Fundamental Homomorphism Theorems for Neutrosophic Extended Triplet Groups

symmetry (moPt, Article Fundamental Homomorphism Theorems for Neutrosophic Extended Triplet Groups Mehmet Celik, Moges Mekonnen Shalla and Necati Olgun *© Department of Mathematics, Gaziantep University, Gaziantep 27310, Turkey; mathcelik@gmail.com (M.C.); ms09268@mail2.gantep.edu.tr (M.M.S.) * Correspondence: olgun@gantep.edu.tr; Tel.: +90-536-321-4006 Gg check for Received: 17 July 2018; Accepted: 31 July 2018; Published: 3 August 2018 updates Abstract: In classical group theory, homomorphism and isomorphism are significant to study the relation between two algebraic systems. Through this article, we propose neutro-homomorphism and neutro-isomorphism for the neutrosophic extended triplet group (NETG) which plays a significant role in the theory of neutrosophic triplet algebraic structures. Then, we define neutro-monomorphism, neutro-epimorphism, and neutro-automorphism. We give and prove some theorems related to these structures. Furthermore, the Fundamental homomorphism theorem for the NETG is given and some special cases are discussed. First and second neutro-isomorphism theorems are stated. Finally, by applying homomorphism theorems to neutrosophic extended triplet algebraic structures, we have examined how closely different systems are related. Keywords: neutro-monomorphism; neutro-epimorphism; neutro-automorphism; fundamental neutro-homomorphism theorem; first neutro-isomorphism theorem; and second neutro -isomorphism theorem 1. Introduction Groups are finite or infinite set of elements which are vital to modern algebra equipped with an operation (such as multiplication, addition, or composition) that satisfies the four basic axioms of closure, associativity, the identity property, and the inverse property. Groups can be found in geometry studied by “Felix klein in 1872” [1], characterizing phenomenality like symmetry and certain types of transformations. Group theory, firstly introduced by “Galois” [2], with the study of polynomials has applications in physics, chemistry, and computer science, and also puzzles like the Rubik’s cube as it may be expressed utilizing group theory. Homomorphism is both a monomorphism and an epimorphism maintaining a map between two algebraic structures of the same type (such as two groups, two rings, two fields, two vector spaces) and isomorphism is a bijective homomorphism defined as a morphism, which has an inverse that is also morphism. Accordingly, homomorphisms are effective in analyzing and calculating algebraic systems as they enable one to recognize how intently distinct systems are associated. Similar to the classical one, neuro-homomorphism is the transform between two neutrosophic triplet algebraic objects N and H. That is, if elements in N satisfy MWR some algebraic equation involving binary operation “*”, their images in H satisfy the same algebraic equation. A neutro-isomorphism identifies two algebraic objects with one another. The most common use of neutro-homomorphisms and neutro-isomorphisms in this study is to deal with homomorphism theorems which allow for the identification of some neutrosophic triplet quotient objects with certain other neutrosophic triplet subgroups, and so on. The neutrosophic logic and a neutrosophic set, firstly made known by Florentin Smarandache [3] in 1995, has been widely applied to several scientific fields. This study leads to a new direction, exploration, path of thinking to mathematicians, engineers, computer scientists, and Symmetry 2018, 10, 321; doi:10.3390/sym10080321 www.mdpi.com/journal/symmetry Symmetry 2018, 10, 321 2 of 14 many other researchers, so the area of study grew extremely and applications were found in many areas of neutrosophic logic and sets such as computational modelling [4], artificial intelligence [5], data mining [6], decision making problems [7], practical achievements [8], and so forth. Florentin Smarandache and Mumtazi Ali investigated the neutrosophic triplet group and neutrosophic triplet as expansion of matter plasma, nonmatter plasma, and antimatter plasma [9,10]. By using the concept of neutrosophic theory Vasantha and Smarandache introduced neutrosophic algebraic systems and N-algebraic structures [11] and this was the first neutrosofication of algerbraic structures. The characterization of cancellable weak neutrosophic duplet semi-groups and cancellable NTG are investigated [12] in 2017. Florentin Smarandache and Mumtaz Ali examined the applications of the neutrosophic triplet field and neutrosophic triplet ring [13,14] in 2017. Sahin Mehmet and Abdullah Kargin developed the neutrosophic triplet normed space and neutrosophic triplet inner product [15,16]. The neutrosophic triplet G-module and fixed point theorem for NT partial metric space are given in literature [17,18]. Similarity measures of bipolar neutrosophic sets and single valued triangular neutrosophic numbers and their appliance to multi-attribute group decision making investigated in [19,20]. By utilizing distance-based sim