# New Results on Neutrosophic Extended Triplet Groups Equipped with a Partial Order
## 2 Preliminaries  
In this section, we recall some basic notions and results which will be used in this paper as indicated below.  
**Definition 1**: ([13]) _Let \(G\) be a non-empty set together with a binary operation \(*\). Then \(G\) is called a neutrosophic extended triplet set if for any \(a\in G\), there exist a neutral of \"\(a\)\" (denoted by \(neut(a)\)) and an opposite of \"\(a\)\" (denoted by \(anti(a)\)), such that \(neut(a)\in G,\ anti(a)\in G,\) and_  
\[a*neut(a)=neut(a)*a=a;\]  
\[a*anti(a)=anti(a)*a=neut(a).\]  
_The triplet \((a,\ neut(a),\ anti(a))\) is called a neutrosophic extended triplet._  
**Definition 2**: ([13]) _Let \((G,*)\) be a neutrosophic extended triplet set. If \((G,*)\) is a semigroup, then \(G\) is called a neutrosophic extended triplet group (for short, NETG)._  
**Proposition 1**: _([[16] Theorems 1 and 2])  Let \((G,*)\) be a NETG. The following properties hold: \(\forall a\in G\)_  
\((1)\)__\(neut(a)\) _is unique;_  
\((2)\)__\(neut(a)*neut(a)=neut(a)\)_;_  
\((3)\)__\(neut(neut(a))=neut(a)\)_._  
_Notice that \(anti(a)\) may be not unique for every element \(a\) in a NETG \((G,*)\). To avoid confusion, we use the following notations:_  
\(anti(a)\) _denotes any certain one opposite of_ \(a\) _and_ \(\{anti(a)\}\) _denotes the set of all opposites of_ \(a\)_._  
**Proposition 2**: _([[19], Theorem 1])  Let \((G,*)\) be a NETG. The following properties hold: \(\forall\ a\in G,\ \forall\ p,\ q\in\{anti(a)\}\)_  
\((1)\)__\(p*neut(a)\in\{anti(a)\}\)_;_  
\((2)\)__\(p*neut(a)=q*neut(a)=neut(a)*q\)_;_  
\((3)\)__\(neut(p*neut(a))=neut(a)\)_;_  
\((4)\)__\(a\in\{anti(p*neut(a))\}\)_;_  
\((5)\)__\(anti(p*neut(a))*neut(p*neut(a))=a\)_._  
**Definition 3**.: _([16]) Let \((G,*)\) be a NETG. If a \(*\)\(neut(b)=neut(b)*a\)\((\forall a\in G,\ \forall b\in G)\), then \(G\) is called a weak commutative neutrosophic extended triplet group ( WCNETG)._  
**Proposition 3**.: _([16], Theorem 2) Let \((G,*)\) be a NETG. Then \(G\) is a WCNETG iff \(G\) satisfies the following conditions: \(\forall a\in G,\ \forall b\in G\)_  
1. \(neut(a)*neut(b)=neut(b)*neut(a)\)_;_
2. \(neut(a)*neut(b)*a=a*neut(b)\)_._  
**Proposition 4**.: _([16], Theorem 3]) Let \((G,*)\) be a WCNETG. The following properties hold: \(\forall a\in G,\ \forall b\in G\)_  
1. \(neut(a)*neut(b)=neut(b*a)\)_;_
2. \(anti(a)*anti(b)\in\{anti(b*a)\}\)_._  
**Definition 4**.: _([29]) A semigroup \((S,*)\) will be called completely regular if there exists a unary operation \(a\mapsto a^{-1}\) on \(S\) with the properties:_  
\[(a^{-1})^{-1}=a,\ a*a^{-1}*a=a,\ a*a^{-1}=a^{-1}*a.\]  
**Proposition 5**.: _([19], Theorem 2) Let \((G,*)\) be a groupoid. Then \(G\) is a NETG iff it is a completely regular semigroup._  
**Note 1.** In semigroup theory, \(a^{-1}\) is called the inverse element of \(a\) and it is unique. However, in a NETG, \(anti(a)\) is called an opposite element of \(a\) and it may not be unique. From Proposition 5, we get that for arbitrary element \(a\) of a NETG \((G,*)\), if we define a unary operation \(a\mapsto a^{-1}\) by \(a^{-1}=anti(a)*neut(a)\), then \((G,*)\) is a completely regular semigroup.  
In the following, we will regard all NETGs as completely regular semigroups, in which \(a^{-1}=anti(a)*neut(a)\) for arbitrary element \(a\). Then by Proposition 2, we have in a NETG \((G,*)\), for each \(a\in G,\ a^{-1}\in\{anti(a)\}\) and \(a^{-1}*a=a*a^{-1}=neut(a)\).