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Article

Neutrosophic Triplet Non-Associative Semihypergroups with Application

1
Department of Mathematics & Statistics, Hazara University, Mansehra 21130, Pakistan
2
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi Selangor DE, Malaysia
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Symmetry 2018, 10(11), 613; https://doi.org/10.3390/sym10110613
Submission received: 8 October 2018 / Revised: 23 October 2018 / Accepted: 27 October 2018 / Published: 8 November 2018

Abstract

:
In this paper, we extended the idea of a neutrosophic triplet set to non-associative semihypergroups and define neutrosophic triplet LA -semihypergroup. We discuss some basic results and properties. At the end, we provide an application of the proposed structure in Football.

1. Introduction

The study of origin and features of neutralities lies in the scope of a new branch of philosophy known as Neutrosophy. In 1995, Smarandache (for the first time) used the idea of Neutrosophy and developed neutrosophic logic which is a more practical and realistic approach, to handle imprecise and vague information. He introduced the concept of (T-truth, I-indeterminacy, F-falsity) memberships. According to Smarandache Neutrosophic, logics generalizes the all previous logics such as fuzzy logic [1], intuitionistic fuzzy logic [2] and interval valued fuzzy logic [3]. Kandasamy and Smarandache [4] developed many neutrosophic algebraic structures, neutrosophic bigroup, neutrosophic vector space, neutrosophic groups and so on, based on neutrosophic logic. For practical applications, we refer the readers to [5,6,7,8,9]. For neutrosophic triplet sets, we refer the readers [10,11,12,13]. In 2016, Smarandache and Ali [14] gave the concept of Neutrosophic triplet groups which is a very useful addition in the theory of groups.
Hyperstucture theory was brought-out by Marty [15] in 1934, when he defined hypergroup, set about analyzing their properties and exerted them to a group. Several papers and books have been compiled in this direction, see references [16,17,18]. In 1990, in Greece, a congress was organized by Thomas Vougiouklis on hyperstructure, which was first named algebraic hyper structures and its applications algebraic hyper structures(AHA); however actually was the fourth, because there had been three more congresses in Italy by Corsini, on the same topic but random names. During this congress, Vougiouklis [19] presented the concept of weak structure, presently known as Hv-structure. A number of writers have gone through various aspects of Hv-structure. For instance, references [20,21,22,23,24,25,26,27]. Another book by Davvaz and Fotea in 2007 has been devoted especially to the study of hyperring theory [28].
Kazim and Naseeruddin [29] in 1970, presented the concept of left almost semigroups (LA-semigroups) and shifted the discussion toward non-associative structures. According to them, a groupoid S is called LA -semigroups, if it is satisfies the left invertive law: ( w 1 w 2 ) w 3 = ( w 3 w 2 ) w 1 for all w 1 , w 2 , w 3 S . After that, researchers started working in this direction such as, references [30,31,32] and Yusuf gave the idea of left almost rings [33]. Hila and Dine [34] in 2011, shifted the non-associative structures to non-associative hyperstructures and furnished the idea of LA -semihypergroup, which is generalization of semigroup, semihypergroup, LA -semigroup by using left invertive law with the help of Marty’s hyperoperation.Yaqoob et al. [35] expanded the work of Hila and Dine. Yousafzai et al. in [36] and Amjad et al. [37] tried to generalize different aspects of left almost semihypergroups. The concept of Hv- LA -semigroup was laid by Gulistan et al. [38] in 2015. The idea of partially ordered left almost semihypergroups was developed by Naveed et al. [39] in 2015. Rehman et al. [40], initiated the study of LA -hyperrings and discussed its hyperideals and hypersystems in 2017. Nawaz et al. introduced the concept of left almost semihyperrings [41]. Yaqoob et al. [42] gave the idea of left almost polygroups in 2018.
In this paper, we extended the idea of neutrosophic triplet set to non-associative semihypergroups. We define neutrosophic triplet LA -semihypergroup. In neutrosophic triplet LA -semihypergroup every element “w” has left neut(w) and left anti(w). In neutrosophic triplet LA -semihypergroup left neut(w) of an element “w” may or may not be equal to left identity. We also defined the neutro-homomorphism on LA -semihypergroups. At the end, we present an application of the proposed structure in football.

2. Preliminaries

This section of paper consists of some basic definitions, which are directly used in our work.
Definition 1
([34]). Let H be a non void set and : H H P H be a hyperoperation, where P H is the family non-void subset of H . The pair H , is called hypergroupoid.
For any two non-void subsets W 1 and W 2 of H , then
W 1 W 2 = w 1 W 1 , w 2 W 2 w 1 w 2 .
Definition 2
([34]). An LA -semihypergroup is the hypergroupoid H , with
w 1 w 2 w 3 = w 3 w 2 w 1
for all, w 1 , w 2 , w 3 H . The equation (1) is called left invertive law.
Definition 3
([35]). An element e of an LA-semihypergroup H is called left identity (resp., pure left identity) if for all w 1 H , w 1 e w 1 (resp., w 1 = e w 1 ). An element e of an LA-semihypergroup H is called right identity (resp., pure right identity) if for all w 1 H , w 1 w 1 e (resp., w 1 = e w 1 ). An element e of an LA-semihypergroup H is called identity (resp., pure right identity) if for all w 1 H , w 1 w 1 e e w 1 (resp., w 1 = w 1 e e w 1 ).
Definition 4
([35]). An LA -smihypergroup with pure left identity satisfies the following property
w 1 ( w 2 w 3 ) = w 2 ( w 1 w 3 ) .
Definition 5
([14]). Let N be a non-void set with a binary operationand w 1 N . Then w 1 is said to be neutrosophic triplet if there exist an element n e u t w 1 N such that
w 1 n e u t w 1 = n e u t w 1 w 1 = w 1 ,
where n e u t w 1 is different from unity element. Also there exist a n t i w 1 N such that
w 1 a n t i w 1 = a n t i w 1 w 1 = n e u t w 1 .
If there are more a n t i w 1 s for a given w 1 , one takes that a n t i ( w 1 ) = w 2 that a n t i w 1 in its turn forms a neutrosophic triplet, i.e., there exists n e u t w 2 and a n t i w 2 . We denote the neutrosophic triplet w 1 by w 1 , n e u t w 1 , a n t i w 1 . By n e u t w 1 , we means neutral of w 1 .
Example 1
([14]). Consider Z 6 under multiplication modulo 6 . Then 2 is a neutrosophic triplet, because n e u t 2 = 4 , as 2 × 4 = 8 . Similarly a n t i 2 = 2 because 2 × 2 = 4 . Thus 2 is a neutrosophic triplet, which is denoted by 2 , 4 , 2 . Similarly 4 is a neutrosophic triplet because n e u t a = a n t i 4 = 4 . So 4 is represented by as ( 4 , 4 , 4 ) . 3 is not a neutrosophic triplet as n e u t ( 3 ) = 5 but a n t i ( 3 ) does not exist in Z 6 and 0 is a trivial neutrosophic triplet as n e u t 0 = a n t i 0 = 0 . This is denoted by ( 0 , 0 , 0 ) .

3. Neutrosophic Triplet LA -Semihypergroups

In this section, we defined the neutrosophic triplet LA -semihypergroup and some results on neutrosophic triplet LA -semihypergroup are provided.
Definition 6.
Let H be a non void set with a binary hyperoperationand w 1 H . Then H is called
  • left neutrosophic triplet set if for every w 1 H , there exist n e u t w 1 and a n t i ( w 1 ) such that
    w 1 n e u t w 1 w 1 , n e u t ( w 1 ) a n t i w 1 w 1 .
  • right neutrosophic triplet set if for every w 1 H , there exist n e u t w 1 and a n t i ( w 1 ) such that
    w 1 w 1 n e u t w 1 , n e u t ( w 1 ) w 1 a n t i w 1 .
  • neutrosophic triplet set if for every w 1 H , there exist n e u t w 1 and a n t i ( w 1 ) such that
    w 1 ( n e u t w 1 w 1 ) ( w 1 n e u t w 1 ) , n e u t ( w 1 ) ( a n t i w 1 w 1 ) ( w 1 a n t i w 1 ) .
Definition 7.
Let H be a set with a binary hyperoperationand w 1 H . Then H is called
  • pure left neutrosophic triplet set if for every w 1 H , there exist n e u t w 1 and a n t i w 1 such that
    w 1 = n e u t w 1 w 1 , n e u t ( w 1 ) = a n t i w 1 w 1 .
  • pure right neutrosophic triplet set if for every w 1 H , there exist n e u t w 1 and a n t i ( w 1 ) such that
    w 1 = w 1 n e u t w 1 , n e u t ( w 1 ) = w 1 a n t i w 1 .
  • pure neutrosophic triplet set if for every w 1 H , there exist n e u t w 1 and a n t i ( w 1 ) such that
    w 1 = ( n e u t w 1 w 1 ) ( w 1 n e u t w 1 ) , n e u t w 1 = a n t i w 1 w 1 w 1 a n t i w 1 .
Example 2.
Let H = w 1 , w 2 , w 3 be a se with hyperoperation defined as follows:
Symmetry 10 00613 i001
A Cayley table 1
Here w 1 , w 2 , w 2 , w 2 , w 2 , w 2 and w 3 , w 2 , w 3 are neutrosophic triplets.
Definition 8.
Let H , be a left (resp., right, left pure, right pure) neutrosophic triplet set. Then H is called left (resp., right, left pure, right pure) neutrosophic triplet LA -semihypergroup, if the following conditions are satisfied:
  • H , is well defined.
  • H , satisfies the left invertive law.
Example 3.
Let H = w 1 , w 2 , w 3 , w 4 , w 5 be a set with the hyperoperation defined as follows:
Symmetry 10 00613 i002
A Cayley table 2
Here H , is an LA -semihypergroup, as the element of H satisfies the left invertive law. Here w 1 , w 1 , w 1 , w 2 , w 5 , w 5 , w 3 , w 3 , w 3 , w 4 , w 4 , w 4 and w 5 , w 2 , w 5 are left neutrosophic triplets. Hence H , is a left neutrosophic triplet LA -semihypergroup.
Definition 9.
Let H , be neutrosophic (resp., pure neutrosophic) triplet set. Then H is said to be neutrosophic (resp., pure neutrosophic) triplet LA -semihypergroup, if the following condition are satisfied:
  • H , is a well defined.
  • H , satisfies the left invertive law.
Example 4.
Let H = w 1 , w 2 , w 3 and the hyperoperation defined in the table as follows:
Symmetry 10 00613 i003
A Cayley table 3
Here H , is an LA -semihypergroup, as the element of H satisfies the left invertive law. Here ( w 1 , w 2 , w 1 ) , ( w 2 , w 1 , w 2 ) and ( w 3 , w 3 , w 3 ) are neutrosophic triplets. Hence H , is a neutrosophic triplet LA -semihypergroup.
Remark 1.
Neut( w 2 ) of an element “ w 2 ” is not unique under the hyperoperationin H and depend on elements and hyperoperation. By the Example 4 neut w 2 = w 1 , w 2 . Similarly a n t i ( w 2 ) = w 1 , w 2 of an element “ w 2 ” is not unique and depends on the element and the hyperoperation.
Remark 2.
Left neut of an element is could be different from left identity.
Definition 10.
Let H , be a neutrosophic LA -semihypergroup. An element w 1 H , then there exist pure left n e u t ( w 1 ) such that w 1 = n e u t w 1 w 1 and pure left a n t i ( w 1 ) such that n e u t ( w 1 ) = a n t i w 1 w 1 .
Proposition 1.
Let H , be a pure left neutrosophic triplet LA -semihypergroup with pure left identity. Then w 2 w 1 = w 3 w 1 if and only if
n e u t w 1 w 2 = n e u t w 1 w 3 .
Proof. 
Suppose that w 2 w 3 = w 3 w 1 for w 1 , w 2 , w 3 H . Since H , is a pure left neutrosophic LA semihypergroup, so a n t i w 1 H . Multiply a n t i w 1 to the right side of w 2 w 1 = w 3 w 1
w 2 w 1 a n t i w 1 = w 3 w 1 a n t i w 1 a n t i a w 1 w 2 = a n t i w 1 w 1 w 3 n e u t w 1 w 2 = n e u t w 1 w 3 .
Conversely, let n e u t w 1 w 2 = n e u t w 1 w 3 . Multiply to both right sides by w 1
n e u t w 1 w 2 w 1 = n e u t w 1 w 3 w 1 w 2 n e u t w 1 w 1 = w 3 n e u t w 1 w 1 w 2 w 1 = w 3 w 1 .
This completes the proof. ☐
Proposition 2.
Let H , be a pure right neutrosophic triplet LA -semihypergroup with pure left identity. Then w 2 n e u t w 1 = w 3 n e u t w 1 if w 2 a n t i w 1 = w 3 a n t i ( w 1 ) for all w 1 , w 2 , w 3 H .
Proof. 
Suppose H , is a pure right neutrosophic triplet LA -semihypergroup with pure left identity and w 2 a n t i w 1 = w 3 a n t i ( w 1 ) for w 1 , w 2 , w 3 H . Multiply w 1 to the left side of w 2 a n t i w 1 = w 3 a n t i ( w 1 ) ,
w 1 ( w 2 a n t i w 1 ) = w 1 ( w 3 a n t i ( w 1 ) ) w 2 ( w 1 a n t i w 1 ) = w 3 ( w 1 a n t i ( w 1 ) ) w 2 n e u t ( w 1 ) = w 3 n e u t w 1 ( because n e u t ( w 1 ) = w 1 a n t i w 1 ) .
Therefore,
w 2 n e u t w 1 = w 3 n e u t ( w 1 ) .
 ☐
Proposition 3.
Let H , be a pure right neutrosophic triplet LA -semihypergroup. Then n e u t ( w 1 ) w 2 = n e u t w 1 w 3 if w 2 a n t i w 1 = w 3 a n t i ( w 1 ) for all w 1 , w 2 , w 3 H .
Proof. 
Suppose H , is a pure right neutrosophic triplet LA -semihypergroup and w 2 a n t i w 1 = w 3 a n t i ( w 1 ) for w 1 , w 2 , w 3 H . Multiply w 1 to the right side of w 2 a n t i w 1 = w 3 a n t i ( w 1 ) ,
( w 2 a n t i w 1 ) w 1 = ( w 3 a n t i ( w 1 ) ) w 1 ( w 1 a n t i w 1 ) w 2 = ( w 1 a n t i ( w 1 ) ) w 3 n e u t ( w 1 ) w 2 = n e u t w 1 w 3 ( because n e u t ( w 1 ) = w 1 a n t i w 1 ) .
Therefore,
n e u t ( w 1 ) w 2 = n e u t w 1 w 3 .
 ☐
Theorem 1.
Let H , be a pure right neutrosophic triplet idempotent LA -semihypergroup. Then n e u t ( w 1 ) n e u t w 1 = n e u t w 1 .
Proof. 
Consider n e u t ( w 1 ) n e u t w 1 = n e u t w 1 . Multiply first with w 1 to the right and then again multiply with w 1 to the right, i.e.,
( ( n e u t ( w 1 ) n e u t w 1 ) w 1 ) w 1 = ( n e u t w 1 w 1 ) w 1 ( ( w 1 n e u t w 1 ) n e u t ( w 1 ) ) w 1 = ( w 1 w 1 ) n e u t w 1 ( w 1 n e u t ( w 1 ) ) ( w 1 n e u t w 1 ) = w 1 n e u t w 1 w 1 w 1 = w 1 w 1 = w 1 .
This shows that
n e u t ( w 1 ) n e u t w 1 = n e u t w 1 .
 ☐
Theorem 2.
Let H , be a pure right neutrosophic triplet idempotent LA -semihypergroup with pure left identity. Then n e u t ( w 1 ) a n t i w 1 = a n t i w 1 .
Proof. 
Let H , be a pure right neutrosophic triplet LA -semihypergroup with pure left identity. Multiply w 1 to the left of both side n e u t ( w 1 ) a n t i w 1 = a n t i w 1 , i.e.,
w 1 ( n e u t ( w 1 ) a n t i w 1 = w 1 a n t i w 1 n e u t ( w 1 ) ( w 1 a n t i w 1 ) = n e u t ( w 1 ) n e u t ( w 1 ) n e u t ( w 1 ) = n e u t w 1 n e u t w 1 = n e u t w 1 ( By Theorem 1 )
This shows that
n e u t ( w 1 ) a n t i w 1 = a n t i w 1 .
 ☐
Theorem 3.
Let H , be a pure right neutrosophic triplet LA -semihypergroup. Then
  • n e u t ( w 1 ) n e u t w 2 = n e u t ( w 1 w 2 ) for all w 1 , w 2 H .
  • a n t w 1 a n t i w 2 = a n t i w 1 w 2 for all w 1 , w 2 H .
Proof. 
1. Consider the left hand side n e u t ( w 1 ) n e u t w 2 . Multiply first with w 2 to the right and then again multiply with w 1 to the right, i.e.,
( ( n e u t ( w 1 ) n e u t w 2 ) w 2 ) w 1 = ( ( w 2 n e u t w 2 ) n e u t ( w 1 ) ) w 1 = ( w 2 n e u t ( w 1 ) ) w 1 = ( w 1 n e u t ( w 1 ) ) w 2 = w 1 w 2 .
So
( ( n e u t ( w 1 ) n e u t w 2 ) w 2 ) w 1 = w 1 w 2
Now consider the right side n e u t ( w 1 w 2 ) . Multiply first with w 2 to the right and then again multiply with w 1 to the right, i.e.,
( n e u t ( w 1 w 2 ) w 2 ) w 1 = ( w 1 w 2 ) n e u t ( w 1 w 2 ) = w 1 w 2 .
So
( n e u t ( w 1 w 2 ) w 2 ) w 1 = w 1 w 2
From the Equations (2) and (3) it is clear that n e u t ( w 1 ) n e u t w 2 = n e u t ( w 1 w 2 ) .
2. Consider the left hand side a n t i ( w 1 ) a n t i w 2 . Multiply first with w 2 to the right and then again multiply with w 1 to the right, i.e.,
( ( a n t i ( w 1 ) a n t i w 2 ) w 2 ) w 1 = ( ( w 2 a n t i w 2 ) a n t i ( w 1 ) ) w 1 = ( n e u t ( w 2 ) a n t i ( w 1 ) ) w 1 = ( w 1 a n t i ( w 1 ) ) n e u t ( w 2 ) = n e u t ( w 1 ) n e u t ( w 2 ) = n e u t ( w 1 w 2 ) .
So
( ( a n t i ( w 1 ) a n t i w 2 ) w 2 ) w 1 = n e u t ( w 1 w 2 )
Now consider the right side a n t i ( w 1 w 2 ) . Multiply first with w 2 to the right and then again multiply with w 1 to the right, i.e.,
( a n t i ( w 1 w 2 ) w 2 ) w 1 = ( w 1 w 2 ) a n t i ( w 1 w 2 ) = n e u t ( w 1 w 2 )
So
( a n t i ( w 1 w 2 ) w 2 ) w 1 = n e u t ( w 1 w 2 )
From the Equations (4) and (5) it is clear that
a n t i ( w 1 ) a n t i w 2 = a n t i ( w 1 w 2 ) .
 ☐
In the following example, we show that in a left neutrosophic triplet LA -semihypergroup
n e u t ( w 1 ) n e u t w 2 n e u t ( w 1 w 2 )
and a n t i ( w 1 ) a n t i w 2 a n t i ( w 1 w 2 ) .
Example 5.
Let H = w 1 , w 2 , w 3 , w 4 be a set with the hyperoperation defined as follow
Symmetry 10 00613 i004
A Cayley table 4
All the elements of H satisfies the left invertive law. Here ( w 1 , w 3 , w 4 ) , ( w 2 , w 4 , w 1 ) , ( w 3 , w 1 , w 4 ) and ( w 4 , w 2 , w 3 ) are left neutrosophic triplets. Hence H , is a left neutrosophic triplet LA -semihypergroup. Now
n e u t ( w 1 ) n e u t w 2 n e u t ( w 1 w 2 ) w 3 w 4 n e u t ( { w 2 , w 4 } ) { w 2 , w 3 , w 4 } n e u t ( w 2 ) n e u t ( w 4 ) { w 2 , w 3 , w 4 } { w 2 , w 4 } .
Also
a n t i ( w 1 ) a n t i w 2 a n t i ( w 1 w 2 ) w 4 w 1 a n t i ( { w 2 , w 4 } ) H a n t i ( w 2 ) a n t i ( w 4 ) H { w 1 , w 3 } .
Hence this shows that n e u t ( w 1 ) n e u t w 2 n e u t ( w 1 w 2 ) and a n t i ( w 1 ) a n t i w 2 a n t i ( w 1 w 2 ) .
Theorem 4.
Let H , be a pure left neutrosophic LA -semihypergroup. Then n e u t a n t i w 1 = n e u t w 1 .
Proof. 
Let n e u t a n t i w 1 = n e u t w 1 . If we put a n t i w 1 = w 2 , then
n e u t w 2 = n e u t w 1 . Post multiply by w 2 n e u t w 2 w 2 = n e u t w 1 w 2 w 2 = n e u t w 1 w 2 a n t i ( w 1 ) = n e u t w 1 a n t i ( w 1 ) , as w 2 = a n t i w 1 a n t i w 1 = a n t i w 1 . By Theorem 1 n e u t w 1 a n t i ( w 1 ) = a n t i ( w 1 )
Hence n e u t a n t i w 1 = n e u t w 1 .  ☐
Theorem 5.
Let H , be a pure left neutrosophic LA -semihypergroup. Then a n t i a n t i w 1 = w 1 .
Proof. 
Consider a n t i a n t i w 1 = w 1 . Post multiplying both sides a n t i w 1
a n t i a n t i w 1 a n t i w 1 = w 1 a n t i w 1 n e u t a n t i w 1 = n e u t w 1 w 1 a n t i w 1 n e u t a n t i w 1 = a n t i w 1 w 1 n e u t w 1 by left invertive law n e u t a n t i w 1 = n e u t w 1 n e u t w 1 n e u t a n t i w 1 = n e u t w 1 by Theorem 1 n e u t w 1 n e u t w 1 = n e u t w 1 n e u t w 1 = n e u t w 1 by Theorem 4
Hence a n t i a n t i w 1 = w 1 .  ☐
Definition 11.
Let H , be a neutrosophic LA -semihypergroup and let K be a subset of H . Then, K is called neutrosophic triplet LA -subsemihypergroup, if K itself is a neutrosophic triplet LA -semihypergroup.
Example 6.
Let H = w 1 , w 2 , w 3 , w 4 , w 5 be a set with the hyperoperation defined in the table as follow
Symmetry 10 00613 i005
A Cayley table 5
Here H , is an LA -semihypergroup, because the element of H satisfies the left invertive law. Here ( w 1 , w 1 , w 1 ) , ( w 2 , w 3 , w 3 ) , ( w 3 , w 2 , w 3 ) , ( w 4 , w 4 , w 4 ) and ( w 5 , w 4 , w 4 ) are neutrosophic triplet. Hence H , is a neutrosophic triplet LA -semihypergroup. Let K = w 1 , w 2 , w 3 be subset of H . As K is a neutrosophic LA -semihypergroup under the . Then K is called neutrosophic triplet LA -subsemihypergroup of H .
Lemma 1.
Let K be a non-empty subset of a neutrosophic triplet LA -semihypergroup H . The following are equivalent.
  • K is a neutrosophic triplet LA -semihypergroup.
  • For all w 1 , w 2 K , w 1 w 2 K .
Proof. 
The proof is straightforward. ☐
Definition 12.
Let H 1 , 1 and H 2 , 2 are two neutrosophic triplet LA -semihypergroups. Let f : H 1 H 2 be a mapping. Then f is called neutro-homomorphism if for all w 1 , w 2 H 1 , we have
  • f ( w 1 1 w 2 ) = f w 2 2 f w 2 ,
  • f n e t w 1 = n e u t f w 1 ,
  • f a n t i w 1 = a n t i f w 1 .
Theorem 6.
Let f : H 1 H 2 be a neutro-homomorphism. Where H 1 and H 2 are two neutrosophic triplet LA -semihypergroup. Let
  • The image of f is a neutrosophic triplet LA -subsemihypergroup of H 2 .
  • The inverse image of f is a neutrosophic LA -subsemihypergroup of H 1 .
Proof. 
The proof is straightforward. ☐
Remark 3.
We have the following key points;
  • Every neutrosophic triplet LA -semihypergroup is an LA -semihypergroup, but the reverse may or may not true.
  • In neutrosophic triplet LA -semihypergroup, every element must have a left n e u t . , but in an LA -semihypergroup the left n e u t . of an element may or may not exist.
  • In neutrosophic LA -semihypergroup, every element must have left a n t i . , but in an LA -semihypergroup the element may or may not have semihypergroup.
  • In neutrosophic LA -semihypergroup pure left n e u t . is not equal to pure left Identity.

4. Application

Neutrosophic triplet LA -semihypergroups has many applications in different areas. Here, we present an application of neutrosophic triplet LA -semihypergroup in football. We can use different versions of neut and anti elements like left, right, pure left and pure right that we may see in different situations. The interesting prospect of this newly defined structure is that it is not comutative, so any change from the left and same types of change from the rigth of a certain element may affect the final results with respect to neut and anti.
Consider a Football team; the centre midfield player “ C m ” having a degree of performance d 1 . The players “ C m l 2 ” and “ C m r 1 ” are the midfield player having degree of performance d 1 . Thus using Definition 6, the n e u t C m C m l 2 , C m r 1 . The players “ C m l 1 ” and “ C m r 2 ” are having better degree of performance d 2 , thus using Definition 6, the n e u t C m a n t i C m C m C m a n t i C m = C m l 1 C m C m C m r 2 = d 1 , d 2 Neutrosophic triplet LA -semihypergroup can help the coach to select the players for filling the position in the playground, when a player gets injured. The major advantage of neutrosophic triplet LA -semihypergroup is that if we have a centre mid player and this player has the other players having the same performance on the right side as neut of it and it has one player on the left having better performance than it as shown in the following Figure 1.
If the performance of a player playing on the left side and right side of a centre mid player is equal to performance of a centre mid player then the structure reduces to a duplet structure. Similarly, we can find many applications in different directions.

5. Conclusions

In this paper, we apply the idea of neutrosophic triplet sets at the very useful non-associative hyperstructures, namely LA -semihypergroups. We define neutrosophic triplet set (left, right, pure left, pure right). We discuss some basic results and an application of the proposed structure at the end. In future, we are aiming to extend this idea and give more interesting results.

Author Contributions

All authors contributed equally.

Funding

Universiti Kebangsaan Malaysia- Grant No. GUP-2017-105.

Acknowledgments

We are indebted to Universiti Kebangsaan Malaysia for providing financial support under the grant GUP-2017-105, and authors are highly grateful to referees for their valuable comments and suggestions.

Conflicts of Interest

The authors declare no conflict of interest.

References

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Figure 1. A view of football match.
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Gulistan, M.; Nawaz, S.; Hassan, N. Neutrosophic Triplet Non-Associative Semihypergroups with Application. Symmetry 2018, 10, 613. https://doi.org/10.3390/sym10110613

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Gulistan M, Nawaz S, Hassan N. Neutrosophic Triplet Non-Associative Semihypergroups with Application. Symmetry. 2018; 10(11):613. https://doi.org/10.3390/sym10110613

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Gulistan, Muhammad, Shah Nawaz, and Nasruddin Hassan. 2018. "Neutrosophic Triplet Non-Associative Semihypergroups with Application" Symmetry 10, no. 11: 613. https://doi.org/10.3390/sym10110613

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