Neutrosophic Triplet Cosets and Quotient Groups

SS symmetry Article

Neutrosophic Triplet Cosets and Quotient Groups Mikail Bal, Moges Mekonnen Shalla and Necati Olgun *

ID

Faculty of Arts and Sciences, Department of Mathematics, Gaziantep University, 27310 Gaziantep, Turkey; mikailbal46@hotmail.com (M.B.); moges6710@gmail.com (M.M.S.) * Correspondence: olgun@gantep.edu.tr; Tel.: +90-536-321-4006 Received: 29 March 2018; Accepted: 17 April 2018; Published: 20 April 2018

Abstract: In this paper, by utilizing the concept of a neutrosophic extended triplet (NET), we define the neutrosophic image, neutrosophic inverse-image, neutrosophic kernel, and the NET subgroup.

The notion of the neutrosophic triplet coset and its relation with the classical coset are defined and the properties of the neutrosophic triplet cosets are given. Furthermore, the neutrosophic triplet normal subgroups, and neutrosophic triplet quotient groups are studied.

Keywords: neutosophic extended triplet subgroups; neutrosophic triplet cosets; neutrosophic triplet normal subgroups; neutrosophic triplet quotient groups

1. Introduction Neutrosophy was first introduced by Smarandache (Smarandache, 1999, 2003) as a branch of philosophy, which studied the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra: (A) is an idea, proposition, theory, event, concept, or entity; anti(A) is the opposite of (A); and (neut-A) means neither (A) nor anti(A), that is, the neutrality in between the two extremes. A notion of neutrosophic set theory was introduced by Smarandache in [1]. By using the idea of the neutrosophic theory, Kandasamy and Smarandache introduced neutrosophic algebraic structures in [2,3]. The neutrosophic triplets were first introduced by Florentin Smarandache and Mumtaz Ali [4–10], in 2014–2016. Florentin Smarandache and Mumtaz Ali introduced neutrosophic triplet groups in [6,11]. A lot of researchers have been dealing with neutrosophic triplet metric space, neutrosophic triplet vector space, neutrosophic triplet inner product, and neutrosophic triplet normed space in [12–22].

A neutrosophic extended triplet, introduced by Smarandache [7,20] in 2016, is defined as the neutral of x (denoted by eneut( x) and called “extended neutral”), which is equal to the classical algebraic unitary element (if any). As a result, the “extended opposite” of x (denoted by e anti( x) ) is equal to the classical inverse element from a classical group. Thus, the neutrosophic extended triplet (NET)

has a form x, eneut( x) , e anti( x) for x ∈ N, where eneut( x) ∈ N is the extended neutral of x. Here, the neutral element can be equal to or different from the classical algebraic unitary element, if any, such that: x ∗ eneut( x) = eneut( x) ∗ x = x, and e anti( x) ∈ N is the extended opposite of x, where x ∗ e anti( x) = e anti( x) ∗ x = eneut( x) . Therefore, we used NET to define these new structures.

In this paper, we deal with neutosophic extended triplet subgroups, neutrosophic triplet cosets, neutrosophic triplet normal subgroups, and neutrosophic triplet quotient groups for the purpose to develop new algebraic structures on NET groups. Additionally, we define the neutrosophic triplet image, neutrosophic triplet kernel, and neutrosophic triplet inverse image. We give preliminaries and results with examples in Section 2, and we introduce neutrosophic extended triplet subgroups in Section 3. Section 4 is dedicated to introduing neutrosophic triplet cosets, with some of their properties, and we show that neutrosophic triplet cosets are different from classical cosets. In Section 5, we introduce neutrosophic triplet normal subgroups and the neutrosophic triplet normal subgroup Symmetry 2018, 10, 126; doi:10.3390/sym10040126

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10 of 13 groups and we examine the relationships test. In Section 6, we define the neutrosophic triplet quotient of these structures with each other. In Section 7, we provide some conclusions. sphic extended triplet subgroup of N. Suppose H is neutrosophic extended triplet ∈ N, y ∈ H : Эz ∈2.HPreliminaries : xy = zx. Thus (xy)anti(x) = z ∈ H implying (xH)anti(x) ⊆

In this section, the definition of neutrosophic triplets, NET’s, and the concepts of NET groups have been outlined. e ⩝ x ∈ N :(xH)anti(x) ⊆H. Then for n ∊ N , w e have (nH )anti(n) ⊆H , w hich o, for anti(n) ∊ N2.1. , w eNeutrosophic have anti(n)H (anti[anti{n}]) = anti(n)Hn ⊆ H, which Triplet efore, nH = H n, meaning that H  N. Let U be a universe of discourse, and (N, ∗) a set included in it, endowed with a well-defined binary law ∗.from a neutrosophic extended triplet group N to a → H be a neutro-homomorphism

let group H, Kerf

 N.

Definition 1 ([1–3]). A neutrosophic triplet has a form (x, neut(x), anti(x)), for x in N, where neut(x) and ad to show that a(anti[b]) meant kerf was neutrosophic extended anti(x)∈ kerf.

N areThis neutral andthat opposite to x,a which are different from the classical algebraic unitary element, if any, . If a ∈ kerf, then such that: x ∗ neut( x ) = neut( x ) ∗ x = x and x ∗ anti ( x ) = anti ( x ) ∗ x = neut( x ), respectively. In general, x may have more than one neut’s and anti’s. f(a) = neutH 2.2. NET

Definition 2 ([4,7]). A neutrosophic extended triplet is a neutrosophic triplet, as defined in Definition 1, where neutral of x (denoted by eneut( x) and called extended neutral) is equal to the classical algebraic f(b)the = neut H unitary element, if any. As a consequence, the extended opposite of x (denoted by e anti( x) ) is also equal to

f(a(anti[b]) = neutH. (f is neutro-homomorphism) neut ( x ) anti ( x ) the classical inverse element from a classical group. Thus, an NET has a form x, e , for x , e nti(b)) ∈ N, where eneut(x) and e anti(x) in N are the extended neutral and opposite of x, respectively, such that: x ∗ eneut( x) = eneut( x) ∗ x = x, which can be equal to or different from the classical algebraic unitary element, if any, and x ∗ e anti( x) = e anti( x) ∗ x = eneut( x) . In general, for each x ∈ N there are many eneut( x) ’s and e anti( x) ’s.

f.

Definition 3 ([1–3]). The element y in ( N, ∗) is the second coordinate of a neutrosophic extended triplet ∊ kerf. We had to show that nas . a neut(y) . (anti(n)) kerf. (f is neutro-homomorphism) (denoted of ∈ a neutrosophic triplet), if there are other elements exist, x and z ∈ N such that: ) . f(a) . f(anti(n)) x ∗ y = y ∗ x = x and x ∗ z = z ∗ x = y. The formed neutrosophic triplet is (x, y, z). The element z ∈ ( N, ∗), as the third coordinate, can be defined in the same way.

∈ kerf

Example 1. Let X = (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11), enclosed with the classical multiplication law, (x) modulo 12, which is well defined on X, with the classical unitary element 1. X iss an NET “weak commutative set” see “Table 1”.

Table 1. Neutrosophic triplets of (x) modulo 12.

hic triplet subgroup H of N is a neutrosophic triplet normal subgroup of N if, and hic triplet coset of H in ∗N is a right of H4∈ N. 5 0 neutrosophic 1 2triplet coset 3 6 0

0

0

0

0

0

0

0

phic triplet normal subgroup of N, then xH(anti[x])=H, ⩝ x ∈ N ⇒ xH(anti[x])x 1 0 1 2 3 4 5 6 H x, ⩝x ∈ N, since each2 left neutrosophic neutrosophic 0 2 triplet 4coset xH6 is the right 8 10 0

3 0 3 6 9 0 3 6 4 0 4 8 0 4 8 0 5 0 of H in 5 N be a 10right neutrosophic 3 8 1 coset 6 h left neutrosophic triplet coset triplet 6 0 6 0 6 0 6 0 at if x is any element of N, then the left neutrosophic triplet coset xH is also a 7 0 7 2 9 4 11 6 et coset. Now neut(x)8∈ H, therefore x0 0 8 x ∗ neu4t(x) = x0 ∈ xH.8Consequently 4 t right neutrosophic triplet coset, which is equal to 3left neutrosophic 9 0 9 6 0 9 triplet 6 10 0 and10needs to 8 contain 6 one common 4 2element 0 a left neutrosophic triplet coset 11 0 11 10 9 8 7 6

. Therefore, Hx is the unique right neutrosophic triplet coset which is equal c triplet coset xH. Therefore, we have xH = xH, ⩝x ∈ N ⇒ xH (anti(x)) = H (anti(x)) = H , ⩝x∈ N, since H is a neutrosophic triplet normal subgroup of

7

8

9

10

11

0 7 2 9 4 11 6 1 8 3 10 5

0 8 4 0 8 4 0 8 4 0 8 4

0 9 6 3 0 9 6 3 0 9 6 3

0 10 8 6 4 2 0 10 8 6 4 2

0 11 10 9 8 7 6 5 4 3 2 1

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The formed NETs of X are: (0, 0, 0), (0, 0, 1), (0, 0, 2), . . . , (0, 0, 11), (1, 1, 1), (3, 9, 3), (3, 9, 7), (3, 9, 11), (4, 4, 4), (4, 4, 7), (4, 4, 10), (5, 1, 5), (7, 1, 7), (8, 4, 2), (8, 4, 5), (8, 4, 8), (8, 4, 11), (9, 9, 5), (9, 9, 9), (11, 1, 11).

Here, 2, 6, and 10 did not give rise to a neutrosophic triplet, as neut(2) = 1 and 7, however anti(2) did not exist in Z12 . In addition, neut(6) = 1, 3, 5, 7, 9, and 11, however anti(6) did not exist in Z12 .

The neut(10) = 1, however anti(10) did not exist in Z12 .

Definition 4 ([4,7]). The set N is called a strong neutrosophic extended triplet set if, for any x in N, eneut( x) ∈ N and e anti( x) ∈ N exists.

Example 2. The NET’s of (x) modulo 12 were as follows: (0, 0, 0), (0, 0, 1), (0, 0, 2), . . . , (0, 0, 11), (1, 1, 1), (3, 9, 3), (3, 9, 7), (3, 9, 11), (4, 4, 4), (4, 4, 7), (4, 4, 10), (5, 1, 5), (7, 1, 7), (8, 4, 2), (8, 4, 5), (8, 4, 8), (8, 4, 11), (9, 9, 5), (9, 9, 9), (11, 1, 11).

Definition 5 ([4,7]). The set N is called an NET weak set if, for any x ∈ N, an NET y, eneut(y) , e anti(y) included in N exists, such that: x = y or x = eneut(y) or x = e anti(y) .

Definition 6. A neutrosophic extended triplet (x, y, z) for x, y, z ∈ N, is called a neutrosophic perfect triplet if both (z, y, x) and (y, y, y) are also neutrosophic triplets.

Example 3. The neutrosophic perfect triplets of (x) modulo 12 are described in “Table 1” as follows: Here, (0, 0, 0), (1, 1, 1), (3, 9, 3), (4, 4, 4), (5, 1, 5), (7, 1, 7), (8, 4, 8), (9, 9, 9), (11, 1, 11) are neutrosophic perfect triplets of (x) modulo 12.

Definition 7. An NET (x, y, z) for x, y, z ∈ N, is called a neutrosophic imperfect triplet if at least one of (z, y, x) or (y, y, y) is not a neutrosophic triplet(s).

Example 4. The neutrosophic imperfect triplets of (x) modulo 12, from the above table, were as follows: (0, 0, 1), (0, 0, 2), . . . , (0, 0, 11), (3, 9, 7), (3, 9, 11), (4, 4, 7), (4, 4, 10), (8, 4, 2), (8, 4, 5), (8, 4, 11), (9, 9, 5).

2.3. Neutrosophic Triplet Group (NTG) Definition 8 ([1–3]). Let (N, ∗) be a neutrosophic strong triplet set. Then, (N, ∗) is called a neutrosophic strong triplet group, if the following classical axioms are satisfied: (1) (2)

(N, ∗) is well-defined, that is, for any x, y ∈ N, one has x ∗ y ∈ N. (N, ∗) is associative, that is, for any x, y, z ∈ N, one has x ∗ (y ∗ z) = ( x ∗ y) ∗ z.

Example 5. We let Y = (Z12 , ×) be a semi-group under product 12. The neutral elements of Z12 were 4 and 9.

The elements (8, 4, 8), (4, 4, 4), (3, 9, 3), and (9, 9, 9) were NETs. NTG, in general, was not a group in the classical sense, because it might not have had a classical unitary element, nor the classical inverse elements. We considered that the neutrosophic

neral structure of groups.

Just as group in a classical f the central concepts of classical theory n thestructure theory ofofneutrosophic triplet neral groups. Just extended as in a classical ofthe neutrosophic triplet quotient group and its n theory of neutrosophic extended triplet of neutrosophic triplet quotient group and its Symmetry 2018, 10, 126

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oup and H  N is a neutrosophic triplet normal H  xH: N isx a∈neutrosophic triplet normal Houp hasand elements N, the neutrosophic triplet neutrals replaced the classical H has elements xH: x ∈ N, the neutrosophic triplet unitary element, and the neutrosophic opposites replaced the classical inverse elements. let quotient groups for the dihedral group D3.

1 ([3]). uotient set D 3/N is Proposition a neutrosophic triplet group if,) be an NTG with respect to ∗ and a, b, c ∈ N: let quotient groups for the dihedral groupLet D 3.(N, ∗ mal subgroups are D 3 itself .

We always have the trivial uotient set D3/N is(1) a neutrosophic a ∗ b = atriplet ∗ c ⇔group neut(if, a) ∗ b = neut( a) ∗ c. mal D3 of itselfindex . We always have is thenormal. trivial â&#x;Š = subgroups (1, r, đ?‘&#x;đ?‘&#x; 2 ) isarethat 2 and thus (2) b ∗ a = c ∗ a ⇔ b ∗ neut( a) = c ∗ neut( a).

2 D3 is a different neutrosophic triplet oup.

If r, N đ?‘&#x;đ?‘&#x; â&#x;Š = (1, ) is that of index 2 and thus is normal.

(3) if anti ( a) ∗ b = anti ( a) ∗ c, then neut( a) ∗ b = neut( a) ∗ c. oup.

N đ?‘ đ?‘ is 2aâ&#x;Š.different neutrosophic triplet  =D3â&#x;¨đ?‘ đ?‘ đ?‘&#x;đ?‘&#x; â&#x;¨đ?‘ đ?‘ đ?‘ đ?‘ â&#x;Š. If or However, none of them are (4) if b ∗ anti ( a) = c ∗ anti ( a), then b ∗ neut( a) = c ∗ neut( a).

2 neutrosophic triplet quotient the only non-triavial â&#x;¨đ?‘ đ?‘ đ?‘ đ?‘ â&#x;Š. or đ?‘ đ?‘ = â&#x;¨đ?‘ đ?‘ đ?‘&#x;đ?‘&#x; â&#x;Š. However, none of them are FORonly PEERnon-triavial REVIEW 10 of 13 the neutrosophic triplet quotient to of ∗ 13 and a, b ∈ N: FOR PEER REVIEW Theorem 1 ([3]). Let (N, ∗) be a commutative NET, with respect 10 neutrosphic extended triplet of N. Suppose H is neutrosophic extended triplet and H be a neutrosophic tripletsubgroup normal subgroup of (i) neut( a) ∗ neut(b) = neut( a ∗ b); en âŠ?H x be ∈aN, yextended ∈ the H :operation Đ­z ∈ Hsubgroup :normal = zx.

Thus (xy)anti(x) ∈ H implyingextended (xH)anti(x) ⊆ iplet group under ofxy(xH)(yH) = xyH. neutrosphic triplet of N. Suppose H =isz neutrosophic triplet and neutrosophic triplet subgroup of (ii) anti ( a) ∗ anti (b) = anti ( a ∗ b); en âŠ? group x ∈ N, y ∈ the H :operation Đ­z ∈ H :ofxy(xH)(yH) = zx. Thus (xy)anti(x) = z ∈ H implying (xH)anti(x) ⊆ iplet under = xyH.

suppose � x ∈ N Theorem :(xH)anti(x) ⊆H. Let Then n ∊a Ncommutative , w e have (nH )anti (n) ⊆H , wtohi 2 ([3]).

(N,for ∗) be NET, with respect ∗ch and a ∈ N: Hsuppose n. A lso,âŠ? for ∊ N , w e⊆ have anti(n)H anti(n)Hn x ∈anti(n) N :(xH)anti(x) H. Then for n(anti ∊ N[anti{n}]) , w e have= (nH )anti(n)⊆ ⊆HH,, wwhich hich (i) neut( a) ∗ neut( a) = neut( a); . Therefore, = H n, ∊meaning that Hanti(n)H  N. (anti[anti{n}]) = anti(n)Hn ⊆ H, which H n.

A lso, for nH anti(n) N , w e have (ii) anti ( a) ∗ neut( a) = neut( a) ∗ anti ( a) = anti ( a); H . Therefore, nH = H n, meaning that H  N. s, xH ∗ H = xH ∗ neut(x)H = x ∗ neut(x)H = xH. let f: N → H be a neutro-homomorphism from a neutrosophic extended triplet group N to a nti(x)H, since anti(x)H= x= ∗(xneut(x)H ∗ anti(x)H) = s, xH ∗ H = xHxH∗ ∗ neut(x)H = xH. nded group Kerf  N. Definition 9 ([3]). Anfrom NETa (N, ∗) is calledextended to be cancellable, if it N satisfies let f: triplet N→H be a H, neutro-homomorphism neutrosophic triplet group to a the following conditions: nti(x)H, since xH∗ anti(x)H = (x ∗ anti(x)H) = nded triplet group H, Kerf  N. H(yz)H = xH(yHzH), âŠ? x, x, z∈ erf, we had to show (a) that a(anti[b]) kerf. y, y,z∈ N,N. xThis âˆ—â–Ą y meant = y ∗ that z ⇒kerf y was = za. neutrosophic extended oup of N. If a ∈ kerf, then H(yz)H = xH(yHzH), âŠ? x, x, z∈ erf, we had to show (b) that a(anti[b]) kerf. y, y,z∈ N,N. yThis âˆ—â–Ąx meant = z ∗ that x ⇒kerf y was = za. neutrosophic extended oup of N. If a ∈ kerf, then f(a) = neutH

he neutrosophic extended triplets and Let N betoan NTG and x ∈ N. N is then called a neutro-cyclic triplet group if N = h ai.

Definition 10 ([3]). f(a) = neut H then to neutrosophic introduce theextended neutrosophic cosets, he to We can saytriplets that atriplet isand the then neutrosophic triplet generator of N. n e to neutrosophic triplet quotient group. also introduce the neutrosophic tripletWe cosets, n created structures and their application =let neut Example 6.f(b) We N H= (2,also 4,to6) be an NTG with respect to (Z8 , .). Then, N was clearly a neutro-cyclic triplet e neutrosophic triplet quotient group.

We rther defined the and neutrosophic group as N f(b) = h aapplication Therefore, 2towas the neutrosophic triplet generator of N. created structures =i.neut H kernel, wed that f(a(anti[b]) = neutH.their (f is neutro-homomorphism) trosophic extended triplets.

As a further rther defined the neutrosophic kernel, f(a)that . f(anti(b)) wed f(a(anti[b]) = neutH. (f is neutro-homomorphism) lled Neutrosophic Triplet Structures trosophic extended triplets.

As a (namely, further (b)) f(a) . f(anti(b)) 2.4. Neutrosophic Extended Triplet Group (NETG) normal subgroup,Triplet and neutrosophic triplet lled Neutrosophic Structures (namely, i(neutH) (b)) normal subgroup, and neutrosophic triplet tH H) i(neut Definition 11 ([4,7]). Let (N, ∗) be an NET strong set. Then, (N, ∗) is called an NETG, if the following classical tyHto this paper. The individual responsibilities and axioms are satisfied: )) ∈ofkerf. dea this whole paper was putresponsibilities forward by Mikail y to this paper.

The individual and (1) (N, ∗ ) is well-defined, that is, for any x, y ∈ N, one has x ∗ y ∈ N. )) ∈ kerf.

Moges Mekonnen Shalla analyzed the work whole putthat forward Mikail Ndea andofathis ∊ kerf.

Wepaper had towas show nexisting . a .by (anti(n)) ∈ kerf. (f is neutro-homomorphism) nt group and wrote part of the paper.

The revision (2) (N, ∗ ) is associative, that is, for any x, y, z ∈ N, one has Moges Mekonnen Shalla analyzed the existing work N a ∊ .kerf.

We had to show that n . a . (anti(n)) ∈ kerf. (f is neutro-homomorphism) (n)and = f(n) f(a)wrote . f(anti(n)) un. nt group and part of the paper. The revision ti(f(n)) x ∗ (y ∗ z) = ( x ∗ y) ∗ z. (n) = f(n) . f(a) . f(anti(n)) un. rest. nti(h)) ti(f(n)) rest.

For NETG, the neutrosophic extended neutrals replaced the classical unitary element, and the nti(h)) neutrosophic extended opposites replaced the classical inverse elements. In the case where NETG anti(n)) ∈ kerf included a classical group, then NETG enriched the structure of a classical group, since there might N. anti(n)) ∈ kerf have been elements with more extended neutrals and more extended opposites.

N. eutrosophic triplet subgroup H of N is a neutrosophic triplet normal subgroup of N if, and Definition 12.is of a triplet settriplet X isnormal acoset function is one to one and onto, that is, a bijective eutrosophic triplet coset of H in right neutrosophic of H Ďƒ: ∈xN.→ subgroup HN of NAaispermutation a neutrosophic subgroup of xNthat if, and map.ofPermutation maps,neutrosophic being bijective, havecoset anti of neutrals and the maps combine neutrally under composition eutrosophic triplet coset H in N is a right triplet H ∈ N. of maps,subgroup which are There is natural neutral Ďƒ: x → x, X = (1, 2, 3, . . . , n), which âŠ? x∈ N ⇒permutation xH(anti[x])x neutrosophic triplet normal of associative.

N, then xH(anti[x])=H, is Ďƒ(k) = k.

Therefore, all of the permutations of a set X = (1, 2, 3, . . . , ⇒ xH = H x, �x ∈ N, since subgroup each left neutrosophic triplet coset xH neutrosophicn) form an NETG under composition. � xis∈theN right neutrosophic triplet normal of N, then xH(anti[x])=H, ⇒ xH(anti[x])x Sn This group is left called the symmetric NETG of the degree n. neutrosophic ⇒ xH = H x, �x ∈ N, since each neutrosophic triplet coset(exH) is right

let each left neutrosophic triplet coset of H in N be a right neutrosophic triplet coset means thatleft if xneutrosophic is any element of N,coset thenof the neutrosophic triplet coset triplet xH is also a let each triplet Hleft in N be a right neutrosophic coset hic triplet Nowelement neut(x)of∈N,H,then therefore ∗ neu t(x) = x triplet ∈ xH.coset Consequently means that coset. if x is any the leftxneutrosophic xH is also x a g to that right neutrosophic triplet coset, which is equal to left neutrosophic triplet hic triplet coset. Now neut(x) ∈ H, therefore x ∗ neu t(x) = x ∈ xH. Consequently x

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Example 7. We let A = (1, 2, 3). The elements of symmetric group of S3 were as follows: !

σ0 =

123 123

!

, σ1 =

123 231

!

µ1 =

123 123

!

, µ2 =

123 123

, σ2 =

, µ3 =

123 312

!

123 123

!

The operartion of S3 is defined in Table 2 as follows: 1.

2.

(S3 , # ) is well-defined, that is, for any σi , µi ∈ S3 , i = 1,2,3 one has σi # µi ∈ S3 .

(S3 , #) is associative, that is, for any σ1 , µ1 , µ3 ∈ S3, one has the following: (σ1 # µ1 ) # µ3 = σ1 # (µ1 # µ3 ) (µ1 # µ3 ) = (σ1 # σ1 ) = σ2 .

Table 2. Neutrosophic triplets of X. #

σ0 σ1 σ2 µ1 µ2 µ3

σ0

σ1

σ2

µ1

µ2

µ3

σ0 σ1 σ2 µ1 µ2 µ3

σ1 σ2 σ0 µ2 µ1 µ2

σ2 σ0 σ1 µ3 µ3 µ1

µ1 µ2 µ3 σ0 σ1 σ2

µ2 µ3 µ1 σ2 σ0 σ1

µ3 µ1 µ2 σ1 σ2 σ0

The NET’s of S3 (eS3 ) are as follows: (σ0 , σ0 , σ0 ), (σ1 , σ0 , σ2 ), (σ2 , σ0 , σ1 ), (µ1 , σ0 , µ1 ), (µ2 , σ0 , µ2 ), (µ3 , σ0 , µ3 ).

Hence, (S3, #) is an NET strong group.

Definition 13 ([9–11]). neutro-homomorphism if: (1) (2)

Let (N1 ∗, N2 #) be two NETGs.

A mapping f: N1 → N2 is called a

For any x, y ∈ N1 , we have f ( x ∗ y) = f ( x ) f (y) If (x, neut[x], anti[x]) is an NET from N1 , then, f (neut[ x ]) = neut( f [ x ]) and f ( anti [ x ]) = anti ( f [ x ]).

Example 8. We let N1 be an NETG with respect multiplication modulo 6 in (Z6 , ×), where N1 = (0, 2, 4), and we let N2 be another NETG in (Z10 , ×), where N2 = (0, 2, 4, 6, 8). We let f: N1 → N2 be a mapping defined as f(0) = 0, f (2) = 4, f (4) = 6. Then, f was clearly a neutro-homomorphism, because condition (1) and (2) were satisfied easily.

Definition 14. Let f: N1 → N2 be a neutro-homomorphism from an NETG (N1 , ∗) to an NETG (N2 , ∗).

The neutrosophic image of f is a subset, as follows: Im(f) = (f(g):g ∈ N1 , ∗) of N2 .

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Definition 15. Let f: N1 → N2 be a neutro-homomorphism from an NETG (N1 , ∗) to an NETG (N∗, #) and B ⊆ N2 . Then Symmetry Symmetry 2018, 2018, 10,Symmetry x FOR 10, xPEER FOR 2018, PEER REVIEW 10, xREVIEW FOR PEER REVIEW f −1 (B) = (x ∈ N1 : f(x) ∈ B)

22. 22.

Smarandache, Smarandache, 22. F.Smarandache, Neutrosophic F. Neutrosophic F.Perspectives: Neutrosophic Perspectives: Triplet Persp Tr Hedge HedgeAlgebras.

Hedge Algebras.And Algebras.

AndApplications; Applications And A https://www.google.com/books?hl=en&lr=&id=4803 https://www.google.com/books?hl=en&lr=&id= https://www.google.com/books?hl=e Definition 16. Let f: N1 → N2 be a neutro-homomorphism from am NETG (N1 , ∗) to an NETG (N2 , #). erspectives:+Triplets,+Duplets,+Multisets,+Hybrid+O erspectives:+Triplets,+Duplets,+Multisets,+Hyb erspectives:+Triplets,+Duplets,+Mult The neutrosophic kernel off is a subset plications&ots=SzcTAGL10B&sig=kJMEHWsQ_tYjT plications&ots=SzcTAGL10B&sig=kJMEHWsQ plications&ots=SzcTAGL10B&sig=kJ

is the neutrosophic inverse image of B under f.

ker ( f ) = { x ∈ N1 : f ( x ) = neut( x )} of N1 , where neut( x ) denotes the neutral element of N2 .

© 2018 © 2018 by the byauthors. the © 2018 authors.

Submitted by the Submitt autho fo termsterms and and conditions terms conditions and of condit the of tC (http://creativecommons.org/license (http://creativecommons.org/li (http://creativecomm

Example 9. We took D4 , the symmetry NETG of the square, which consisted of four rotations and four reflections. We took a set of the four lines through the origin at angles 0, ᴨ /4, ᴨ /2, and 3 ᴨ /4, numbered 1, 2, 3, 4, respectively. We let S4 be the permutation NETG of the set of four lines. Each symmetry s, of the square in particular, gave a permutation ϕ(s) of the four lines. Then we defined a mapping, as follows: Φ: D4 → S4 whose value at the symmetry s ∈ D4 was the permutation ϕ(s) of the four lines. Such a process would always define a neutro-homomorphism. We found the kernel and image of ϕ. The neutral permutation of the square gave the neutral of the four lines. The rotation (1234) of the square gave the permutation (13)(24) of the four lines; the rotation (13)(24) by 180 degrees gave the neutral permutation eneut of the four lines; the rotation (4321) of the square gave the permutation (13)(24) of the four lines again. Thus, the neutrosophic image of the rotation NET subgroup R4 of D4 was the NET subgroup (neut, [13][24]) of S4 . The reflections of the square were given by the compositions of the rotations of the square with a reflection, for example, the reflection (13). The reflection (13) of the square (in the vertical axis) gave the permutation (24) of the lines. Thus, the homomorphism ϕ took the set of reflections R4 # (13) to the following: ϕ(R4) # φ(13) = (neut, [13][24] # [24]) = ([24], [13]).

The neutrosophic image of ϕ was the union of the neutrosophic image of the rotations and the reflections, 10 of 13 which was Im(ϕ) = (neut, [13][24], [13], [24]) ∈ S4 . In the work above, we saw that the neutrosophic kernel of ϕ was as triplet follows:subgroup of N. Suppose H is neutrosophic extended triplet e a neutrosphic extended ker(ϕ) =H (neut, [13][24]) of D4 ⊆ Then ⩝ x ∈ N, y ∈ H : Эz ∈ H : xy = zx. Thus (xy)anti(x) =z∈ implying (xH)anti(x)

0, x FOR PEER REVIEW

3. Neutrosophic Extended Triplet Subgroup

In this section, definition extended y, suppose ⩝ x ∈ N :(xH)anti(x) ⊆H. aThen for n ∊ofN the , w eneutrosophic have (nH )anti (n) ⊆H ,triplet w hich subgroup and its example have been given. ⊆ H n. A lso, for anti(n) ∊ N , w e have anti(n)H (anti[anti{n}]) = anti(n)Hn ⊆ H, which nH . Therefore, nH = H n, meaning that H  N. Definition 17. Given an NETG (N, ∗), a subset H is called an NET subgroup of N, if it forms an NETG itself ∗. Explicitly, this means following: extended triplet group N to a We let f: N → H be aunder neutro-homomorphism from athe neutrosophic

tended triplet group H, Kerf  N. (1) The extended neutral element eneut( x) lies ∈ H. ∈ kerf, we had to show a(anti[b]) kerf was a neutrosophic extended (2) that For any x, y∈ kerf.

H, x This ∗ y ∈meant H (Hthat is closed under ∗). anti ( x ) group of N. If a ∈ (3) kerf, then If x ∈ H, then e ∈ H (H has extended opposites).

then

H We wrotef(a) H =≤neut N whenever H was an NET subgroup of N. ∅ 6= H ⊆ N, satisfying (2) and (3) of Definition 17, would be an NET subgroup, as we took x ∈ H and then (2) gave e anti( x) ∈ H, after which (3) gave x ∗ e anti( x) = eneut( x) ∈ H.

f(b) = neutH

howed that f(a(anti[b]) = neutH. (f is neutro-homomorphism) ) = f(a) . f(anti(b)) nti(b)) anti(neutH)

n important role in the study of the general structure of groups. Just as in a classical , quotient groups played a similar role in the theory of neutrosophic extended triplet section, we have introduced the notion of neutrosophic triplet quotient group and its e neutrosophic extended triplet group.

Symmetry 2018, 10, 126 7 of 13 1. If N is a neutrosophic extended triplet group and H  N is a neutrosophic triplet normal the neutrosophic triplet quotient group N/H has elements xH: x ∈ N, the neutrosophic triplet N, and an operation Example of (xH)(yH)10. = (xy)H. We let S4 = (neut, Ďƒ1 , Ďƒ2 , . . . , Ďƒ9 , Ď„ 1 , Ď„ 2 , . . . , Ď„ 8 , δ1 , δ2 , . . . , δ6 ) with Ďƒ1 = (1234), Ďƒ2 = (13)(24), Ďƒ3 = (1432), Ďƒ4 = (1243), Ďƒ5 = (14)(23), Ďƒ6 = (1342), Ďƒ7 = (1324), Let’s find all of the possible neutrosophic triplet quotient groups for the dihedral group D3. Ďƒ8 = (12)(34),Ďƒ 9 = (1432),Ď„ 1 = (234),Ď„ 2 = (243), Ď„ 3 = (134), Ď„ 4 = (143), Ď„ 5 = (124), Ď„ 6 = (142), Ď„ 7 = (123), 3 2 , r2, s, sr, đ?‘ đ?‘ đ?‘&#x;đ?‘&#x; 2 ), where đ?‘&#x;đ?‘&#x; = đ?‘ đ?‘ = rsrs = 1.

D3(14), /N isδa neutrosophic triplet group if, Ď„ 8 = (132), δ1 = (12),Aδ2quotient = (13), set δ3 = 4 = (23), δ5 = (24), δ6 = (34). The trivial neutrosophic extended  D3. Then, all of neutrosophic subgroups are Dand 3 itself. We always have the trivial subgroups of triplet S4 werenormal the neutral elements, the non-trivial neutrosophic extended subgroups S4 of order 2 2 2 ≅ 1 and D3/1 ≅ D3.were The as subgroup â&#x;¨râ&#x;Š = â&#x;¨ đ?‘&#x;đ?‘&#x; â&#x;Š = (1, r, đ?‘&#x;đ?‘&#x; ) is that of index 2 and follows: (neut, Ďƒ2 ), (neut, Ďƒ5 ), (neut, Ďƒ8 ), (neut, δ1 ), thus (neut,is δnormal.

2 ), (neut, δ3 ), (neut, δ4 ), (neut, δ5 ), (neut, râ&#x;Š is also a neutrosophic triplet quotient group.

If N  D3 is a different neutrosophic triplet δ6 ), and the neutrosophic extended subgroups, S4 , of order 3 were as follows: up, then â&#x;¨đ?‘ đ?‘ â&#x;Š. = 2, so either N = â&#x;¨đ?‘ đ?‘ â&#x;Š, N = â&#x;¨đ?‘ đ?‘ đ?‘ đ?‘ â&#x;Š. or đ?‘ đ?‘ = â&#x;¨đ?‘ đ?‘ đ?‘&#x;đ?‘&#x; 2 â&#x;Š. However, none of them are L11 = hĎ„ 1neutrosophic i = hĎ„ 2 i = (neut, Ď„ 1quotient , Ď„2 ) (đ?‘ đ?‘ đ?‘ đ?‘ )đ?‘ đ?‘ (đ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Ž(đ?‘ đ?‘ đ?‘ đ?‘ )) = đ?‘ đ?‘ đ?‘&#x;đ?‘&#x; 2 not in â&#x;¨đ?‘ đ?‘ â&#x;Š. Hence, the only non-triavial triplet .

L12 = hĎ„ 3 i = hĎ„ 14 i = (neut, Ď„ 3 , Ď„ 4 ) et N be a neutrosophic extended triplet group and H be a neutrosophic triplet normal subgroup of L13 = hĎ„ 5 i = hĎ„ 6 i = (neut, Ď„ 5 , Ď„ 6 ) /H = xH, x ∈ N is a neutrosophic extended triplet group under the operation of (xH)(yH) = xyH. L14 = hĎ„ 7 i = hĎ„ 8 i = (neut, Ď„ 7 , Ď„ 8 ) N/H → N/H it was straightforward to find the neutrosophic extended subgroups of order 4, 6, 8, and 12 of S4 . and yH = y’H and yh2 = y’, h1, h2∊ H Neutrosophic Triplet Cosets h1yh2H = xh1yH =4.x h 1Hy = xHy = xyH. ral, for any x ∊ H, is neut(x)H = H. That xH ∗ H = xHtriplet ∗ neut(x)H x ∗its neut(x)H = xH. In this section, theis,neutrosophic coset =and properties have been outlined. Furthermore, of a neutrosophicthe triplet coset xH is anti(x)H, since xH∗ anti(x)H = (x ∗ anti(x)H) = one have been given. difference between the neutrosophic triplet coset and the classical H = H. ivity, (xHyH)zH =Definition (xy)HzH =18. (xy)zH xH(yz)H = xH(yHzH), y, the z ∈set N.xh/ â–Ą h ∈ H, is denoted by xH, analogously, Let N= be an NETG and H ⊆ N. âŠ? x ∈x,N, as follows: ns Hx = hx/h ∈ H

n theme of this paper and was to introduce the neutrosophic extended triplets and then to neutrosophic extended triplets in order to introduce the neutrosophic triplet∈ cosets, (xH)anti(x) = (xh)anti(x)/h H. triplet normal subgroup, and finally, the neutrosophic triplet quotient group. We also When h ≤ N,newly xH is called the structures left neutrosophic coset of H ∈ N e interesting properties of these created and triplet their application to containing x, and Hx is called the right neutrosophic coset ofdefined H ∈ N containing x. In this case,kernel, the element x is called the neutrosophic extended triplet group. Wetriplet further the neutrosophic triplet coset representative of xH or Hx. | xH | and | Hx | are used to denote the number of elements in xH -image, and inverse image for neutrosophic extended triplets. As a further or Hx, n, we created a new fieldrespectively. of research, called Neutrosophic Triplet Structures (namely, phic triplet cosets, neutrosophic triplet normal subgroup, and neutrosophic triplet Example 11. When N = S3 and H = ([1], [12]), the “Table 3� lists the left and right neutrosophic triplet p).

H-cosets of every element of the NETG.

butions: All authors have contributed equally to this paper. The individual responsibilities and 3. whole Neutrosophic triplet and right cosets of S3 . all authors can be described as follows: the ideaTable of this paper was putleft forward

by Mikail mpleted the preparatory work of the paper. Moges Mekonnen Shalla analyzed the existing work gHpaper. The revision Hg 92516 neutrosophic triplet coset and quotient groupg and wrote part of the n of this paper was completed by Necati Olgun.

(1) ([1], [12]) ([1], [12]) (12) ([1], [12]) ([1], [12]) terest: The authors declare no conflict of interest. (13) ([13], [123]) ([13], [132]) (23) ([23], [132]) ([23], [123]) (123) ([13], [123]) ([23], [123]) (132) ([23], [132]) ([23], [123])

First of all, cosets were not usually neutrosophic extended triplet subgroups (some did not even contain the extended neutral). In addition, since (13) 6= H(13), a particular element could have different left and right neutrosophic triplet H-cosets. Since (13)H = H(13), different elements could have the same left neutrosophic triplet H-cosets.

11 of 13 Symmetry 2018, 10, 126

8 of 13

f the central concepts of classical group theory ExampleJust 12. as Weincalculated the neutrosophic triplet cosets of N = (Z4 , +) under addition and let H = (0, 2). neral structure of groups. a classical The elements (0, 0, 0), (0, 0, 1), n the theory of neutrosophic extended triplet(0, 0, 2), (0, 0, 3), (1, 1, 1), and (3, 3, 3) were NET’s of Z4 and the classical cosets of N were as follows: of neutrosophic triplet quotient group and its H = H + 0 = H + 2 = (0, 2).

and oup and H  N is a neutrosophic triplet normal H + 1 = H + 3 = (1, 3).

H has elements xH: x ∈ N, the neutrosophic triplet Here, 2 did not give rise to NET, because the neut’s of 2 were 1 and 3, however there were no anti’s. Therefore, we could not obtain the neutrosophic triplet coset of N. In general, classical cosets were not neutrosophic triplet might let quotient groups cosets, for the because dihedralthey group D3. not have satisfied the NET conditions. uotient set D3/N is a neutrosophic triplet group if, Definition mal subgroups are D3 itself. Similarly We alwaysto have the trivial16, we could define neutrosophic triplet cosets as follows:

â&#x;Š = (1, r, đ?‘&#x;đ?‘&#x; 2 ) is that of index 2 and thus is normal.

a neutrosophic triplet group and H ≤ N. We defined a relation ≥ `(modH) on N 19. Let N betriplet oup. If N  D3 is Definition a different neutrosophic as follows: â&#x;¨đ?‘ đ?‘ đ?‘ đ?‘ â&#x;Š. or đ?‘ đ?‘ = â&#x;¨đ?‘ đ?‘ đ?‘&#x;đ?‘&#x; 2 â&#x;Š. However, none of them are if x1 , x2 ∈ N and anti(x1 )x2 ∈ N, Then the only non-triavial neutrosophic triplet quotient x1 = l x2 (modH)

and H be a neutrosophic Or, triplet normal subgroup equivalently, if there of exists an h ∈ H, such that: iplet group under the operation of (xH)(yH) = xyH. anti(x1 ) ∗ x2 = h That is, if x2 = x1 h for some h ∈ H.

Proposition 2. The relation ≥ `(modH) is a neutrosophic triplet equivalence relation. The neutrosophic triplet equivalence class containing x is the set xH = xh/h ∈ H. s, xH ∗ H = xH ∗ neut(x)H = x ∗ neut(x)H = xH. nti(x)H, since xH∗ anti(x)H = (x ∗ anti(x)H) = Proof.

H(yz)H = xH(yHzH), ∈ N. âˆ—â–Ą x = neut(x) ∈ H. Hence, x = `x1 (modH)} and ≥ `(modH) is reflexive.

(1) âŠ?x ∈x,Ny, anti(x) 1, z (2) İf x = `x2 (modH), then anti(x1 ) ∗ x2 ∈ H. However, since an anti of an element of H is also in H, anti(anti[x1 ] ∗ x2 ) = anti(x2 ) ∗ anti(anti[x1 ]) = anti(x2 ) ∗ x1 ∈ H. Thus, x2 = `x1 (modH), hence ≥ `(modH) is symmetric. he neutrosophic extended triplets and then to Finally, if xtriplet (modH) and x2 = `x3 (modH), then anti(x1 ) ∗ x2 ∈ H and anti(x2 ) ∗ x3 ∈ H. Since 1 = `x2cosets, to introduce the(3) neutrosophic H is closed under e neutrosophic triplet quotient group.

Wetaking also products, anti(x1 )x2 anti(x2 )x3 = anti(x1 )x3 ∈ H. Hence, x1 = `x3 (modH) so that `(modH) is transitive.

Thus, ≥ `(modH) is a neutrosophic triplet equivalence relation. created structures and their≥application to

rther defined the neutrosophic kernel,

trosophic extended triplets. As a further 4.1. Properties of Neutrosophic Triplet Cosets lled Neutrosophic Triplet Structures (namely, normal subgroup, and neutrosophic triplet Lemma 1. Let H ≤ N and let x, y ∈ N. Then, (1)

x ∈ xH.

(6) (7) (8) (9)

xH = Hx ⇔ H = (xH)anti(x). xH ⊆ N ⇔ x ∈ H. (xy)H = x(yH) and H(xy) = (Hx)y. |xH | = |YH |.

y to this paper. The individual responsibilities and (2) was xH put = Hforward ⇔ x ∈ H. dea of this whole paper

by Mikail (3) analyzed xH = yHthe ⇔existing x ∈ yH.work Moges Mekonnen Shalla nt group and wrote (4) part of paper.

The∊revision xHthe = yH or xH yH = Ă˜. un.

(5) xH = yH ⇔ anti(x)y ∈ H.

rest.

try 2018, 10, x FOR PEER REVIEW (1) x ∈ xH.

10 of 13

(2) xH = H ⇔ x ∈ H. f Let H be a neutrosphic triplet subgroup of N. Suppose H is neutrosophic extended triplet (3)extended xH = yH⇔ x∈ yH. oup of N. Then ⩝ x ∈ N, y ∈ H : Эz ∈ H : xy = zx.

(4) xH = yH or xH ∩ yH = Ø.Thus (xy)anti(x) = z ∈ H implying (xH)anti(x) ⊆ Symmetry 126 anti(x)y (5) 2018, xH=10, yH⇔

∈ H.

9 of 13

(6) xH = Hx⇔ H = (xH)anti(x).

Conversly, suppose ⩝ x ∈ N :(xH)anti(x) ⊆H. Then for n ∊ N , w e have (nH )anti(n) ⊆H , w hich (7) xH ⊆ N ⇔ x ∈ H. Proof. es nH ⊆ H n. A lso, for anti(n) ∊ N , w e have anti(n)H (anti[anti{n}]) = anti(n)Hn ⊆ H, which (8) (xy)H = x(yH) and H(xy) = (Hx)y. es Hn ⊆ nH . Therefore, nH = H n, meaning that H  N. = ∣yH∣.∈ xH (1) (9) x = ∣xH∣ x(neut(x)) Symmetry 2018, 10, xxFOR PEER REVIEW 10 of 13 xH = H. Then = x(neut(x)) ∈ xHextended = H. mple 15. We let f: N(2) → H⇒ beSuppose a neutro-homomorphism from a neutrosophic triplet group N to a Proof  N. osophic extended triplet group H, Kerf ⇐ Now assume xLet in H H. be Since H is closed, xH ⊆ H. Proof a neutrosphic extended triplet subgroup of N. Suppose H is neutrosophic extended triplet (1) x = x(neut(x)) ∈ xH subgroup of N. Then ⩝ x ∈ N, y ∈ H : Эz ∈neutrosophic H : xy = zx. Thus (xy)anti(x) = z ∈ H implying (xH)anti(x) ⊆ If ∀ a, b ∈ kerf, we had (2) toNext, show that a(anti[b]) kerf.

H,Then soThis anti(x)h ∈that H, kerf since H a≤ assume h∈ Suppose xH = H. xmeant = x(neut(x)) ∈was xH = N. H. Then, extended ⇒also H. □ triplet subgroup of N. If a ∈ kerf, then assume x in H. Since H is closed, xH ⊆ H. ⇐ Now h = neut(x)h = x ∗ anti(x)h = x(anti[x])h ∈ xH, Next, also assume ∊ H, f(a) = hneut H so anti(x)h ∈ H, since H ≤ N. Then, Conversly, suppose ⩝ x ∈ N :(xH)anti(x) ⊆H. Then for n ∊ N , w e have (nH )anti(n) ⊆H , w hich h =Aneut(x)h = x ∗ anti(x)h ∈ xH, implies nH ⊆ H n. lso, for anti(n) ∊ N , w= ex(anti[x])h have anti(n)H (anti[anti{n}]) = anti(n)Hn ⊆ H, which and So H

By By mutual inclusion, xH xH = h.= h.

So⊆HxH. ⊆implies xH. mutual inclusion, Hn ⊆ nH .

Therefore, nH = H n , meaning that H N.  b ∈ kerf, then (3) xH = Yh (3) xH = Yh f(b) = neut H 15. let f: N → H be a neutro-homomorphism from a neutrosophic extended triplet group N to a ⇒ x⇒ = x(neut(x)) ∈ xH yH. x =Example x(neut(x)) ∈=We xH = yH.  =N. neutrosophic extended triplet group H, ∈ xneutro-homomorphism) =where yh, where ∈ hH ∈Kerf H, xH (yh)H = y(hH) Then,we showed that f(a(anti[b]) =xneut H. (f⇒ ⇐ x⇐ ∈ yH ⇒yH x = is yh, h ∈ Hh ⇒ ∈⇒H,h xH = (yh)H = y(hH) = yH.= yH. (a(anti(b)) = f(a) . f(anti(b)) (1)that If xH ∀ a,∩byH ∈∅. we had to show a(anti[b]) meant that kerf was a neutrosophic extended (4) (4) Suppose that xH ∩ yH 6= ∃a∃a ∈∈ xHxH ∩that yH ⇒⇒∃∃h h1 h1h22∈∊ kerf.

H ∍ aaThis = xh Suppose ≠kerf, ∅.Then, Then, ∩ yH H = xh 11 = f(a) . f(anti(b)) triplet subgroup of N. If a ∈ kerf, then and and = neutH . anti(neutH) a = yh2. Thus, x = a(anti(h1)) = yh2(antih1) and xH = yh2(anti(h1))H f(a) = neutH = neutH . neutH a = =yhyh Thus, x1)H) = a(anti(h = yh (antih1 )1.and xH = yh2 (anti(h1 ))H 2 .2(anti(h 1 ))(2) = yH by of 2Lemma = neutH and =HyH by (2) of Lemma 2 (anti(h = yH 1 )H) = anti(x)yH ⇔b ∈ ⇔ (2)1.of Lemma 1, anti(x)y ∈ H. ⇒ a(anti(b)) ∈ kerf. (5)= yhxH kerf, then (5) (6) xH xH = yH ⇔ H⇐= anti(x)yH ⇔=(2) of Lemma=1,H(x anti(x)y ∈ H. = Hx (xH)anti(x) (Hx)anti(x) ∗ anti(x) = H ⇐ xH(anti(x)) = H. We let n ∈ N and a ∊ kerf. We had to show that n . a . (anti(n)) ∈ kerf. (f is neutro-homomorphism) f(b) =xH(anti(x)) neutH (6) (7) xH (That = Hx ⇐ (xH)anti(x) = (Hx)anti(x) = H(x ∗ anti(x) = H ⇐ = H. is, xH = H) (n . a . (anti(n) = f(n) . f(anti(n)) (7). f(a) (That is, xH = H) Then,we that f(a(anti[b]) = neut H. (f subgroup is neutro-homomorphism) Suppose thay xH is ashowed neutrosophic extended triplet of N. Then = f(n) f(a) anti(f(n)) f(a(anti(b)) = f(a) . f(anti(b)) xH contains so xHextended = H by triplet (3) of subgroup Lemma 1,ofwhich holds ⇔ x ∈ H by (2) of Suppose thay xHthe is aidentity, neutrosophic N. Then = h neutH (anti(h)) = f(a) . f(anti(b)) Lemma 1.

Symmetry 10, x FOR REVIEW 9 of 13 xH 2018, contains the=PEER identity, so xH =HH = neutH neutH . anti(neut ) by (3) of Lemma 1, which holds ⇔ x ∈ H by (2) of Lemma 1. ∈ kerf ⇒ n . a . (anti(n)) Symmetry 2018, 10, x FOR PEERneut REVIEW 9 of 13 H . neut H = H ≤ N by (2) of Lemma 1.

Conversely, Conversely, ifif xx=∈ ∈H, H,then thenxH xH = H ≤ N by (2) of Lemma 1. ⇒ kerf  N. = neut H (8) (xy)H = x(yH) and H(xy) = (Hx)y follows from the associative if x ∈and H, H(xy) then xH =∈ H ≤kerf.

N by (2)from of Lemma 1.

(8)Conversely, (xy)H = x(yH) = (Hx)y follows the associative a(anti(b)) ⇒ rem 5. A neutrosophic triplet subgroup H ofmultiplication.

N is a neutrosophic triplet normal subgroup of N if, and property of property of group group multiplication.

(8) (xy)H = x(yH)(2) andWe H(xy)n=∈ (Hx)y follows from the associative N neutrosophic and a ∊ kerf.

We hadcoset to show n . a . (anti(n)) ∈ kerf. (f is neutro-homomorphism) f, each left neutrosophic of H is axH right triplet of Hthat ∈ N. (9) (Find aa map α: xHNlet xH that is one one to one one and onto) (9) triplet (Findcoset map α: in xH ⟶ that is to and onto) property of group multiplication.

Consider α: xHf(n ⟶. xH defined=by α (xh) =f(anti(n)) yh. This is clearly onto yH. Suppose α (xh1) a . (anti(n) f(n) f(a) and .= Consider α: xH xH defined by α .(xh) yh.onto) This is N clearly onto yH. Suppose α (xh1 ) (9) (Find a map α: xH ⟶ xH that is one to one ⩝ x∈ f Let H be a neutrosophic triplet normal subgroup of N, then xH(anti[x])=H, ⇒ = α (xh2). Then =yhf(n) 1 = yh xh1 = xh2, therefore ⍺ is one to one. ⇒ h1 = h2 by left cancellation ⇒ xH(anti[x])x f(a)2 anti(f(n)) Consider α: xH ⟶ xH defined by α (xh) = yh.

This is clearly onto yH. Suppose α (xh1) ⩝ x ∈ N ⇒ xH = Since H x, ⩝x N,2 ).since each left neutrosophic triplet coset xH isxH theand right neutrosophic =⍺provides α∈(xh Then yh cancellation ⇒yH, xh = xh=2∣yH∣. , therefore α is one to one. a one one between □ 2 ⇒ h1 = h 2 by left 1= 1∣xH∣ =yh htoneut H correspondence (anti(h)) = α (xh 2). Then yh1 = yh2 ⇒ h1 = h2 by left cancellation ⇒ xh1 = xh2, therefore ⍺ is one to one. t coset Hx. □ Since α provides a one to one correspondence between xH and yH, | xH | = | yH | .

= neut Since ⍺provides a one to one Hcorrespondence between xH and yH, ∣xH∣ = ∣yH∣. □ In classical group theory, cosets were used in the construction of vitali sets (a type of ⇒ n . a . (anti(n)) ∈ kerf Conversely, let each left neutrosophic triplet coset of H in N be a right neutrosophic triplet coset non-measurable set),⇒ andkerf in computational group theory cosets were used to decode received data in cosets N. InifIn classical group theory, were used in in thethe construction of of vitali sets (a (a type of of group theory, cosets were used construction sets type in N. This means that x error-correcting isclassical any element of N, then left neutrosophic triplet coset is also a vitali linear codes, tothe prove Lagrange’s theorem.

The xH neutrosophic triplet coset plays a non-measurable set), andand in theory cosets were usedused to decode received data in non-measurable set), in computational group cosets were to decode received data neutrosophic triplet coset.role Now ∈ computational H, x group ∗ neu t(x) =theory x triplet ∈ xH. Consequently similar in neut(x) the theory ofAtherefore neutrosophic extended group, in thex classical group theory.

Theorem 5. neutrosophic triplet subgroup H of N is as a neutrosophic triplet normal subgroup of N if, and linear error-correcting codes, to prove Lagrange’s theorem.

The neutrosophic triplet coset plays a in linear error-correcting codes, to prove Lagrange’s theorem.

The neutrosophic triplet coset plays also belong to that right neutrosophic triplet could coset, be which tosuch left neutrosophic triplet Neutrosophic triplet cosets usedistriplet inequal areas, as neutrosophic computational modelling, only if, each left neutrosophic coset of H in N is a right neutrosophic triplet coset of H ∈ N. role in the of neutrosophic extended triplet group, as in the classical group theory. a similar role in theory the theory of neutrosophic extended triplet group, as in the classical group theory. xH. However, similar x is left neutrosophic triplet coset and needs to contain one common element toaprove Lagrange’s theorem in the neutrosophic extended triplet, etc.

Neutrosophic could ininareas, neutrosophic computational Neutrosophic triplet cosets couldbebeused used areas,such suchas ascoset neutrosophic computational modelling, modelling, to e they are identical.

Therefore,triplet Hx iscosets the unique neutrosophic triplet which equal Proof Let H be aright neutrosophic triplet normal subgroup of N,isthen xH(anti[x])=H, ⩝ x ∈ N ⇒ xH(anti[x])x to prove Lagrange’s theorem in the neutrosophic extended triplet, etc. prove Lagrange’s theorem in the neutrosophic extended triplet, etc. e left neutrosophic triplet cosetand Therefore, wexHhave =∈ ⩝x ∈each N ⇒ (anti(x)) =triplet coset xH is the right neutrosophic 4.2.

The Index Lagrange’s Theorem: ∣⩝x di vixH, des ∣ =xH. Hx, ⩝x∈N ⇒ =∣H H x,xH N,∣N since leftxH neutrosophic nti(x), ⩝x ∈ N ⇒ xH (anti(x)) = H ,triplet ⩝x∈ N, since H□ is a neutrosophic triplet normal subgroup of coset Hx. Index Lagrange’s Theorem: | divides N| ∣ 4.2.4.2.

TheThe Index andand Lagrange’s Theorem: ∣H|H∣ divides|∣N Theorem 3. If N is a finite neutrosophic extended triplet group and H ≤ N, then ∣H ∣/∣N ∣.M oreover, the number of the distinctConversely, left neutrosophic triplet of H in N is ∣N ∣ /∣Hof ∣. let each leftcosets neutrosophic triplet coset H in/∣N N be a Mright neutrosophic triplet coset ∣. oreover , the Theorem 3. If NNis isa finite Theorem 3 If a finiteneutrosophic neutrosophicextended extendedtriplet tripletgroup groupand and H H ≤≤N, N,then then∣H | H|∣ /| N|. Moreover, the number of H in N. This means that if x is any element of N, then the left neutrosophic triplet coset xH is also a number the distinct left neutrosophic cosets Hisin| N ∣. of the of distinct leftxneutrosophic triplettriplet cosets of H inofleft N N|is/| ∣N H|. ∣/∣H 2H, …, xrH denote the distinct neutrosophic triplet cosets of H in N. Then, ⩝ x ∈ Proof Let x1H, right neutrosophic triplet coset. Now neut(x) ∈ H, therefore x ∗ neu t(x) = x ∈ xH. Consequently x N. xH = xiH formust some i = 1,belong 2, …, r.

(1) of Lemma 1, x ∈coset, xH. Thus, N =equal x1H ∪tox2left H ∪neutrosophic …∪ toConsidering that right which triplet rH denote the distinct leftneutrosophic neutrosophictriplet triplet cosets of H is in N. Then, ⩝x∈ Proof Let x1H, x2H, …, xalso xrH. Considering (4) of Lemma 1, this union is disjointed: coset xH. However, x is a left neutrosophic triplet coset and needs to contain one common element N. xH = xiH for some i = 1, 2, …, r. Considering (1) of Lemma 1, x ∈ xH. Thus, N = x1H ∪ x2H ∪ … ∪ before they are identical. xrH. Considering (4) of Lemma 1, this is disjointed: ∣N∣union = ∣x1Therefore, H∣ + ∣x2H∣ +Hx ... +is∣xthe rH unique = r∣H∣. right neutrosophic triplet coset which is equal to the left neutrosophic triplet coset xH. Therefore, we have xH = xH, ⩝x ∈ N ⇒ xH (anti(x)) = ∣xr.

1H∣ Therefore: ∣xiH∣ = ∣xH∣ for i = 1,∣N∣ 2, =…, □ + ∣x2H∣ + ... + ∣xrH = r∣H∣.

Hx(anti(x), ⩝x ∈ N ⇒ xH (anti(x)) = H , ⩝x∈ N, since H is a neutrosophic triplet normal subgroup of N. for i = 1, 2, …, r. □ Therefore: ∣xiH∣ = ∣xH∣ Example 13. We let H = ([1], [12]), it had three left neutrosophic triplet cosets in S3, see example 11, [S3:H] = 3 = (H, [13]H, [23]H) = (H, [13]H, [23]H).

Example 13. We let H = ([1], [12]), it had three left neutrosophic triplet cosets in S3, see example 11, [S3:H] = 3

tion 21. If N is a neutrosophic extended triplet group and H  N is a neutrosophic triplet normal 1. group xH N/H = x’Hhas and yH = y’H up, then the neutrosophic triplet quotient elements xH: x ∈ N, the neutrosophic triplet Xh 1 = x’ and yh2 = y’, h1, h2∊ H f H in N, and an operation of (xH)(yH) = (xy)H. 10, x FOR PEER REVIEW 13 x’y’H = xh1yh2H = xh1yH = x h1Hy = xHy11 = of xyH. 2. Thetriplet neutral, for any x ∊ H, neut(x)H H. That = x ∗ neut(x)H = xH ple 16. Let’s find allSymmetry of the possible neutrosophic quotient groups foristhe dihedral =group D3. is, xH ∗ H = xH ∗ neut(x)H 2018, 10, 126 10 of 13 hic Triplet Quotient 2 (Factor)3Groups 2 3.

2 An anti of a neutrosophic triplet coset xH is anti(x)H, since xH∗ anti(x)H = (x ∗ anti(x)H) = 3 = (1, r, r , s, sr, đ?‘ đ?‘ đ?‘&#x;đ?‘&#x; ), where đ?‘&#x;đ?‘&#x; = đ?‘ đ?‘ = rsrs = 1. A quotient set D3/N is a neutrosophic triplet group if, = H. y if,of Nquotient  D3. Then, all of neutrosophic triplet normal subgroups areof D3classical itself. We always the trivial on (factor) groups was one ofneut(x)H the central concepts grouphave theory of 13 2 denote 11 2 distinct 4.

Associativity, (xHyH)zH = (xy)HzH = (xy)zH = xH(yz)H xH(yHzH), Proof.

Let x H, x H, . . . , x H the left neutrosophic triplet cosets of=H in N. Then, âŠ? x ∈x, y, z ∈ N. â–Ą r 1 2 3/Dimportant 3 = 1 ≅ 1 and D3in /1 ≅the D3study . The subgroup â&#x;¨râ&#x;Š = â&#x;¨đ?‘&#x;đ?‘&#x;structure â&#x;Š = (1, r, of đ?‘&#x;đ?‘&#x; )groups. is that ofJust index thus is normal. an role of the general as 2inand a classical N. xH = x H for some i = 1, 2, . . . , r.

Considering (1) of Lemma 1, x ∈ xH. Thus, N = x 1 H âˆŞ x2 H âˆŞ, . . . , re,quotient D3/â&#x;¨râ&#x;Š isgroups also a neutrosophic tripletrole quotient group.

If Nof  D3 is a different neutrosophic y, played ai similar in the theory neutrosophic extended triplet triplet or) Groups 7.

Conclusions âˆŞintroduced x=r H. this union is 213disjointed: ofquotient â&#x;¨đ?‘ đ?‘ â&#x;Š,of then â&#x;¨đ?‘ đ?‘ â&#x;Š.

2, Considering so either = (4) = â&#x;¨đ?‘ đ?‘ đ?‘ đ?‘ â&#x;Š. 1,or đ?‘ đ?‘ triplet = 11â&#x;¨đ?‘ đ?‘ đ?‘&#x;đ?‘&#x; â&#x;Š. However, none of its them are s subgroup, section, we have the N notion ofNLemma neutrosophic group and ups was one of the central concepts of classical group theory 2 (đ?‘ đ?‘ đ?‘ đ?‘ )đ?‘ đ?‘ (đ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Ž(đ?‘ đ?‘ đ?‘ đ?‘ )) = triplet đ?‘ đ?‘ đ?‘&#x;đ?‘&#x; notgroup. e, since neutrosophic extended in â&#x;¨đ?‘ đ?‘ â&#x;Š. Hence, the only non-triavial neutrosophic triplet quotient The main theme of this paper was to introduce the neutrosophic extended triplets and then to |N |classical = |x1 H| + |x2 H| + ... + |xr H = r|H|. udy general structure of groups.

Just as in a st (Factor) D3/of â&#x;¨râ&#x;Š.theGroups utilize these neutrosophic extended triplets in order to introduce the neutrosophic triplet cosets a1.similar in the theory of neutrosophic extended triplet Ifgroups N is role a was neutrosophic extended triplet group Htriplet N is atheory neutrosophic triplet normal the neutrosophic triplet quotient group. We also group neutrosophic and finally, or) one of the central concepts of |and classical Therefore: |xi H | = |xH for i = 1,and 2,normal . its . . , r. subgroup,

ced the notion of neutrosophic triplet quotient group em Let N of be athe neutrosophic extended triplet group andinteresting H be aaneutrosophic tripletofnormal subgroup of n the6.study neutrosophic triplet quotient group N/H hassome elements xH: xclassical ∈ properties N, the neutrosophic triplet studied these newly created structures and their application to the general structure of groups.

Just as in iplet group. he set N/H = xH, x ∈ N is a neutrosophic extended triplet group under the operation of (xH)(yH) = xyH. N, and an operation of (xH)(yH) = (xy)H. neutrosophic extended triplet group. We further the neutrosophic kernel layed a similar role in the theory neutrosophic extended Example 13. Weoflet H = ([1], [12]), it had triplet three left neutrosophic triplet cosets indefined S3, see example 11, neutrosophic-image, and inverse image for neutrosophic extended triplets.

As a further ntroduced the notion of neutrosophic triplet quotient group and its [S3:H] 3 =N(H, [23]H) =triplet (H, [13]H, [23]H). ndedĂ—find triplet group and H = is a[13]H, neutrosophic normal N/H N/H Let’s all→group. ofN/H the possible neutrosophic triplet quotient groups for the group D3. generalization, we created a dihedral new field of research, called Neutrosophic Triplet Structures (namely nded triplet 2 3 2xH: x ∈ N, the neutrosophic triplet 2, s, tient group N/H has elements r, r sr, đ?‘ đ?‘ đ?‘&#x;đ?‘&#x; ), where đ?‘&#x;đ?‘&#x; = đ?‘ đ?‘ = rsrs = 1.

A quotient set D 3 /N is a neutrosophic triplet group if, the neutrosophic triplet cosets, neutrosophic triplet normal subgroup, and neutrosophic triple H = x’H and yH = y’H yH) =D(xy)H.  3. Then,triplet all of 5. neutrosophic triplet normal subgroups are D3 itself . We always have the trivial Neutrosophic Triplet Normal Subgroups quotient group). hic extended group and H N is a neutrosophic triplet normal  h1 = x’ and yh2 = y’, h1, h2∊ H 2 2 1y’H ≅ quotient 1=and 3group /1 ≅ DN/H 3. Thehas subgroup = â&#x;¨xđ?‘&#x;đ?‘&#x;=∈ â&#x;ŠxyH. =N, (1,the r, neutrosophic đ?‘&#x;đ?‘&#x; ) is that of triplet index 2 and thus is normal. plet elements xh1D yh 2H = xh 1yHIn = groups xthis h1Hy =â&#x;¨râ&#x;ŠxH: xHy section, the neutrosophic triplet normal subgroup and neutrosophic triplet normal eutrosophic triplet quotient for the dihedral group DD 3.

Author Contributions: All authors have contributed equally to this paper. The individual responsibilities and â&#x;¨he r(xH)(yH) â&#x;Š is also a neutrosophic triplet quotient group.

If N 3 is a different neutrosophic triplet  (xy)H. anysubgroup x ∊set H,D is3/N neut(x)H =been H. That is, xHgroup ∗ H = if, xH ∗ neut(x)H = x ∗ neut(x)H = xH. 2 neutral,=for test outlined. = rsrs = 1. A quotient is have a neutrosophic triplet 2 contribution of all authors can be described as follows: the idea of this whole paper was put forward by Mikai â&#x;¨đ?‘ đ?‘ â&#x;Š, oup, then â&#x;¨đ?‘ đ?‘ â&#x;Š. = 2, so either N = N = â&#x;¨đ?‘ đ?‘ đ?‘ đ?‘ â&#x;Š. or đ?‘ đ?‘ = â&#x;¨đ?‘ đ?‘ đ?‘&#x;đ?‘&#x; â&#x;Š.

However, none of them are n anti of normal a neutrosophic triplet coset is anti(x)H, since xH∗ anti(x)H = (x ∗ anti(x)H) = phic triplet subgroups are D3 itself .Bal, WexH always have the trivial he also completed the preparatory work of the paper. Moges Mekonnen Shalla analyzed the existing work â&#x;¨đ?‘ đ?‘ â&#x;Š. (đ?‘ đ?‘ đ?‘ đ?‘ )đ?‘ đ?‘ (đ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Ž(đ?‘ đ?‘ đ?‘ đ?‘ )) đ?‘ đ?‘ đ?‘&#x;đ?‘&#x; 2 not in20.

Hence, the only non-triavial quotient extended triplet group N is called ssible neutrosophic quotient groups for the dihedral group Dneutrosophic 3subgroup . eut(x)H = H. 2 =triplet A neutrosophic extended triplet H oftriplet a neutrosophic Definition 2 of symmetry 292516 neutrosophic triplet coset and quotient group and wrote part of the paper. The revision group â&#x;¨râ&#x;Š = â&#x;¨ đ?‘&#x;đ?‘&#x; â&#x;Š = (1, r, đ?‘&#x;đ?‘&#x; ) is that of index 2 and thus is normal.

3 2 . đ?‘&#x;đ?‘&#x; = đ?‘ đ?‘ = rsrs(xHyH)zH eâ&#x;Šssociativity, = 1. aAneutrosophic quotient set D 3/N isnormal a neutrosophic triplet = (xy)HzH =and (xy)zH =subgroup xH(yz)H xH(yHzH), âŠ? x ∈x,N y,and zNecati ∈weN.denote â–Ą triplet of=N, ifgroup xH =if,Hx, it as H E N. submission of this paper was completed by Olgun. plet quotienttriplet group.

If N  D3 is a are different triplet utrosophic normal subgroups D3 itself.neutrosophic We always have the trivial 2 â&#x;¨đ?‘ đ?‘ â&#x;Š, N N= be N = â&#x;¨đ?‘ đ?‘ đ?‘ đ?‘ â&#x;Š. 2or đ?‘ đ?‘ = â&#x;¨đ?‘ đ?‘ đ?‘&#x;đ?‘&#x; â&#x;Š. However, none them areauthors Conflicts Interest: The declare noof conflict ofof interest. et a neutrosophic group HS be aofwas neutrosophic triplet normal subgroup The subgroup â&#x;¨râ&#x;Š = â&#x;¨Example đ?‘&#x;đ?‘&#x; â&#x;Šextended = (1, r, đ?‘&#x;đ?‘&#x; 2The ) isset that index 2 Ďƒand thus isa normal. clusions 14.triplet Anof=and Ďƒ ∈of even normal subgroup Sn . It was called the alternating neutrosophic n/ â&#x;¨đ?‘ đ?‘ â&#x;Š. in Hence, the only non-triavial neutrosophic triplet quotient /H xH, x quotient ∈ N is aextended neutrosophic extended group the operation of that (xH)(yH) xyH. Since |Sn |= n!, thus, hic =triplet group.

Iftriplet N  D3 is on atriplet different neutrosophic triplet group n letters.

Itunder was enough to notice An = =ker(sgn). he main theme of this paper was to introduce the neutrosophic extended triplets and then to o either N = â&#x;¨đ?‘ đ?‘ â&#x;Š, N = â&#x;¨đ?‘ đ?‘ đ?‘ đ?‘ â&#x;Š. or đ?‘ đ?‘ = â&#x;¨đ?‘ đ?‘ đ?‘&#x;đ?‘&#x; 2 â&#x;Š. However, none of them are these extended triplets in order to introduce the neutrosophic triplet cosets, → neutrosophic N/H N/H |An | = n!/2. đ?‘&#x;đ?‘&#x; 2 not in â&#x;¨đ?‘ đ?‘ â&#x;Š. Hence, the only non-triavial neutrosophic triplet quotient sophic andtriplet finally, the neutrosophic ed triplettriplet groupnormal and H besubgroup, a neutrosophic normal subgroup of triplet quotient group. We also Symmetry 2018, 10, x FOR PEER REVIEW 10 of 13 H and yHinteresting = y’H d some properties ofoperation these newly created structures and their phic extended triplet group under the of (xH)(yH) = xyH. Sn /An = n!/n!/2 = 2. application to Symmetry 10 of 13 and yh 2 = y’, h 1, h2∊ H2018, 10, x FOR PEER REVIEW sophic extended triplet group. We further defined the neutrosophic kernel, Proof Let H be a neutrosphic triplet subgroup of N. Suppose H is neutrosophic extended triple extended group a neutrosophic triplet normal subgroup extended of xh 1yh2H =triplet xh1yH =Neutrosophic x hand 1HyH=be xHy = xyH. Triplet Normal Subgroup Test extended triplets. As a further sophic-image, and inverse image for neutrosophic Proof Let H be a neutrosphic extended triplet subgroup of N. Suppose is=neutrosophic extended= ztriplet subgroup of N. Then âŠ? x ∈ N, y ∈ H : Đ­z ∈ H :H=xyxH. zx. Thus (xy)anti(x) ∈ H implying (xH)anti(x) ⊆ eutrosophic extended triplet group under the operation of (xH)(yH) = xyH. tral, for any ∊ H, is aneut(x)H = H. is, xHcalled ∗ H = Neutrosophic xH ∗ neut(x)HTriplet = x ∗ neut(x)H lization, we xcreated new field of That research, Structures (namely, subgroup of N.coset Then âŠ? xH. âˆˆâ–Ą N, y extended ∈ H since : Đ­ztriplet ∈xH∗ H subgroup : xy = zx. Thus (xy)anti(x) = z =∈if,Hand implying (xH)anti(x) ⊆ of a neutrosophic triplet xH is anti(x)H, anti(x)H =of(xN ∗is anti(x)H) 4 A neutrosophic Hand normal in N utrosophic tripletTheorem cosets, neutrosophic triplet normal subgroup, neutrosophic triplet only if, anti(x)Hx ⊆ H, H. â–Ą Hnt=group).

H. Conversly, suppose âŠ? xâŠ?∈ :(xH)anti(x) =ivity, xHy (xHyH)zH = xyH. = (xy)HzH = (xy)zH = xH(yz)H = xH(yHzH), x,N. y, z ∈ N. â–Ą ⊆H. Then for n ∊ N , w e have (nH )anti(n) ⊆H , w hich x ∈N implies nH ⊆ H n.

A lso, for anti(n) (anti =of13 anti(n)Hn ⊆ H, which Conversly, suppose âŠ? x ∈ N :(xH)anti(x) ⊆ H. Then for ∊n ∊N N, w , we ehave haveanti(n)H (nH )anti (n)[anti{n}]) ⊆H , w 10 hi ch x)H = H. That is, xH ∗ H2018, =2018, xH ∗xneut(x)H =REVIEW x ∗ neut(x)H = xH. Symmetry 10, xFOR FORPEER PEERREVIEW 10of 13 10, All authors have contributed equally to this paper. The individual responsibilities and HContributions: Symmetry implies Hn nH . Therefore, nH = H n, meaning that H  N. implies nH ⊆xH∗ H n. anti(x)H A lso, for anti(n) ∊ N , w e= have anti(n)H (anti[anti{n}]) = anti(n)Hn ⊆ H, which coset xH isauthors anti(x)H, since (x idea ∗⊆anti(x)H) ns ution can be described as follows:=the of this whole paper was put forward by Mikail h1Hyof= all xHy = xyH. implies Hn ⊆ nH .

Therefore, nH = H n , meaning that H N.  Proof Let H be a neutrosphic extended triplet subgroup of Suppose H is neutrosophic extended triplet Proof.

Let H be a neutrosphic extended triplet subgroup of N. Suppose H is neutrosophic extended triplet Proof Let H be a neutrosphic extended triplet subgroup of N. Suppose H is neutrosophic extended triplet also completed preparatory paper. Moges Mekonnen= Shalla analyzed the existing work H.the That is, was xH ∗toHwork = xHof∗the neut(x)H = x ∗ neut(x)H xH. triplets nneut(x)H theme of= this paper introduce the neutrosophic extended and then to Example 15.

We let f: N → H be a neutro-homomorphism from a neutrosophic extended subgroup N.Then Then H :: Đ­z ∈HHHwrote :xy xy===part zx.Thus Thus(xy)anti(x) (xy)anti(x) ∈H implying(xH)anti(x) (xH)anti(x)⊆⊆H.triplet group N to a H = (xy)zH = xH(yz)H = xH(yHzH), âŠ?x∈ x,N, y, ∈ â–Ąz∈∈ subgroup of N. Then x∈∈ N, yz∈ ∈ H:N. zx.

Thus (xy)anti(x) zz∈∈ HHimplying implying (xH)anti(x) subgroup ofof N. âŠ?âŠ?xand N, y y∈ Hgroup Đ­z ::xy zx. ===zrevision metry 292516 neutrosophic triplet coset quotient paper. The riplet coset xH is anti(x)H, since xH∗ anti(x)H = (x ∗ and anti(x)H) = of the neutrosophic extended triplets in order to introduce the neutrosophic triplet cosets,  N. extended triplet group N to a extended triplet groupfrom H, Kerf We let f: by Nneutrosophic → H beOlgun. a neutro-homomorphism a neutrosophic H.H.â–Ą â–Ąwas15. bmission of thisExample paper completed Necati triplet normal subgroup,extended and finally, the neutrosophic neutrosophic triplet group H, Kerf triplet N. quotient group. We also of Interest: The authors declare no conflict interest. xy)HzH = (xy)zH = xH(yz)H = xH(yHzH), âŠ?a,xbx, z:(xH)anti(x) ∈we N.had â–Ą and (1) Ifofcreated ∀ kerf, to show a(anti[b]) Thishave meant(nH)anti(n) that kerf was⊆a H, neutrosophic extended Conversly, suppose ∈∈y, N ⊆ H.that Then for n ∈tokerf.

N, we ets interesting properties of these newly structures Conversly, suppose :(xH)anti(x) . Thenapplication forn n∊ ∊NN, w , we ehave have(nH (nH)anti(n) )anti(n)⊆⊆ , which hich Conversly, suppose ��x x∈∈NN:(xH)anti(x) ⊆⊆HH. their Then for HH, w to extended introduce (1) the neutrosophic extended triplets and then to triplet subgroup of N. If a ∈ kerf, then If ∀ a, b ∈ kerf, we had to show that a(anti[b]) ∈ kerf.

This meant that kerf was a neutrosophic extended which implies nH ⊆ Hn. Also, for anti(n) N, we have anti(n)H(anti[anti{n}]) = anti(n)Hn ⊆ H, triplet group.

We further defined neutrosophic kernel, impliesnH nH⊆⊆HHn. n.AAlso, lso, foranti(n) anti(n) we ehave have anti(n)H(anti (anti [anti{n}])= =anti(n)Hn anti(n)Hn⊆⊆H,H,which which implies for ∊ ∊NN, ,wthe anti(n)H [anti{n}]) plets in order to introduce the for neutrosophic triplet cosets, triplet subgroup of N. If a ∈ kerf, then which implies Hn ⊆neutrosophic nH. Therefore, nH = Hn, triplets. meaning As that aH E N. -image, and implies inverse image extended further impliesHn Hn⊆⊆nH nH. Therefore, . Therefore,nH nH= =HHn, nmeaning , meaningthat thatHH N. N. f(a) = neutH and finally, theaneutrosophic triplet quotient We alsoTriplet Structures (namely, n, was we created newthe field of research, called group.

Neutrosophic er to introduce neutrosophic extended triplets andf(a) then to H = neut fphic these newly created structures and their application to and neutrosophic and Example 15.

We let f: N → H be a neutro-homomorphism from a neutrosophic extended triplet group N to a triplet cosets, neutrosophic triplet normal subgroup, triplet extended ded triplets inExample order to 15. introduce neutrosophic triplet cosets, from Example 15.We Weletletf:the f:NN→ →HHbebea aneutro-homomorphism neutro-homomorphism froma aneutrosophic neutrosophic extendedtriplet tripletgroup groupNNtotoa a up.

We further defined the neutrosophic kernel, b ∈ kerf, then and neutrosophic extended triplet group H, Kerf E N. up). roup, and finally, the neutrosophic triplet quotient group.

We neutrosophic extendedtriplet tripletgroup groupH,H,Kerf KerfN. N. also neutrosophic extended age for neutrosophic extended triplets. As a further b ∈ kerf, then rties of these newly structures and their that application f(b)that = neut (1) created If ∀ a, b ∈ kerf, we had to show a(anti[b]) to ∈ kerf. This meant kerfHwas a neutrosophic extended of research, called Triplet Structures (namely, (1) Neutrosophic If∀∀ a,b b∈∈kerf, kerf, wehad had toshow showpaper. that a(anti[b]) kerf.responsibilities Thismeant meantthat that kerfwas wasa aneutrosophic neutrosophicextended extended (1) If a,contributed we toto that a(anti[b]) ∈∈kerf.

This kerf butions: All authors have equally this The individual and t group. We further the f(b)kernel, = neutH tripletdefined subgroup of N. Ifneutrosophic a ∈ kerf, then Then,we showed that f(a(anti[b]) = neut H . (f is neutro-homomorphism) f all authors can be described as follows: the idea of this whole paper was put forward by Mikail osophic triplet normal subgroup, and neutrosophic triplet triplet subgroupof ofN.N.If Ifa a∈∈kerf, kerf,then then triplet subgroup se image for neutrosophic extended triplets. As a further f(a) = neut f(a(anti(b)) = f(a) . f(anti(b)) Then,we f(a(anti[b]) = neut H. (f is neutro-homomorphism) mpleted the preparatory workshowed of the that paper.

Moges Mekonnen Shalla analyzed the H existing work w292516 field neutrosophic of research, triplet called coset Neutrosophic Triplet Structures (namely, f(a) = neut f(a) = neut HH and quotient group and wrote part of the paper.

The revision = f(a) . f(anti(b)) f(a(anti(b)) = f(a) . f(anti(b)) and triplet normal subgroup, and neutrosophic triplet nneutrosophic of this paper was by Necati Olgun. ntributed equally tocompleted paper.

The individual responsibilities and = neut H . anti(neut H) =this f(a) . f(anti(b)) and and bHwhole ∈kerf, kerf,then thenHwas as follows: idea declare of this paper put forward byHMikail = neut = bneut H . neut . anti(neut b∈∈ kerf, then nterest: The the authors no conflict of) interest. f(b) = neutH k of the paper. Moges Mekonnen Shalla analyzed the existing work = neutH responsibilities = neut H . neutH have contributed to this paper. individual f(b)=and =neut neut HH set and quotient equally group and wrote partThe of the paper. The revisionf(b) ∈ kerf. ⇒wasa(anti(b)) = neutHof this whole paper as follows: put forward

by Mikail dscribed

by Necati Olgun. the idea Then, we showed that f(a(anti[b]) = neut . (f is neutro-homomorphism) Then,we showed that f(a(anti[b]) neut . (fH neutro-homomorphism) Then,we showed that f(a(anti[b]) = =neut H.H(f isisneutro-homomorphism) a(anti(b)) ∈ kerf. ry work of the paper.⇒ Moges Mekonnen Shalla analyzed existing work (2) We let n ∈ the N and a ∊ kerf.

We had to show that n . a . (anti(n)) ∈ kerf. (f is neutro-homomorphism) oiplet conflict of and interest. f(a(anti(b)) f(a). f(anti(b)) . f(anti(b)) f(a(anti(b)) = =f(a) coset quotient group and wrote part of the paper. The revision (2) We let n ∈ N and a ∊ kerf. We had to show that n . a . (anti(n)) ∈ kerf. (f is neutro-homomorphism) f(n . a . (anti(n) = f(n) . f(a) . f(anti(n)) f(a). f(anti(b)) . f(anti(b)) = =f(a) mpleted by Necati Olgun. = f(n) f(a) anti(f(n)) f(n . aneut . (anti(n) = f(n) . f(anti(n)) . anti(neut = =neut HH . anti(neut H)H.)f(a) clare no conflict of interest. = h neutH (anti(h)) = =f(n) f(a) =neut neut . neut HH .anti(f(n)) neut HH = neutH = =h=neut neut HH(anti(h)) neut H

The notion of quotient (factor) groups was one of the central concepts of classical group theory and an important role the study of theory the general structure of groups. Just as in a classical ups quotient was one (factor) of the groups centralwas concepts oneplayed of of theclassical central group concepts theory of in classical group group theory, role in the theory of neutrosophic extended triplet portant udy of the rolegeneral in the study structure of the of groups. general structure Justquotient as in of a groups classical groups.played Just asainsimilar a classical 2018, 10, xrole FORplayed PEER 10 of 13 group. thistheory section, have introduced the triplet notion of neutrosophic triplet quotient group and its aient similar groups in theREVIEW theory a similar of role neutrosophic inInthe extended of we neutrosophic triplet extended relation to the neutrosophic extended triplet group. on, ed the we notion have introduced of neutrosophic the notion triplet of quotient neutrosophic group triplet and its quotient group and its Symmetry 2018, 10, 126 11 of 13 Let Hgroup. be aextended neutrosphic extended triplet subgroup of N. Suppose H is neutrosophic extended triplet rosophic plet triplet group. p of N. Then âŠ? x ∈ N, y ∈ HDefinition : Đ­z ∈ H : xy zx.NThus = zextended ∈ H implying (xH)anti(x) 21.= If is a (xy)anti(x) neutrosophic triplet group and H⊆ N is a neutrosophic triplet normal Symmetry 2018, 10, x FOR PEER REVIEW nded is a neutrosophic triplet group extended and Hf(a(anti(b)) and H neutrosophic N is group a=neutrosophic N is normal a neutrosophic triplet normal triplet triplet f(a)then . f(anti(b)) subgroup, the triplet quotient group N/H has elements xH: x 11 ∈ofN,13the neutrosophic triplet W 11 of 13 11 of 13 cosets of N, H N,neutrosophic and an operation = (xy)H. eutrosophic ient group N/H triplet hasquotient elements group xH: xN/H hasin elements xH: x∈ ∈ the triplet N, of the(xH)(yH) neutrosophic triplet f(a):(xH)anti(x) . f(anti(b)) 6. suppose Neutrosophic Quotient (Factor) nversly, âŠ? x ∈=Triplet ⊆H. Then Groups for n ∊ N , w e have (nH )anti(n) ⊆H , w hich H) an =operation (xy)H. of (xH)(yH) =N(xy)H. nt (Factor) nH ⊆ H n.Groups A lso, for = anti(n) N (factor) , w16. e have anti(n)H = anti(n)Hn ⊆ofHquotient , whichgroups )groups Example Let’s find all of (anti the [anti{n}]) possible neutrosophic triplet the dihedral group D3.

The notion ofneut quotient was one of the central concepts classical groupfor theory H ∊. anti(neut H 2 3 2 Hnall ⊆ofand nH Therefore, nH =central H n, meaning that H N.  đ?‘ đ?‘ đ?‘&#x;đ?‘&#x; nd utrosophic the.triplet possible quotient neutrosophic groups triplet for the dihedral groups group D for the dihedral 3), . group D 3theory = quotient (1, r, r2of ,study s, sr, where đ?‘&#x;đ?‘&#x; =structure đ?‘ đ?‘ group = rsrsD=3.of 1. groups.

A quotient setasD3in /Naisclassical a neutrosophic triplet group if, played an role in the of the general Just central tor) groups concepts was one of classical ofimportant the group concepts classical theory = neutH . neutH 2 2 only if, Ngroups.  Then, ofaif,neutrosophic triplet normal are D3 itself . We always have the trivial sr, =the rsrs đ?‘ đ?‘ đ?‘&#x;đ?‘&#x;study =group ),1. where quotient đ?‘&#x;đ?‘&#x; 3 = general đ?‘ đ?‘ 2Just set = rsrs Das 3/N =and isgroups aclassical neutrosophic quotient set D Dtriplet isgroup a all neutrosophic group if, subgroups theory, quotient a33./N similar role in the triplet theory of neutrosophic extended triplet n structure ofAof groups. the structure in1. aA ofplayed Just as in classical le 15.

We let f: N → Hsection, be=triplet athe neutro-homomorphism from a3/1itself neutrosophic extended triplet neut H hic . theory Then, triplet all normal ofneutrosophic neutrosophic are Dnormal 3 itself .of We subgroups have are D the trivial We always have the trivial group.

In subgroups this we have introduced the notion oftriplet neutrosophic its 2 and thus is normal. played aofsimilar role in extended theory ones D 3triplet /D 3 neutrosophic =always 1 ≅ 1 and D 3extended ≅ .D 3. The subgroup â&#x;¨râ&#x;Š =triplet â&#x;¨đ?‘&#x;đ?‘&#x; 2group â&#x;Š = quotient (1,Nr,tođ?‘&#x;đ?‘&#x; 2a) group is that and of index 2 2 2 2  ophic extended triplet group H, Kerf N. relation the extended introduced utrosophic triplet quotient group its triplet its Therefore, D 3/2â&#x;¨ â&#x;Šđ?‘&#x;đ?‘&#x;istriplet also aisgroup. neutrosophic triplet group. If N  D3 is a different neutrosophic triplet group nd D3/1 â&#x;¨râ&#x;Š ≅=Dâ&#x;¨3đ?‘&#x;đ?‘&#x;the . The â&#x;Š to =notion (1, subgroup r,neutrosophic đ?‘&#x;đ?‘&#x;of isâ&#x;¨râ&#x;Š that = â&#x;¨đ?‘&#x;đ?‘&#x;of and â&#x;Šindex (1, r,rand )quotient thus is that normal. ofgroup index 2and and thusquotient is normal. ⇒)neutrosophic a(anti(b)) ∈= kerf. ended triplet group. let lso quotient a neutrosophic group. triplet If N quotient D 3 is agroup. different If N neutrosophic D 3 â&#x;¨đ?‘ đ?‘ â&#x;Š. is a triplet different neutrosophic triplet   â&#x;¨đ?‘ đ?‘ â&#x;Š, normal subgroup, then = 2, so either N = N = â&#x;¨đ?‘ đ?‘ đ?‘ đ?‘ â&#x;Š. or đ?‘ đ?‘ neutro-homomorphism) = â&#x;¨đ?‘ đ?‘ đ?‘&#x;đ?‘&#x; 2 â&#x;Š. However, none of them are ∀ a, b ∈ kerf, we had that a(anti[b]) meant that kerf extended (2)to show We let n∈ N and a∈ kerf. kerf. This We had to show thatwas n . aa .neutrosophic (anti(n)) ∈ kerf. (f is 2 2 Definition 21. Ifđ?‘ đ?‘ =N= isnormal, aNneutrosophic extended triplet group and N isare a neutrosophic triplet normal them â&#x;¨đ?‘ đ?‘ â&#x;Š. â&#x;¨đ?‘ đ?‘ â&#x;Š, â&#x;¨đ?‘ đ?‘ â&#x;Š, since or (đ?‘ đ?‘ đ?‘ đ?‘ )đ?‘ đ?‘ (đ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Ž(đ?‘ đ?‘ đ?‘ đ?‘ )) = đ?‘ đ?‘ đ?‘&#x;đ?‘&#x; 2 not inHof Hence, the only non-triavial neutrosophic triplet quotient en N =â&#x;¨đ?‘ đ?‘ â&#x;Š. =N2,=so either â&#x;¨đ?‘ đ?‘ đ?‘&#x;đ?‘&#x; = â&#x;¨đ?‘ đ?‘ đ?‘ đ?‘ â&#x;Š. đ?‘ đ?‘ none = â&#x;¨đ?‘ đ?‘ đ?‘&#x;đ?‘&#x; â&#x;Š.

However, of them â&#x;Š. However, are none plet subgroup ofâ&#x;¨đ?‘ đ?‘ đ?‘ đ?‘ â&#x;Š.

N. If aorN ∈ kerf, then f(n . a . (anti(n) = f(n) . f(a) . f(anti(n)) 2 nd phic H extended triplet group and H N is a neutrosophic triplet normal N is a neutrosophic triplet normal   subgroup, the neutrosophic quotient N/H has elements xH: x ∈ N, the neutrosophic triplet â&#x;¨đ?‘ đ?‘ â&#x;Š. Hence, â&#x;¨đ?‘ đ?‘ â&#x;Š. group neutrosophic isthe Dtriplet 3/only â&#x;¨râ&#x;Š. non-triavial đ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Ž(đ?‘ đ?‘ đ?‘ đ?‘ )) = đ?‘ đ?‘ đ?‘&#x;đ?‘&#x; in thethen not onlyin non-triavial Hence, tripletgroup quotient neutrosophic triplet quotient f(a) = neut H = (xy)H. cosets of∈ H N, inN/H N, an operation ofx(xH)(yH) elements iplet quotient xH: xgroup has elements xH: theand triplet ∈ N, the neutrosophic triplet =neutrosophic f(n) f(a) anti(f(n)) of (xH)(yH) = (xy)H. Theorem 6.

Let N be a neutrosophic extended triplet group and H be a neutrosophic triplet normal subgroup of d = h neut (anti(h)) H Example 16.

Let’s find all of the possible neutrosophic triplet quotientsubgroup groups for the dihedral groupthe D3.operation of (xH)(yH) = xyH. eedakerf, triplet neutrosophic group and extended H be atriplet neutrosophic triplet benormal a= xH, subgroup normal oftriplet N.group In theand setH N/H x ∈ N is aofneutrosophic extended group under ∈ then 2 3 2 2 ossible otient neutrosophic for triplet dihedral quotient group groups D for the D3. of (xH)(yH) 3where .

3 =the (1,group r, r = , s, sr, đ?‘ đ?‘ đ?‘&#x;đ?‘&#x; ), đ?‘&#x;đ?‘&#x; of = (xH)(yH) đ?‘ đ?‘ dihedral = rsrs 1.

A quotient set D3/N is a neutrosophic triplet group if, H, x extended phic ∈groups N is aDtriplet neutrosophic under extended operation group under the==group operation xyH. = xyH. neut Hthe triplet 2 =neutrosophic neut D3. triplet Then, all off(b) neutrosophic triplet normal subgroups are D3 itself. We always have the trivial nt re set đ?‘&#x;đ?‘&#x; 3 =Dand đ?‘ đ?‘ 3/N = only rsrs is a neutrosophic =if,1.NA quotient set D group 3/N isĂ—if,aN/H triplet group if, → HN/H Proof N/H ⇒ n have . a . (anti(n)) ∈itselfkerf 2 trivial 2 neutrosophic bgroups are DD triplet 3that We normal the D 3neutro-homomorphism) . We always N/H showed ones 3itself /D.3 f(a(anti[b]) = 1 always ≅ 1 subgroups and=Dneut 3/1 ≅are D . The subgroup â&#x;¨râ&#x;Šhave = â&#x;¨đ?‘&#x;đ?‘&#x; the â&#x;Š = (1, r, đ?‘&#x;đ?‘&#x; ) is that of index 2 and thus is normal. en,we H . trivial (f=3is 1. xH x’H and yH = y’H 2 2 2 Therefore, D 3 / â&#x;¨ r â&#x;Š is also a neutrosophic triplet quotient group.

If N  D3 is a different neutrosophic triplet The r, đ?‘&#x;đ?‘&#x; subgroup ) is that of â&#x;¨râ&#x;Š index = â&#x;¨ đ?‘&#x;đ?‘&#x; 2 â&#x;Š and = (1, thus r, đ?‘&#x;đ?‘&#x; is ) normal. is that of index 2 and thus is normal. ⇒ kerf

N. (anti(b)) Xh1 = x’ and yh2 = y’, h1, h2∊ H yH = y’H= f(a) . f(anti(b)) ophic f(a)N. f(anti(b)) triplet D 3 is quotient a different group. neutrosophic If N D triplet 3 is a different neutrosophic triplet   normal subgroup, then â&#x;¨đ?‘ đ?‘ â&#x;Š.x’y’H = 2, so either N = â&#x;¨đ?‘ đ?‘ â&#x;Š, N = â&#x;¨đ?‘ đ?‘ đ?‘ đ?‘ â&#x;Š. or đ?‘ đ?‘ = â&#x;¨đ?‘ đ?‘ đ?‘&#x;đ?‘&#x; 2 â&#x;Š. However, none of them are = xh 1yh2H = xh1yH = x h1Hy = xHy = xyH. h2 = y’, h 1, h2∊ H 2â&#x;¨đ?‘ đ?‘ â&#x;Š, 22 â&#x;¨đ?‘ đ?‘ â&#x;Š.none normal, since đ?‘ đ?‘ đ?‘&#x;đ?‘&#x;â&#x;Š. However, notfor in any the only quotient = x ∗ neut(x)H = xH. so or Nâ&#x;¨đ?‘ đ?‘ đ?‘&#x;đ?‘&#x; N =(đ?‘ đ?‘ đ?‘ đ?‘ )đ?‘ đ?‘ (đ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Ž(đ?‘ đ?‘ đ?‘ đ?‘ )) â&#x;¨đ?‘ đ?‘ đ?‘ đ?‘ â&#x;Š.none or đ?‘ đ?‘ The == â&#x;¨đ?‘ đ?‘ đ?‘&#x;đ?‘&#x; of them are of isthem are non-triavial eut Hđ?‘ đ?‘ . anti(neut H )However, neutral, xHence, ∊ H, neut(x)H = H. Thatneutrosophic is, xH ∗normal H = triplet xH ∗ neut(x)H H xHy =either xh== 1yH xyH. == xâ&#x;Š.h 1Hy = xHy =52. xyH. Theorem . A neutrosophic triplet subgroup H of N is a neutrosophic triplet subgroup of N if, and only 2 â&#x;¨đ?‘ đ?‘ â&#x;Š. group is D 3/ â&#x;¨ r â&#x;Š . đ?‘ đ?‘ đ?‘&#x;đ?‘&#x; nly non-triavial not in neutrosophic Hence, the only triplet non-triavial quotient neutrosophic triplet quotient H . neutH An neutrosophic coset xH= is anti(x)H, since )H reut any = H. x ∊That H, isis,neut(x)H xH H ==left xH H.3. ∗That neut(x)H is, anti xH =triplet ∗of xH∗a=neut(x)H xH neut(x)H xH. ∗ neut(x)H xH. if,∗each neutrosophic coset∗ of H=in N triplet is= x a right neutrosophic triplet coset of HxH∗ ∈ N.anti(x)H = (x ∗ anti(x)H) = eutH xH is anti(x)H, H. neutrosophic coset triplet coset sincexH xH∗ is neut(x)H anti(x)H anti(x)H, ==since (x ∗ xH∗ anti(x)H) anti(x)H = = (x ∗ anti(x)H) = Theorem 6. Let N be a neutrosophic extended triplet group and H be a neutrosophic triplet normal subgroup of ∈ kerf. ⇒ a(anti(b)) 4. H Associativity, (xHyH)zH (xy)HzH = (xy)zHof= N, xH(yz)H = xH(yHzH), âŠ? x x,∈ y,Nz ∈ Proof. Let be a neutrosophic triplet =normal subgroup then xH(anti[x])=H, ⇒ N. â–Ą Hicbe extended a neutrosophic triplet group triplet H be subgroup a neutrosophic of triplet normal of under the operation of (xH)(yH) = xyH. N. In the set N/Hand =normal xH, x∈ N is a neutrosophic extendedsubgroup triplet group H = (xy)zH =xH(yz)H (xy)HzH xH(yHzH), (xy)zH xH(yz)H âŠ?that =y, zneutro-homomorphism) ∈ N. â–Ą left neutrosophic triplet coset xH is the right ==show Hx, x ∈x,nN ⇒ xHN. =â–Ą Hx,∈âŠ?kerf. x ∈x,N, each e(xHyH)zH let n ∈ N =and a ∊ xH(anti[x])x kerf.==We had to .xH(yHzH), az. ∈ (anti(n)) (fy,issince group neutrosophic under the extended operation triplet of (xH)(yH) group under = xyH. the operation of (xH)(yH) = xyH. 7. triplet Conclusions neutrosophic coset Hx. → N/H Proof=N/H N/H. f(anti(n)) . a . (anti(n) f(n) Ă—. f(a) The main theme of this paper was to introduce the neutrosophic extended triplets and then to (n) f(a) anti(f(n)) Conversely, 1. xH = x’H and yH = y’Hlet each left neutrosophic triplet coset of H in N be a right neutrosophic triplet coset utilize these neutrosophic extended triplets order me to introduce of this paper the was neutrosophic to introduce extended the neutrosophic triplets andextended then to triplets and in then to to introduce the neutrosophic triplet cosets, neutH (anti(h)) ofand H inyh N.2 =This that if x is any element of N, then the left neutrosophic triplet coset xH is also Xh1 = x’ y’, hmeans 1, h2∊ H neutrosophic triplet normal subgroup, and finally, the neutrosophic triplet quotient group. We also osophic plets in extended order to introduce triplets in the order neutrosophic to introduce triplet the neutrosophic cosets, triplet cosets, triplet neut(x) ∈ H, therefore x ∗ neut(x) = x ∈ xH. Consequently x x’y’H a= right xh1yhneutrosophic 2H = xh1yH = x h1Hy coset. = xHyNow = xyH. ∊eut HH studied some interesting of these t normal finally, subgroup, the neutrosophic and finally, triplet the quotient neutrosophic group. triplet Weproperties also quotient group.

Wenewly also created structures and their application to . a .2.=(anti(n)) ∈must kerf ⇒and The neutral, for any x ∊ H, is neut(x)H = H. That is, triplet xH ∗ Hcoset, = xH which ∗ neut(x)H = x ∗toneut(x)H = xH. also belong to that right neutrosophic is equal left neutrosophic triplet x hn1Hy xHy = xyH. neutrosophic extended triplet group.

We further defined the neutrosophic kernel, f resting these newly properties created of these structures newly and created their structures application and to their application to  kerf ⇒is∗ neut(x)H 3. ∗N.neut(x)H An of triplet is anti(x)H, since and xH∗needs anti(x)H = (x ∗ anti(x)H) = element H = xH = H.anti That =is,xaxH. xH ∗neutrosophic neut(x)H ∗ H = xH= ∗xH. = x ∗xH neut(x)H = xH. coset coset However, xneut(x)H is a leftcoset neutrosophic triplet to contain one common neutrosophic-image, inverse image for neutrosophic extended triplets. As a further ended up. Wetriplet further group. defined We the further neutrosophic defined and the kernel, neutrosophic kernel, neut(x)H = H. H, triplet sincecoset xH∗ xH anti(x)H is anti(x)H, = they (x ∗ are anti(x)H) since xH∗=anti(x)H = (xHx∗ is anti(x)H) = right neutrosophic triplet coset which is equal to before identical.

Therefore, the unique generalization, created a triplets. new field of research, called Triplet Structures (namely, age e, 5.and for inverse neutrosophic image for neutrosophic triplets. extended a further As a subgroup further m A neutrosophic tripletextended subgroup H of N iswe aAs neutrosophic triplet normal of N Neutrosophic if, and 4. Associativity, (xHyH)zH = triplet (xy)HzH = (xy)zH = xH(yz)H xH(yHzH), ∈ N. â–Ą the left neutrosophic coset xH. Therefore, we =have xH = xH, âŠ?x ∈x,Ny,⇒z xH(anti(x)) = Hx(anti(x), the neutrosophic triplet cosets, neutrosophic triplet normal subgroup, and neutrosophic triplet of created research, a new called field Neutrosophic of research, called Triplet Neutrosophic Structures (namely, Triplet Structures (namely, each neutrosophic coset Hâ–Ąin N is a right neutrosophic triplet coset of H ∈ N. (xy)HzH H = left xH(yHzH), = (xy)zHâŠ?=xtriplet xH(yz)H z xH(anti(x)) ∈=of N. xH(yHzH), z ∈ N. â–Ą a neutrosophic ∈x,Ny,⇒ = H, âŠ?x ∈x,N,y,since H is triplet normal subgroup of N. quotient group). sophic riplet cosets, triplet neutrosophic normal subgroup, tripletand normal neutrosophic subgroup,triplet and neutrosophic triplet 7. Conclusions et H be a neutrosophic triplet normal subgroup of N, then(Factor) xH(anti[x])=H, 6. Neutrosophic Triplet Quotient GroupsâŠ? x ∈ N ⇒ xH(anti[x])x Author Contributions: All authors contributed equally to this paper.and The then individual responsibilities and The main theme of this paper was to introduce the neutrosophic extended triplets to x ∈ N ⇒ xH = H x, âŠ?x ∈ N, since each left neutrosophic triplethave coset xH is the right neutrosophic contribution of all authors can be described as of follows: the idea of this whole paper was put theory forward by Mikail The notion of quotient (factor) groups was one the central concepts of classical group ns: ntributed All authors equally have to this contributed paper.

The equally individual to this responsibilities paper.

The individual and responsibilities and utilize these neutrosophic extended triplets in order to introduce the neutrosophic triplet cosets, eutrosophic per was to introduce extended the triplets neutrosophic and then extended to triplets and then to oset Hx. â–Ą Bal, he also completed the preparatory work of the paper.

Moges Mekonnen Shalla analyzed the existing work as thors follows: can be the described idea of this as follows: whole paper the idea was of put this forward whole paper by Mikail was put forward by Mikail and played an important role in the study of the general structure of groups.

Just as in a classical group neutrosophic triplet normal subgroup, and finally, the cosets, neutrosophic triplet quotient group. We also ntroduce nded triplets the neutrosophic in order to introduce triplet cosets, the neutrosophic triplet ofShalla symmetry 292516 neutrosophic triplet coset and quotient group and wrote part of the paper.

The revision d of the the preparatory paper.

Moges work Mekonnen of the paper.

Moges analyzed Mekonnen the existing Shalla work analyzed the existing work theory, quotient groups played a similar role in the theory of neutrosophic extended triplet group. some interesting properties of of these newly created structures triplet and their trosophic group,studied and triplet finally, quotient theneutrosophic neutrosophic group. We also triplet quotient group. alsoneutrosophic nversely, let each left triplet coset H in N be aWe right cosetapplication to and submission of this paper was completed by Necati Olgun. set neutrosophic and quotient triplet group coset and and wrote quotient part of group the paper. and wrote The part revision of the paper.

The revision Inifthis section, wetriplet have introduced the notion of neutrosophic triplet quotient and its relation to neutrosophic group.

We further the ated erties structures of thesethat and newly their application structures and theirleft application todefined N. This means x extended iscreated any element oftoN, then the neutrosophic triplet coset xHneutrosophic is also a groupkernel, sby paper Necati was Olgun. completed

by Necati Olgun. the neutrosophic extended triplet group.

Conflicts of Interest: The authors declare no conflict of interest. neutrosophic-image, inverse forx ∗neutrosophic triplets. xAs a further et defined group. the We neutrosophic further defined kernel, neutrosophic eutrosophic triplet coset.

Nowand neut(x) ∈the H,image therefore neukernel, t(x) = x ∈extended xH. Consequently The conflict authors of interest. no conflict generalization, we created a new field of research, Neutrosophic Triplet triplet Structures (namely, erse phic image extended for triplets. neutrosophic Asof interest. a extended further triplets.

As isacalled further so belong todeclare that right neutrosophic triplet coset, which equal to left neutrosophic Definition 21.

If N is a neutrosophic extended triplet group and H E N is a neutrosophic triplet normal the neutrosophic triplet cosets, neutrosophic triplet normal subgroup, and neutrosophic triplet Neutrosophic w field of research, Triplet called Structures Neutrosophic (namely, Triplet Structures (namely, H. However, x is a left neutrosophic triplet coset and needs to contain one common element subgroup, then the neutrosophic triplet quotient group N/H has elements xH: x ∈ N, the neutrosophic triplet group). mal ,hey neutrosophic subgroup, and triplet neutrosophic normal triplet andright neutrosophic triplet arequotient identical.

Therefore, Hxsubgroup, is the unique neutrosophic triplet coset which is equal cosets of H in N, and an operation of (xH)(yH) = (xy)H. eft neutrosophic triplet coset xH. Therefore, we have xH = xH, �x ∈ N ⇒ xH (anti(x)) = equally to this paper.

The individual and (x), �x Author ∈ N ⇒Contributions: xH (anti(x)) = All H , authors �x∈ N,have sincecontributed H is a neutrosophic triplet normal subgroupresponsibilities of Example 16. paper.

Let’s find all of theaspossible neutrosophic triplet quotient the dihedral group D3 . contribution of all responsibilities authors can be described follows: the idea of this whole paper groups was putfor forward

by Mikail his have paper. contributed The individual equally to this The and individual responsibilities and 2 ), where 3 = s2 =Moges Bal, he alsowas completed the Shalla existing worktriplet group D3forward = the (1, r,whole r2 ,Mikail s, paper sr, srwork rpaper. rsrs = Mekonnen 1. A quotient setanalyzed D3 /N is the a neutrosophic fescribed this whole as follows: paper theput idea of thispreparatory by wasof put forward by Mikail of of symmetry neutrosophic triplet coset and quotient group and wrote part of the paper.

The. We revision ory Mekonnen work Shalla the paper. analyzed Moges the Mekonnen existing work Shalla analyzed the existing work if,292516 and only if, N E D .

Then, all of neutrosophic triplet normal subgroups are D always have the 3 3 itself 2 2 and submission of this paper was completed by Necati Olgun. âˆź âˆź up triplet andcoset wrote and part quotient of the paper. group The and revision wrote part of the paper.

The revision trivial ones D3 /D3 = 1 = 1 and D3 /1 = D3 . The subgroup hri = hr i = (1, r, r ) is that of index 2 and thus is

ompleted by Necati Olgun.

Conflicts of Interest: The authors declare no conflict of interest. eclare no conflict of interest.

subgroup, then the neutrosophic triplet quotient group N/H has elements xH: x ∈ N, the neutrosophic triplet cosets of H in N, and an operation of (xH)(yH) = (xy)H.

Example 16. Let’s find all of the possible neutrosophic triplet quotient groups for the dihedral group D3.

D3 =2018, (1, r,10,r2126 , s, sr, đ?‘ đ?‘ đ?‘&#x;đ?‘&#x; 2 ), where đ?‘&#x;đ?‘&#x; 3 = đ?‘ đ?‘ 2 = rsrs = 1. A quotient set D3/N is a neutrosophic triplet group Symmetry 12 ofif, 13 and only if, N  D3. Then, all of neutrosophic triplet normal subgroups are D3 itself. We always have the trivial Symmetry 2018, 10, x FOR PEER REVIEW 10 of 13 ones D3/D3 = 1 ≅ 1 and D3/1 ≅ D3. The subgroup â&#x;¨râ&#x;Š = â&#x;¨đ?‘&#x;đ?‘&#x; 2 â&#x;Š = (1, r, đ?‘&#x;đ?‘&#x; 2 ) is that of index 2 and thus is normal. normal. Therefore, rineutrosophic is also a neutrosophic triplet quotient If 3NisEa D is a different neutrosophic 3 /ha D3/â&#x;¨rextended â&#x;Š is D also triplet If group.

N  D neutrosophic triplet

different 3 triplet Proof Let H beTherefore, a neutrosphic triplet subgroup of N.quotient Supposegroup.

H is neutrosophic extended 2 . However, N sr triplet normal subgroup, then . = 2, so either N = h s i , N = . or N = sr none of them are h i h i 2 metry 2018, 10, x FOR PEER REVIEW 10 of 13 â&#x;¨đ?‘ đ?‘ â&#x;Š, N = â&#x;¨đ?‘ đ?‘ đ?‘ đ?‘ â&#x;Š. normal â&#x;¨đ?‘ đ?‘ â&#x;Š.∈=H2,: xy so =either N = (xy)anti(x) = â&#x;¨đ?‘ đ?‘ đ?‘&#x;đ?‘&#x; (xH)anti(x) â&#x;Š. However,⊆ none of them are subgroup of N. Then âŠ? xsubgroup, ∈ N, y ∈then H : Đ­z zx. Thus = z ∈orHđ?‘ đ?‘ implying normal, since (sr )s( anti (sr )) = sr22 not in hsi. Hence, the only non-triavial neutrosophic triplet quotient â&#x;¨đ?‘ đ?‘ â&#x;Š. normal, since (đ?‘ đ?‘ đ?‘ đ?‘ )đ?‘ đ?‘ (đ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Ž(đ?‘ đ?‘ đ?‘ đ?‘ )) = đ?‘ đ?‘ đ?‘&#x;đ?‘&#x; not in Hence, the only non-triavial neutrosophic triplet quotient H. â–Ą group is D3/ hri. triplet subgroup of N. Suppose H is neutrosophic extended triplet of Let H be a neutrosphic group is extended D3/â&#x;¨râ&#x;Š. roup of Conversly, N. Then âŠ? xsuppose ∈ N, y ∈ : Đ­z H : xy = zx.⊆Thus (xy)anti(x) implying (xH)anti(x) âŠ? xH∈ N∈ :(xH)anti(x) H. Then for n ∊=Nz ,∈ w eHhave (nH )anti (n) ⊆H ⊆ , w hich Theorem 6 Let N be a neutrosophic extended triplet group and H be a neutrosophic triplet normal subgroup of Let N be a neutrosophic extended triplet(anti group and H be a neutrosophic normal subgroup of implies nH ⊆ Theorem H n. A lso,6. for anti(n) ∊ N , w e have anti(n)H [anti{n}]) = anti(n)Hn ⊆ triplet H, which N. In the set N/H = xH, x ∈ N is a neutrosophic extended triplet group under the operation of (xH)(yH) = xyH. N. In the set N/H = xH, x ∈ N is a neutrosophic extended triplet group under the operation of (xH)(yH) = xyH. implies Hn ⊆ nH . Therefore, nH = H n, meaning that H  N. Conversly, suppose âŠ? x ∈ N :(xH)anti(x) ⊆H. Then for n ∊ N , w e have (nH )anti(n) ⊆H , w hich Proof. Ă— N/H lies nH ⊆ H n. AProof lso, forN/H anti(n) ∊ NN/H ,→w N/H e have anti(n)H (anti[anti{n}]) = anti(n)Hn ⊆ H, which Example 15. We let f: N/H N →×HN/H be a→neutro-homomorphism from a neutrosophic extended triplet group N to a lies Hn ⊆ nH . Therefore, nH =0 H n, meaning that H  N. 0 neutrosophic extended triplet H, Kerf 1. xH xH xgroup Hand and yH yH N. 1. ==x’H yH = =y’H

= ax’x and 2 = y’, h , h,2∊h H 0 show mple Wea,let f: N →Xh H neutro-homomorphism from extended triplet group N toextended a (1) 15.

If ∀ b∈ kerf, we had to that kerf.

This meant that kerf was a neutrosophic Xh1be andyhyh y0 ,1h H a neutrosophic 1= 2 =a(anti[b]) 1 2∈ x’y’H = xh 1yh2H = xh1yH = x h1Hy = xHy = xyH.  rosophictriplet extended tripletofgroup H, Kerf N. subgroup N. If a ∈ kerf, then x0 y0neutral, H = xh1 yh = xxh∊1 yH x h1 Hy = =xHy = xyH. 2. The for2 H any H, is= neut(x)H H. That is, xH ∗ H = xH ∗ neut(x)H = x ∗ neut(x)H = xH. f(a) = neut H If ∀ a, b ∈ kerf, we had to show that a(anti[b]) ∈ kerf.

This meant that kerf was a neutrosophic 2.

The neutral, for any x H, is neut(x)H = H. is, xH ∗ since H = xH ∗extended neut(x)H ∗ neut(x)H = xH. 3. An anti of a neutrosophic triplet coset xH That is anti(x)H, xH∗ anti(x)H==x (x ∗ anti(x)H) = triplet and subgroup of3.N. Ifneut(x)H a ∈anti kerf, An of=then aH. neutrosophic triplet coset xH is anti(x)H, since xH∗ anti(x)H = (x ∗ anti(x)H) = neut(x)H = H. 4.4. Associativity, ==(xy)HzH ==(xy)zH ∈ N. Associativity,(xHyH)zH (xHyH)zH (xy)HzH (xy)zH==xH(yz)H xH(yz)H= =xH(yHzH), xH(yHzH), âŠ?x, y,x,zy,∈zN. â–Ą b ∈ kerf, then f(a) = neut H f(b) = neutH and Conclusions 7.7.Conclusions b ∈ kerf, then showed that f(a(anti[b]) = neutH. (f is neutro-homomorphism) Then,we Themain maintheme themeofofthis thispaper paperwas wastoto introduce neutrosophic extended triplets then The introduce thethe neutrosophic extended triplets andand then to f(a(anti(b))utilize =tof(a) . f(anti(b)) f(b) extended = neut H utilize these neutrosophic extended triplets order introducethe theneutrosophic neutrosophictriplet tripletcosets, cosets, these neutrosophic triplets in in order totointroduce = f(a) . f(anti(b)) neutrosophic subgroup,and andfinally, finally,the theneutrosophic neutrosophictriplet tripletquotient quotientgroup. group.We Wealso also neutrosophic subgroup, Then,we showed that f(a(anti[b]) triplet =triplet neutHnormal .normal (f is neutro-homomorphism) = neutH . anti(neut H) some interesting properties of these newly created structures and their application to studied studied some interesting properties of these newly created structures and their application to f(a(anti(b)) = f(a) . f(anti(b)) = neutH . neut H neutrosophic extended triplet group.

We further defined the neutrosophic kernel, neutrosophic-image, neutrosophic extended triplet group.

We further defined the neutrosophic kernel, = f(a) . f(anti(b)) = neutH and inverse image forand neutrosophic extended As a further generalization, inverse image for triplets. neutrosophic extended triplets. Aswea created furthera = neutH . anti(neutneutrosophic-image, H) ∈field kerf.of research, ⇒ a(anti(b)) new called Neutrosophic Triplet Structures (namely, the neutrosophic triplet cosets, generalization, we created a new field of research, called Neutrosophic Triplet Structures (namely, = neutH . neutH neutrosophic triplet normal subgroup, and neutrosophic triplet quotient group). neutrosophic cosets, subgroup, and neutrosophic triplet N and a ∊ kerf. We triplet had to show thatneutrosophic n . a . (anti(n))triplet ∈ kerf.normal (f is neutro-homomorphism) = (2) neutHWe let n ∈the quotient group). ∈ Author kerf. = f(n) ⇒ a(anti(b)) Contributions: All authors have contributed equally to this paper. The individual responsibilities and f(n . a . (anti(n) . f(a) . f(anti(n)) contribution of all authors can be described as follows: the idea of this whole paper was put forward by Mikail anti(f(n)) We let =n f(n) ∈ Nf(a) and a ∊ kerf.Contributions: Wecompleted had to show n . a have . (anti(n)) ∈ kerf.equally (f isMoges neutro-homomorphism) Author Allthat authors contributed to this paper. The individual and Bal, he also the preparatory work of the paper.

Mekonnen Shalla analyzedresponsibilities the existing work of = h neutH (anti(h)) contribution of all authors can be described as follows: the idea of this whole paper was put forward by Mikail symmetry 292516 neutrosophic triplet coset and quotient group and wrote part of the paper.

The revision and f(n . a . (anti(n) = f(n) . f(a) . f(anti(n)) submission of this paper completed

by Necati he also completed thewas preparatory work of the Olgun. paper. Moges Mekonnen Shalla analyzed the existing work = neutH Bal, = f(n) f(a) anti(f(n)) of symmetry 292516 neutrosophic triplet coset and group and wrote part of the paper. The revision ⇒ n . a . (anti(n)) Conflicts∈ of kerf Interest: The authors declare no conflictquotient of interest. = h neutH (anti(h))and submission of this paper was completed by Necati Olgun. ⇒ kerf  N. = neutH References and Note Conflicts of Interest: The authors declare no conflict of interest. n . a . (anti(n)) ∈ kerf triplet subgroup H of N is a neutrosophic triplet normal subgroup of N if, and ⇒ Theorem 5. A neutrosophic kerf  N. 1. Smarandache, F. A Unifying Field in Logics: Neutrosophic Logic; Philosophy; American Research Press: Santa Fe, ⇒ only if, each left neutrosophic triplet coset of H in N is a right neutrosophic triplet coset of H ∈ N. NM, USA, 1999; pp. 1–141.

Kandasamy Vasantha, Basic Neutrosophic Structures orem 5. Let A neutrosophic triplet subgroup H ofW.B.; N is Smarandache, a neutrosophic triplet normal subgroup N if, and and Their Application to � xAlgebraic ∈ N of⇒ xH(anti[x])x Proof H be a 2. neutrosophic triplet normal subgroup of N, thenF.xH(anti[x])=H, Fuzzy and Neutrosophic Models; ProQuest Information &coset Learning: Ann Harbor, MI, USA, 2004. if,=each left neutrosophic triplet coset of H in N is a right neutrosophic triplet of H ∈ N. Hx, � x ∈ N ⇒ xH = H x, �x ∈ N, since each left neutrosophic triplet coset xH is the right neutrosophic 3.

Kandasamy Vasantha, W.B.; Smarandache, F. Some Neutrosophic Algebraic Structures and Neutrosophic

triplet coset Hx. â–Ą Structures; ProQuest Information & Learning: MI, USA, 2006. âŠ? x Ann ∈ NHarbor, of Let H be a neutrosophicN-Algebraic triplet normal subgroup of N, then xH(anti[x])=H, ⇒ xH(anti[x])x Smarandache, F.; each Mumtaz, A. Neutrosophic triplet group.

Neural Comput.

Appl. 2018, 29, 595–601. [CrossRef] x, � x ∈Conversely, N ⇒ xH =4.H x, �x ∈ N, since left neutrosophic triplet coset xH is the right neutrosophic let each left neutrosophic triplet coset of H in N be a right neutrosophic triplet coset 5.

Smarandache, F.; Mumtaz, A. Neutrosophic triplet as extension of matter plasma, unmatter plasma, et of coset Hx. â–Ą H in N. This means that if x is any element of N, then the left neutrosophic triplet coset xH is also a and antimatter plasma. In Proceedings of the 69th Annual Gaseous Electronics Conference on APS Meeting

right neutrosophic triplet coset. Now neut(x) ∈ H, therefore x ∗ neu t(x) = x ∈ xH. Consequently x Abstracts, Bochum, Germany, Conversely, let each left neutrosophic triplet coset10–14 of HOctober in N be2016. a right neutrosophic triplet coset must also belong to that right neutrosophic triplet coset, which is equal to left neutrosophic triplet F.; Mumtaz, GroupxHand Its aApplication to Physics. in N. This means6.that Smarandache, if x is any element of N, thenA.theThe left Neutrosophic neutrosophic Triplet triplet coset is also coset xH. However, Available x is a leftonline: neutrosophic triplet coset and needs to contain one common element https://scholar.google.com.hk/scholar?hl=en&as_sdt=0%2C5&q=The+Neutrosophic+ t neutrosophic triplet coset.

Now neut(x) ∈ H, therefore x ∗ neu t(x) = x ∈ xH. Consequently x before they are identical.

Therefore, Hx is the unique right neutrosophic triplet coset is equal Triplet+Group+and+Its+Application+to+Physics&btnG= on 20which March 2018). t also belong to that right neutrosophic triplet coset, which is equal to left(accessed neutrosophic triplet to the left neutrosophic triplet coset xH. Therefore, we have xH = xH, �x ∈ N ⇒ xH (anti(x)) = Smarandache, F. Neutrosophic Extended Triplets; Special Collections.

Arizona State University: Tempe, AZ, t xH. However, x7.is a left neutrosophic triplet coset and needs to contain one common element Hx(anti(x), �x ∈ N ⇒ xH (anti(x)) = H , � x ∈ N, since H is a neutrosophic triplet normal subgroup of USA, 2016.Hx is the unique right neutrosophic triplet coset which is equal re they are identical. Therefore, N. he left neutrosophic triplet coset xH. Therefore, we have xH = xH, �x ∈ N ⇒ xH (anti(x)) = anti(x), �x ∈ N ⇒ xH (anti(x)) = H , �x∈ N, since H is a neutrosophic triplet normal subgroup of

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Available online: https://www.google.com/books?hl=en&lr=&id=4803DwAAQBAJ&oi=fnd&pg=PA15&dq= Neutrosophic+Perspectives:+Triplets,+Duplets,+Multisets,+Hybrid+Operators,+Modal+Logic,+Hedge+ Algebras.+And+Applications&ots=SzcTAGL10B&sig=kJMEHWsQ_tYjTJZbOk5TaNahnt0 (accessed on 20 March 2018). © 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).