Left (Right)-Quasi Neutrosophic Triplet Loops (Groups) and Generalized BE-Algebras

Abstract

The new notion of a neutrosophic triplet group (NTG) is proposed by Florentin Smarandache; it is a new algebraic structure different from the classical group. The aim of this paper is to further expand this new concept and to study its application in related logic algebra systems. Some new notions of left (right)-quasi neutrosophic triplet loops and left (right)-quasi neutrosophic triplet groups are introduced, and some properties are presented. As a corollary of these properties, the following important result are proved: for any commutative neutrosophic triplet group, its every element has a unique neutral element. Moreover, some left (right)-quasi neutrosophic triplet structures in BE-algebras and generalized BE-algebras (including CI-algebras and pseudo CI-algebras) are established, and the adjoint semigroups of the BE-algebras and generalized BE-algebras are investigated for the first time.

The Decomposition Theorems of AG-Neutrosophic Extended Triplet Loops and Strong AG-(l, l)-Loops

“…Remark 1. In [30,32], the name of neutrosophic triplet loop is used. To be more rigorous with and echo the neutrosophic extended triplet group (NETG), the name of the neutrosophic extended triplet loop (NET-loop) is used in this paper.…”

“…The contrast between the neutrosophic set and the neutrosophic triplet group are as shown in Figure 1. And in [30], sorts of general neutrosophic triplet structures were pointed out, and their basic properties were investigated. For the structure of NETG, some research papers are published with a series of results [31][32][33][34][35].…”

Neutrosophic Triplets in Neutrosophic Rings

“…Related theories of neutrosophic triplet, duplet, and duplet set were developed by Smarandache [18]. Neutrosophic duplets and triplets have fascinated several researchers who have developed concepts such as neutrosophic triplet normed space, fields, rings and their applications; triplets cosets; quotient groups and their application to mathematical modeling; triplet groups; singleton neutrosophic triplet group and generalization; and so on [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]. Computational and combinatorial aspects of algebraic structures are analyzed in [37].…”

On Homomorphism Theorem for Perfect Neutrosophic Extended Triplet Groups

“…For the nature and structure of NTG, recently, some new results have been published: cancellable NTGs are discussed in [17]; several homomorphism theorems of commutative NTGs are proved in [18]; some new properties of NTGs are obtained in [19]; the relationships between generalized NTGs and logical algebraic systems are investigated in [20]. In these papers, the name "neutrosophic triplet group" essentially refers to "neutrosophic extended triplet group", which is illustrated by the authors of [17][18][19][20].…”