Neutrosophic extended triplet group is a new algebra structure and is different from the classical group. In this paper, the notion of generalized neutrosophic extended triplet group is proposed and some properties are discussed. In particular, the following conclusions are strictly proved: (1) an algebraic system is a generalized neutrosophic extended triplet group if and only if it is a quasi-completely regular semigroup; (2) an algebraic system is a weak commutative generalized neutrosophic extended triplet group if and only if it is a quasi-Clifford semigroup; (3) for each $n\in Z^{+},n\ge 2,(Z_n,\otimes )$ is a commutative generalized neutrosophic extended triplet group; (4) for each $n\in Z^{+},n\ge 2,(Z_n,\otimes )$ is a commutative neutrosophic extended triplet group if and only if $n=p_1{p}_2\cdots p_m$ , i.e., the factorization of n has only single factor.

Neutrosophic extended triplet group (NETG) is a novel algebra structure and it is different from the classical group. The major concern of this paper is to present the concept of a partially ordered neutrosophic extended triplet group (po-NETG), which is a NETG equipped with a partial order that relates to its multiplicative operation, and consider properties and structure features of po-NETGs. Firstly, in a po-NETG, we propose the concepts of the positive cone and negative cone, and investigate the structure features of them. Secondly, we study the specificity of the positive cone in a partially ordered weak commutative neutrosophic extended triplet group (po-WCNETG). Finally, we introduce the concept of a po-NETG homomorphism between two po-NETGs, construct a po-NETG on a quotient set by providing a multiplication and a partial order, then we discuss some fundamental properties of them.

The research is supported by the National Natural Science Foundation of China (Grant nos. 11501435, 11301316), the Natural Science Program for Basic Research of Shaanxi Province (Grant no. 2015JM1020) and the Fundamental Research Funds for the Centra

This article aims to investigate the Zariski topology on the set of prime ideals of a weak commutative neutrosophic extended triplet group (NETG) N , denoted by Prim ( N ) . First, by giving an equivalent characterization of idempotent weak commutative NETGs, we show that a topological space X is an S S -space if and only if X is homeomorphic to the space Prim ( N ) of some weak commutative NETG. In addition, we prove that there exists an adjunction between the dual category of weak commutative NETGs and the category of S S -spaces. Finally, we further study the categorical relation between idempotent weak commutative NETGs and that of S S -spaces, which leads to a conclusion that the category of idempotent weak commutative NETGs is equivalent to that of commutative idempotent semigroups.