of MathematicsQuaid-i-Azam UniversityIslamabadPakistan

Personalised recommendations