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Refined neutrosophic quadruple (po-)hypergroups and their fundamental group.

1 Introduction

In 1934, Marty [19] introduced the concept of hypergroups by considering the quotient of a group by its subgroup. And this was the birth of an interesting new branch of Mathematics known as "Algebraic hyperstructures" which is considered as a generalization of algebraic structures. In algebraic structure, the composition of two elements is an element whereas in algebraic hyperstructure, the composition of two elements is a nonempty set. Since then, many different kinds of hyperstructures (hyperrings, hypermodules, hypervector spaces, ...) were introduced. And many studies were done on the theory of algebraic hyperstructures as well on their applications to various subjects of Sciences (see [12, 13, 30]). Later, in 1991, Vougioklis [28] generalized hyperstuctures by introducing a larger class known as weak hyperstructures or [H.sub.v]-structures. For more details about [H.sub.v]-structures, see [28, 29, 30, 31].

In 1965, Zadeh [32] extended the classical notion of sets by introducing the notion of Fuzzy sets whose elements have degrees of membership. The theory of fuzzy sets is mainly concerned with the measurement of the degree of membership and non-membership of a given abstract situation. Despite its wide range of real life applications, fuzzy set theory can not be applied to models or problems that contain indeterminancy. This is the reason that arose the importance of introducing a new logic known as neutrosophic logic that contains the concept of indeterminancy. It was introduced by F. Smarandache in 1995, studied annd developed by him and by other authors. For more details about neutrosophic theory, we refer to [17, 22, 23, 25]. Recently, many authors are working on the applications of this important concept. For example in [2], Abdel-Baset et al. offered a novel approach for estimating the smart medical devices selection process in a group decision making in a vague decision environment and used neutrosophics in their methodology. Moreover, in [7], R. Alhabib et al. worked on some neutrosophic probability distribution. Other interesting applications of it are found in [1, 3, 4, 15, 20].

In 2015, Smarandache [22] introduced the concept of neutrosophic quadruple numbers and presented some basic operations on the set of neutrosophic quadruple numbers such as, addition, subtraction, multiplication, and scalar multiplication. After that, a connection between neutrosophy and algebraic structures was established where Agboola et al. [5] considered the set of neutrosophic quadruple numbers and used the defined operations on it to discuss neutrosophic quadruple algebraic structures. More results about neutrosophic algebraic structures are found in [11, 26]. A generalization of the latter work was done in 2016 where Akinleye et al. [6] considered the set of neutrosophic quadruple numbers and defined some hyperoperations on it and discussed neutrosophic quadruple hyperstructures. More specifically, the latter papers introduced the notions of neutrosophic groups, neutrosophic rings, neutrosophic hypergroups and neutrosophic hyperrings on a set of real numbers and studied their basic properties.

The authors in [9] discussed neutrosophic quadruple [H.sub.v]-groups and studied their properties. Then in [10], they found the fundamental group of neutrosophic quadruple [H.sub.v]-groups and proved that it is a neutrosphic quadruple group. This paper is an extension to the above mentioned results. In Section 2, some definitions related to weak hyperstructures have been presented while section 3 involves the refined neutrosophic quadruple hypergroup and the studying of it's properties. As for section 4, an order on refined neutrosophic quadruple hypergroups is defined and some examples on refined neutrosophic quadruple po-hypergroups are presented. Finally, in section 5, the fundamental refined neutrosophic quadruple group of refined neutrosophic quadruple hypergroups with some important theorems, corollaries and propositions have been submitted.

2 Preliminaries

In this section, some definitions and theorems related to both: hyperstructure theory and neutrosophic theory are presented. (See [12, 13, 30].)

2.1 Basic notions of hypergroups

Definition 2.1. Let H be a non-empty set. Then, a mapping [omicron] : H x H [right arrow] P*(H) is called a binary hyperoperation on H, where P*(H) is the family of all non-empty subsets of H. The couple (H, [omicron]) is called a hypergroupoid.

In this definition, if A and B are two non-empty subsets of H and x [member of] H, then:

[mathematical expression not reproducible]

Definition 2.2. A hypergroupoid (H, [omicron]) is called a:

1. semihypergroup if for every x,y,z [member of] H, we have x o (y [omicron] Z) = (x [omicron] y) o z;

2. quasi-hypergroup if for every x [member of] H, x [omicron] H = H = H [omicron] x (The latter condition is called the reproduction axiom);

3. hypergroup if it is a semihypergroup and a quasi-hypergroup.

Definition 2.3. [13] Let (H, *) and (K, *') be two hypergroups. Then f: H [right arrow] K is said to be hypergroup homomorphism if f (x*y) = f (x)*' f (y) for all x,y [member of] H. (H,*) and (K,*') are called isomorphic [H.sub.v]-groups, and written as H [congruent to] K, if there exists a bijective function f: R [right arrow] S that is also a homomorphism. The set of all isomorphism of (H, *) is denoted as Aut (H).

T. Vougiouklis, the pioneer of [H.sub.v]-structures, generalized the concept of algebraic hyperstructures to weak algebraic hyperstructures [28]. The latter concept is known as "weak" since the equality sign in the definitions of [H.sub.v]-structures is more likely to be replaced by non-empty intersection. The concepts in [H.sub.v]-structures are mostly used in representation theory [29].

A hypergroupoid (H, [omicron]) is called an [H.sub.v]-semigroup if (x [omicron] (y [omicron] z)) [intersection] ((x [omicron] y) [omicron] z) [not equal to] [empty set] for all x,y,z [member of] H. An element 0 [member of] H is called an identity if x [member of] (0 [omicron] x [intersection] x [omicron] 0) for all x [member of] H and it is called a scalar identity if x = 0 [omicron] x = x [omicron] 0 for all x [member of] H. If the scalar identity exists then it is unique. A hypergroupoid (H, [omicron]) is called an [H.sub.v]-group if it is a quasi-hypergroup and an [H.sub.v]-semigroup. A non empty subset S of an [H.sub.v]-group (H, [omicron]) is called [H.sub.v]-subgroup of H if (S, [omicron]) is an [H.sub.v]-group.

Definition 2.4. [27] A hypergroup is called cyclic if there exist h [member of] H such that H = h [union] [h.sup.2] [union]... [union] [h.sub.i] [union]... with i [member of] N. If there exists s [member of] N such that H = h [union] [h.sup.2] [union]... [union] [h.sup.s] then H is a cyclic hypergroup with finite period. Otherwise, H is called cyclic hypergroup with infinite period. Here, [mathematical expression not reproducible].

Definition 2.5. [27] A hypergroup is called a single power cyclic hypergroup if there exist h [member of] H and s [member of] N such that H = h [union] [h.sup.2] [union]... [union] [h.sup.s] [union]... and h [union] [h.sup.2] [union]... [union] [h.sup.m-1] C [h.sup.m] for every m [greater than or equal to] 1. In this case, h is called a generator of H.

2.2 Refined neutrosophic quadruple hypergroups

Let T, I, F, represent the neutrosophic components truth, indeterminacy, and falsehood respectively. Symbolic (or Literal) Neutrosophic theory is referring to the use of these symbols in neutrosophics. In 2013, F. Smarandache [24] introduced the refined neutrosophic components. Where the neutrosophic literal components T, I, F can be split into respectively the following neutrosophic literal subcomponents:

[T.sub.1],... [T.sub.p]; I,...,[I.sub.r];[F.sub.1],...,[F.sub.s],

where p, r, s are positive integers with max{p, r, s} [greater than or equal to] 2.

Definition 2.6. [25] Let X be a nonempty set and p, r, s [member of] N with (p, r, s) [not equal to] (1, 1, 1). A refined neutrosophic quadruple X-number is a number having the following form:

a + [p.summation over (i = 1)][b.sub.i][T.sub.i] + [r.summation over (j = 1)][c.sub.j][I.sub.j] + [s.summation over k=1][b.sub.k][F.sub.k],

where a, [b.sub.i], [c.sub.j], [d.sub.k] [member of] X and T, I, F have their usual neutrosophic logic meanings, and [T.sub.i], [I.sub.j], [F.sub.k] are refinements of T, I, F respectively.

The set of all refined neutrosophic quadruple X-numbers is denoted by RNQ(X), that is,

RNQ(X) - {a|[p.summation over (i = 1)][b.sub.i][T.sub.i]|[r.summation over (j = 1)][c.sub.j][I.sub.j]|[s.summation over k=1][d.sub.k][F.sub.k]:a, [b.sub.i], [c.sub.j], [d.sub.k] [member of] X}.

For simplicity, we write a + [p.summation over (i = 1)] [b.sub.i][T.sub.i] + [r.summation over (j = 1)] [c.sub.j] [I.sub.j] + [s.summation over k = 1] [d.sub.k][F.sub.k] as

(a, [p.summation over (i = 1)][b.sub.i][T.sub.i], [r.summation over (j = 1)][c.sub.j][I.sub.j], [s.summation over (k = 1)][b.sub.k][F.sub.k]).

In what follows, [T.sub.i], [I.sub.j], [F.sub.k] are refinements of T, I, F respectively with 1 [less than or equal to] i [less than or equal to] p, 1 [less than or equal to] j [less than or equal to] r and 1 [less than or equal to] k [less than or equal to] s. Let (H, +) be a hypergroupoid with identity "0" and 0 + 0 = 0 and define "[symmetry]" on RNQ(H) as follows:

(a, [p.summation over (i = 1)][b.sub.i][T.sub.i], [r.summation over (j = 1)][c.sub.j][I.sub.j], [s.summation over (j = 1)][c.sub.j][I.sub.j], [r.summation over (j = 1)][d.sub.k][F.sub.k]) [symmetry] (a', [p.summation over (i = 1)][b'.sub.i][T.sub.i], [r.summation over (j = 1)][c'.sub.j][I.sub.j], [s.summation over k=1][d'.sub.k][F.sub.k]) -{(x, [p.summation over (i = 1)][y.sub.i][T.sub.i], [r.summation over (i - 1)][z.sub.j][I.sub.j], [s.summation over (k = 1)][w.sub.k][F.sub.k]):x [member of] a + a', [y.sub.i] [member of] [b.sub.i] + [b'.sub.i], [z.sub.j] [member of] [c.sub.j] + [c'.sub.j], [w.sub.k] [member of] [d.sub.k] + [d'.sub.k]}.

3 New properties of refined neutrosophic quadruple hypergroups

In this section, refined neutrosophic quadruple hypergroups are defined and their properties are studied. Proposition 3.1. Let (H, +) be a hypergroupoid with 0 [member of] H and [T.sub.i], [I.sub.j], [F.sub.k] are refinements of T, I, F respectively. Then (RNQ(H), [symmetry]) is a quasi-hypergroup with identity [bar.0] = (0, [p.summation over i=1] 0[T.sub.i], [r.summation over (j=1)] 0[I.sub.j], [s.summation over (k = 1)] 0[F.sub.k]) if and only if (H, +) is a quasi-hypergroup with identity 0.

Proof. Let (H, +) be a quasi-hypergroup. We prove now that (RNQ(H), [symmetry]) satisfies the reproduction axiom. That is, [bar.x] [symmetry] RNQ(H) = RNQ(H) [symmetry] [bar.x] = RNQ(H) for all [bar.x] = (a,[p.summation over i=1)] [b.sub.i][T.sub.i], [r.summation over (j=1)] [c.sub.j] [I.sub.j], [s.summation over k=1] [d.sub.k] [F.sub.k]) [member of] RNQ(H). We prove [bar.x] [symmetry] RNQ(H) = RNQ(H) and the proof of RNQ(H) [symmetry] [bar.x] = RNQ(H) is done in a similar manner. Let [bar.y] =(a', [p.summation over i=1] [b'.sub.i][T.sub.i]} [r.summation over (j=1)] [c'.sub.j] [I.sub.j], [s.summation over k=1] [d'.sub.k] [F.sub.k]) [member of] RNQ(H), we have [bar.x] [symmetry] [bar.x] = (a + a', [p.summation over i=1]([b.sub.i] + [b'.sub.i])[T.sub.i], [r.summation over (j=1)] ([c.sub.j]+ [c'.sub.j])[I.sub.j],[s.summation over k=1] ([d.sub.k] + [d'.sub.k])[F.sub.k]) [??] RNQ(H) as (a + a') [union] ([b.sub.i] + [b'.sub.i]) [union] ([c.sub.j] + [c'.sub.j]) [union] ([d.sub.k] + [d'.sub.k]) [??] H. Thus [bar.x] [symmetry] RNQ(H) [??] RNQ(H). Let [bar.x] = (a',[p.summation over i=1] [b'.sub.i][T.sub.i],[r.summation over (j=1)][c'.sub.j][I.sub.j],[s.summation over (k=1)][d'.sub.k][F.sub.k]) [member of] RNQ(H). Since (H, +) satisfies the reproduction axiom and a ('), [b'.sub.i], [c'.sub.j], [d'.sub.k] [member of] H, it follows that a (') [member of] a + H, [b'.sub.i] [member of] [b.sub.i] + H, [c'.sub.j] [member of] [c.sub.j] + H and [d'.sub.k] [member of] [d.sub.k] + H. The latter implies that there exist a*,b*,[c*.sub.j], [d*.sub.k] [member of] H such that a' [member of] a + a*, [b.sub.i] [member of] [b.sub.i] + [b*.sub.i], [c'.sub.j] [member of] [c.sub.j] + [c*.sub.j] and [d.sub.k] [member of] [d.sub.k] + [d*.sub.k]. It is clear that [bar.y] [member of] [bar.x] [symmetry] [bar.z] where [bar.z] = (a*,[p.summation over i=1] [b*.sub.i][T.sub.i],[r.summation over (j=1)] [c*.sub.j][I.sub.j],[s.summation over k=1] [d*.sub.k][F.sub.k]) [member of] RNQ(H). Thus, (RNQ(H),[symmetry]) satisfies the reproduction axiom.

Conversely, let (RNQ(H), [symmetry]) be a quasi-hypergroup and a [member of] H. Since 0 [member of] H, it follows that [bar.a] = (a, [p.summation over i=1] 0[T.sub.i], [r.summation over (j=1)] 0[I.sub.j], [s.summation over k=1] 0[F.sub.k]) [member of] RNQ(H). Having (RNQ(H), [symmetry]) a quasi-hypergroup implies that a[theta]RNQ(H) = RNQ(H) [symmetry] [bar.a] = RNQ(H). The latter implies that a + H = H + a = H.

Proposition 3.2. Let (H, +) be a hypergroupoid with 0 [member of] H. Then (RNQ(H), [symmetry]) is a semi-hypergroup with identity element [bar.x] = (0, [p.summation over i=1] 0[T.sub.i], [r.summation over (j=1)] 0[I.sub.j], [s.summation over k=1] 0[F.sub.k]) if and only if (H, +) is a semi-hypergroup with identity element 0.

Proof. Let (H, +) be a a semi-hypergroup and [bar.x], [bar.y], [bar.z] [member of] RNQ(H) with

[bar.x]=(a, [P.summation over i=1] [b.sub.i][T.sub.i] [r.summation over (j=1)] [c.sub.j] [I.sub.j], [s.summation over k=1] [d.sub.k] [F.sub.k]), [bar.y]=(a', [p.summation over i=1] [b'.sub.i] [T.sub.i] [r.summation over (j=1)] [c'.sub.j] [I.sub.j], [s.summation over k=1] [d'.sub.k] [F.sub.k]) and [bar.z] = (a", [p.summation over i=1] [b".sub.i][T.sub.i], [r.summation over (j=1)] [c".sub.j][I.sub.j], [s.summation over k=1] [d".sub.k][F.sub.k]). Having a + (a' + a") = (a + a') + a", [b.sub.i] + ([b'.sub.i] + [b".sub.i]) = ([b.sub.i] + [b'.sub.i]) + [b".sub.i], [c.sub.j] + ([c'.sub.j] + [c".sub.j]) = ([c.sub.j] + [c'.sub.j]) + [c".sub.j] and [d.sub.k] + ([d'.sub.k] + [d".sub.k]) = ([d.sub.k] + [d'.sub.k]) + [d".sub.k] implies that [bar.x] [symmetry] ([bar.x] [symmetry] [bar.z]) = ([bar.x] [symmetry] [bar.x]) [symmetry] [bar.x].

Let (RNQ(H), [symmetry]) be a semi-hypergroup and a,b,c, [member of] H. Then [bar.a],[bar.b],[bar.c] [member of] RNQ(H) with [bar.a] = (a, [summation over i=1] 0[T.sub.i], [p,r.summation over (j=1)] 0[I.sub.j], [s.summation over k=1] 0[F.sub.k]

[bar.b] = (b, [r.summation over i=1]0[T.sub.i], [s.summation over j=1]0[I.sub.j], [summation over k=1] 0[F.sub.k]) and

[bar.c] = (c, [p.summation over i=1] 0[T.sub.i][r.summation over (j=1)] 0[I.sub.j], [s.summation over (k=1)] 0[F.sub.k]). Having [bar.a] [symmetry] ([bar.b] [symmetry] [bar.c]) = ([bar.a] [symmetry] [bar.b]) [symmetry] [bar.c] implies that a + (b + c) = (a + b) + c.

Proposition 3.3. Let (H, +) be a hypergroupoid with 0 [member of] H. Then (RNQ(H), [symmetry]) is an [H.sub.v]-semigroup with identity element [bar.0] = (0, [p.summation over (i=1)] 0[T.sub.i], [R.summation over(j=1)] 0[I.sub.j], [s.summation over(k=1)] 0[F.sub.k]) if and only if (H, +) is an [H.sub.v]-semigroup with identity element 0.

Proof. The proof is similar to that of Proposition 3.2 but instead of equality we have non-empty intersection.

Theorem 3.4. Let (H, +) be a hypergroupoid. Then (RNQ(H), [symmetry]) is a hypergroup with identity element [bar.0] = (0, [p.summation over (i=1)] 0[T.sub.i], [R.summation over(j=1)] 0[I.sub.j], [r.summation over(k=1)] 0[F.sub.k]) if and only if (H, +) is a hypergroup with identity element 0.

Proof. The proof is direct from Propositions 3.1 and 3.2.

Theorem 3.5. Let (H, +) be a hypergroupoid with 0 [member of] H. Then (RNQ(H), [symmetry]) is an [H.sub.v]-group with identity element [bar.0] = (0, [p.summation over (i=1)] 0[T.sub.i], [r.summation over(j=1)] 0[I.sub.j], [s.summation over(k=1)] 0[F.sub.k]) if and only if (H, +) is an [H.sub.v]-group with an identity element 0.

Proof. The proof follows from Propositions 3.1 and 3.3.

Theorem 3.6. Let (H, +) be a hypergroupoid. Then (RNQ(H), [symmetry]) is a commutative hypergroup ([H.sub.v]-group) with identity element [bar.0] = (0, [p.summation over (I=1)] 0[T.sub.i], [R.summation over(j=1)] 0[I.sub.j], [SIGMA] 0[F.sub.k]) if and only if (H, +) is a commutative hypergroup ([H.sub.v] group) with an identity 0.

Proof. The proof is straightforward.

NOTATION 1. Let (H, +) be a hypergroup ([H.sub.v]-group) with identity "0" satisfying 0+0 = 0. Then (RNQ(H), [symmetry]) is called a refined neutrosophic quadruple hypergroup (refined neutrosophic quadruple [H.sub.v]-group).

Corollary 3.7. Let (H, +) be a hypergroup ([H.sub.v]-group) containing an identity element 0 with the property that 0 + 0 = 0. Then there are infinite number of refined neutrosophic quadruple hypergroups ([H.sub.v]-groups).

Proof. Let (H, +) be a hypergroup ([H.sub.v]-group). Theorem 3.4 and Theorem 3.5 implies that (RNQ(H), [symmetry]) is a neutrosophic quadruple hypergroup ([H.sub.v]-group) with identity [bar.0] and [bar.0] [symmetry] [bar.0] = [bar.0]. Applying Theorem 3.4 and Theorem 3.5 on (RNQ(H), [symmetry]), we get RNQ(RNQ(H)) is a neutrosophic quadruple hypergroup ([H.sub.v]-group). Continuing on this pattern, we get RNQ(RNQ(... (RNQ(H))...) is a neutrosophic quadruple hypergroup ([H.sub.v]-group).

Proposition 3.8. Let X be any set with a hyperoperation "+". Then RNQ(X) is a cyclic refined neutrosophic quadruple hypergroup if and only if X is a cyclic hypergroup with an identity element "0 [member of] X" and 0 + 0 = 0.

Proof. Let X be a cyclic hypergroup with identity "0 [member of] X" and 0 + 0 = 0. Then there exist a [member of] X such that a is a generator of X. It is clear that [bar.a] is a generator of RNQ(X) where [bar.a] = (a, [p.summation over (I=1)] a[T.sub.i], [r.summation over (j=1)] a[I.sub.j], [s.summation over(k=1)] a[F.sub.k]) [member of] RNQ(X).

Let RNQ(X) be a cyclic quadruple hypergroup. Then there exist [bar.x] [member of] RNQ(X) such that [bar.x] = (a, [p.summation over (I=1)] [b.sub.i][T.sub.I],[R.summation over(j=1)] [c.sub.j] [I.sub.j], [s.summation over(k=1)] [d.sub.k] [F.sub.k]) is a generator of RNQ(X). It is clear that a is a generator of X.

Example 3.9. Let [T.sub.1], [T.sub.2] be refinements of T, [I.sub.1], [F.sub.1] be refinements of I,F respectively, [H.sub.1] = {0,1} and define ([H.sub.1], [+.sub.1]) as follows:
+1   0   1
0    0   1
1    1   [H.sub.1]


Since ([H.sub.1], [symmetry]) is a commutative hypergroup with an identity 0, it follows by Theorem 3.6 that (RNQ([H.sub.1]), [symmetry]) is a commutative refined neutrosophic quadruple hypergroup with 32 elements and identity [bar.0] = (0, 0[T.sub.1] + 0[T.sub.2], 0[T.sub.1], 0[F.sub.1]). Moreover, having [H.sub.1] = 1 [+.sub.1] 1 implies that 1 is a generator of ([H.sub.1], +) and ([H.sub.1], +) is a single-power cyclic hypergroup of period 2. Theorem 3.8 asserts that (RNQ([H.sub.1]), [symmetry]) is a single power cyclic hypergroup of period 2 and the generator element is (1, 1[T.sub.1] + 1[T.sub.2], 1[F.sub.1]).

It is clear that (1,0[T.sub.1] + 0[T.sub.1], 1[F.sub.1]) [phi] (1,0[T.sub.1] + 1[T.sub.2], 0[I.sub.1], 1[F.sub.1]) = {(1, 0T + 1[T.sub.2], 0[F.sub.1]), (1, 0[T.sub.1] + 1[T.sub.2], 1[F.sub.1])}.

Definition 3.10. Let (H, +) be a hypergroup ([H.sub.v]-group). A subset X of RNQ(H) with the property that [bar.0] = (0, [p.summation over (i=1)] 0[T.sub.i], [r.summation over (j=1)] 0[I.sub.j], [s.summation over (i=1)] 0[F.sub.k]) [member of] X is called a refined neutrosophic subhypergroup ([H.sub.v]-subgroup) of RNQ(H) if there exists S [??] H such that X = RNQ(S) and (X, [symmetry]) is a refined neutrosophic quadruple hypergroup ()[H.sub.v]-group).

Proposition 3.11. Let (H, +) be a hypergroup ([H.sub.v]-group) and S [??] H. A subset X = RNQ(S) [??] RNQ(H) is a refined neutrosophic subhypergroup ([H.sub.v]-subgroup) of RNQ(H) if the following conditions are satisfied:

1. [bar.0] = (0, [p.summation over (i=1)] 0[T.sub.i], [r.summation over (j=1)] 0[I.sub.j], [s.summation over (k=1)] 0[F.sub.k]) [member of] X;

2. [bar.x] [symmetry] X = X [symmetry] [bar.x] = X for all [bar.x] [member of] X.

Proof. The proof is straightforward.

Theorem 3.12. Let (H, +) be a hypergroup ([H.sub.v]-group) with identity "0", S [??] H and 0 [member of] S. Then (RNQ(S), [symmetry]) is a refined neutrosophic quadruple subhypergroup ([H.sub.v]-subgroup) of (RNQ(H), [symmetry]) if and only if (S, +) is a subhypergroup ([H.sub.v]-subgroup) of (H, +).

Proof. The proof is straightforward by applying Proposition 3.11.

Example 3.13. Since ([H.sub.1], [+.sub.1]) in Example 3.9 has only two subhypergroups ({0} and [H.sub.1]), it follows by applying Theorem 3.12 that (RNQ([H.sub.1]), [symmetry]) has only two refined neutrosophic quadruple subhypergroups:

({[bar.0]}, [symmetry]) = (RNQ({0}), [symmetry]) and (RNQ([H.sub.1]), [symmetry]).

Example 3.14. Let [H.sub.2] = {0, 1, 2, 3} and define ([H.sub.2], [+.sub.2]) as follows:
+2   0   1    2       3
 0   0   1    2       3
 1   1   2    3       0
 2   2   3   {0, 2}   1
 3   3   0    1       2


It is clear that ([H.sub.2], [+.sub.2]) is a commutative [H.sub.v]-group that has exactly three non-isomorphic [H.sub.v]-subgroups containing 0: {0}, {0, 2} and [H.sub.2]. We can deduce by Theorem 3.12 that (RNQ([H.sub.2]), [symmetry]) is a commutative refined neutrosophic quadruple [H.sub.v]-group and has three non-isomorphic refined neutrosophic quadruple [H.sub.v]subgroups: RNQ({0}) = {[bar.0]}, RNQ({0, 2}) and RNQ([H.sub.2]).

Proposition 3.15. Let (H, +) be a hypergroup and (S, +) be a subhypergroup of (H, +) containing 0. Then

RNQ(S) [symmetry] RNQ(S) = RNQ(S).

Proof. The proof is straightforward.

Definition 3.16. Let (RNQ(H), [[symmetry].sub.1]) and (RNQ(J), [[symmetry].sub.2]) be refined neutrosophic quadruple hypergroups with [0.sub.H] [member of] H and [0.sub.J] [member of] J.A function [phi] : RNQ(H) [right arrow] RNQ(J) is called refined neutosophic homomorphism if the following conditions are satisfied:

1. [phi]([0.sub.H], [p.summation over (i=1)] [0.sub.H][T.sub.i], [r.summation over (j=1)] [0.sub.H][I.sub.j], [s.summation over (k=1)] [0.sub.H][F.sub.k]) = ([0.sub.J], [p.summation over (i=1)] [0.sub.J][T.sub.i], [r.summation over (j=1)] [0.sub.J]I, [s.summation over (k=1)] [0.sub.J][F.sub.k]);

2. [phi](x [[symmetry].sub.1] y) = [phi](x) [[symmetry].sub.2] [phi](y) for all x, y [member of] RNQ(H).

If [phi] is a refined neutrosophic bijective homomorphism then it is called refined neutrosophic isomorphism and we write RNQ(H) = RNQ(J).

Example 3.17. Let (H, +) be a hypergroup. Then the function f : RNQ(H) [right arrow] RNQ(H) is an isomorphism, where f (x) = x for all x [member of] RNQ(H).

Example 3.18. Let (H, +) be a hypergroup and 0 [member of] H and f : RNQ(H) [right arrow] RNQ(H) be defined as follows:

f ((a, [p.summation over (i=1)] [b.sub.i][T.sub.i], [r.summation over (j=1)] [c.sub.j] [I.sub.j], [s.summation over(k=1)] [d.sub.k][F.sub.k])) = (a, [p.summation over (i=1)] 0[T.sub.i], [r.summation over (j=1)] 0[I.sub.j], [s.summation over(k=1)] 0[F.sub.k]).

Then f is a refined neutrosophic homomorphism.

Proposition 3.19. Let (H, [+.sub.1]) and (J, [+.sub.2]) be hypergroups with [0.sub.H] [member of] H, [0.sub.J] [member of] J. If there exist a homomorphism f : H [right arrow] J with f ([0.sub.H]) = [0.sub.J] then there exist a refined neutrosophic homomorphism from (RNQ(H), [[symmetry].sub.1]) to (RNQ(J), [[symmetry].sub.2]).

Proof. Let [phi] : RNQ(H) [right arrow] RNQ(J) be defined as follows:

[phi]((a, [p.summation over (i=1)] [b.sub.i][T.sub.i], [r.summation over (j=1)] [c.sub.j] [I.sub.j], [s.summation over (k=1)] [d.sub.k] [F.sub.k])) = (f(a), [p.summation over (i=1)] f ([b.sub.i])[T.sub.i], [r.summation over (j=1)] f ([c.sub.j])[I.sub.j], [s.summation over (k=1)] f ([d.sub.k])[F.sub.k]).

It is clear that [phi] is a refined neutrosophic homomorphism.

Corollary 3.20. Let (H, [+.sub.1]) and (J, [+.sub.2]) be isomorphic hypergroups with [0.sub.H] [member of] H, [0.sub.J] E J. Then (RNQ(H), [[symmetry].sub.1]) and (RNQ(J), [[symmetry].sub.2]) are isomorphic refined neutrosophic quadruple hypergroups.

Proof. The proof is straightforward by using Proposition 3.19.

Definition 3.21. Let (H/S, +') be a commutative hypergroup with an identity element "0" and S [??] R be a subhypergroup of H. Then (H/S, +') is a hypergroup with: S as an identity element and S +' S = S. Here "+'" is defined as follows: For all x,y [member of] H,

(x + S) +' (y + S) = (x + y) + S.

Proposition 3.22. Let (S, +) be a subhypergroup of a commutative hypergroup (H, +). Then (RNQ(H/S), [symmetry]) is a hypergroup.

Proof. Since (H, +) is commutative, it follows that "+'" is well defined. The proof follows from having (H/S, +') a hypergroup with S as an identity, S +' S = S and from Theorem 3.4.

Proposition 3.23. Let (S, +) be a subhypergroup of a commutative hypergroup (H, +). Then (RNQ(H/S), [theta]) [congruent to] (RNQ(H)/RNQ(S), [symmetry]').

Proof. Let g : RNQ(H)/RNQ(S) [right arrow] RNQ(H/S) be defined as follows:

[mathematical expression not reproducible]

Then g is a hypergroup isomorphism. This can be proved easily by applying a similar proof to that of Proposition 3.27 that was done by the authors in [9].

Example 3.24. Let [H.sub.3] = {0, 1, 2, 3, 4} and define "+" on [H.sub.3] as follows: x + y = {x, y} for all x,y [member of] [H.sub.3]. It is clear that {0}, {0, 1}, {0, 1, 2}, {0, 1, 2, 3} and [H.sub.3] are the only non-isomorphic subhypergroups of [H.sub.3]. By applying Proposition 3.23, we get RNQ([H.sub.3]/{0, 1}) [congruent to] RNQ([H.sub.3])/RNQ({0, 1}), RNQ([H.sub.3]/{0, 1, 2}) [congruent to] RNQ([H.sub.3])/RNQ({0, 1, 2}) and RNQ([H.sub.3]/{0, 1, 2, 3}) [congruent to] RNQ(H)/RNQ({0, 1, 2, 3}).

4 Ordered refined neutrosophic quadruple hypergroups

In this section, an order on refined neutrosophic quadruple hypergroups is defined and some examples and results on refined neutrosophic quadruple partially ordered hypergroups (po-hypergroups) are presented.

A partial order relation on a set X (Poset) is a binary relation "[less than or equal to]" on X which satisfies conditions reflexivity, antisymmetry and transitivity.

Let (II, [less than or equal to]) be a partial ordered set and define (RNQ(H), [??]) as follows:

(a, [p.summation over (I=1)] [b.sub.i][T.sub.i], [R.summation over(j=1)] [c.sub.j] [I.sub.j], [s.summation over(k=1)] [d.sub.k][F.sub.k]) [??] (a', [p.summation over (I=1)] [b'.sub.i], [R.summation over(j=1)] [c'.sub.j][I.sub.j], [s.summation over(k=1)] [d'.sub.k][F.sub.k])

if and only if a [less than or equal to] a', [b.sub.i] [less than or equal to] [b'.sub.i], [c.sub.j] [less than or equal to] [c'.sub.j] and [d.sub.k] [less than or equal to] [d'.sub.k]. It is clear that (RNQ(H), [??]) is a partial ordered set.

Definition 4.1. [16] An algebraic hyperstructure (H, [omicron], [less than or equal to]) is called a partially ordered hypergroup or po-hypergroup, if (H, [omicron]) is a hypergroup and [less than or equal to] is a partial order relation on H such that the monotone condition holds as follows:

x [less than or equal to] y [??] a [omicron] x [less than or equal to] a [omicron] y for all a, x, y [member of] H.

Let A, B be non-empty subsets of (H, [less than or equal to]). The inequality A [less than or equal to] B means that for any a [member of] A, there exist b [member of] B such that a [less than or equal to] b.

Theorem 4.2. Let (II, +) be a hypergroupoid. Then (RNQ(H), [symmetry], [??]) is a refined neutrosophic quadruple po-hypergroup with identity element [bar.0] = (0, [p.summation over (I=1)] 0[T.sub.i], [R.summation over(j=1)] 0[I.sub.j], [s.summation over(k=1)] 0[F.sub.k]) if and only if (H, +, [less than or equal to]) is a po-hypergroup with identity element 0.

Proof. Let (H, +, [less than or equal to]) be a po-hypergroup, [bar.e] = (e, [p.summation over (I=1)] [[integral].sub.i][T.sub.i], [R.summation over(j=1)] [c.sub.j][I.sub.j], [s.summation over(k=1)] [h.sub.k][F.sub.k]) [member of] RNQ(H) and (a, [p.summation over (I=1)] [b.sub.i][T.sub.i], [R.summation over(j=1)] [c.sub.j][I.sub.j], [s.summation over(k=1)] [d.sub.k][F.sub.k]) [??] (a', [p.summation over (I=1)] [b'.sub.i][T.sub.i], [R.summation over(j=1)] [c'.sub.j] [I.sub.j], [s.summation over(k=1)] [d'.sub.k][F.sub.k]). We need to show that:

[bar.e] [symmetry] (a, [p.summation over (I=1)] [b.sub.i][T.sub.i], [R.summation over(j=1)] [c.sub.j][I.sub.j] [s.summation over(k=1)] [d.sub.k][F.sub.k]) [??] [bar.e] [symmetry] (a', [p.summation over (I=1)] [b'.sub.i][T.sub.i], [R.summation over(j=1)] [c'.sub.i][I.sub.j], [s.summation over(k=1)] [d'.sub.k][F.sub.k]).

Having a [less than or equal to] a', [b.sub.i] [less than or equal to] [b'.sub.i], [c.sub.j] [less than or equal to] [c'.sub.j], [d.sub.k] [less than or equal to] [d'.sub.k] and (H, +, [less than or equal to]) a po-hypergroup implies that e + a [less than or equal to] e + a', [f.sub.i] + [b.sub.i] [less than or equal to] [f.sub.i] + [b'.sub.i], [g.sub.i] + [c.sub.i] [less than or equal to] [g.sub.i] + [c'.sub.i] and [h.sub.k] + [d.sub.k] [less than or equal to] [h.sub.k] + [d'.sub.k]. Let [bar.a*] = (a*, [p.summation over (i=1)] [b*.sub.i], [r.summation over (j=1)] [c*.sub.j],[I.sub.j] [s.summation over (k=1)] [d*.sub.k][F.sub.k]) [member of] [bar.e] [symmetry] (a, [p.summation over (I=1)] [b.sub.i][T.sub.i], [R.summation over(j=1)] [c.sub.j][I.sub.j], [s.summation over(k=1)] [d.sub.k][F.sub.k]). Then a* [member of] e + a, [b*.sub.i] [member of] [f.sub.i] + [b.sub.i], [c*.sub.j] [member of] [g.sub.j] + [c.sub.j] and [d.sub.k] [member of] [h.sub.k] + [d.sub.k]. Having e + a [less than or equal to] e + a', [f.sub.i] + [b.sub.i] [less than or equal to] [f.sub.i] + [b'.sub.i], [g.sub.j] + [c.sub.j] [less than or equal to] [g.sub.j] + [c'.sub.j] and [h.sub.k] + [d.sub.k] [less than or equal to] [h.sub.k] + [d'.sub.k] implies that there exist a*' [member of] e + a', [b*'.sub.i] [member of] [f.sub.i] + [b'.sub.i], [c*'.sub.i] [member of] [g.sub.j] + [c'.sub.i] and [d*'.sub.k] [member of] [h.sub.k] + [d'.sub.k] such that a (*) [less than or equal to] a (*'), [b*.sub.k] [less than or equal to] [b*'.sub.i], [c*.sub.j] [less than or equal to] [c*'.sub.k] and [d*.sub.k] [less than or equal to] [d*'.sub.k]. We get now that [bar.a*] [??] [bar.a*'] where [bar*'.a] = (a*', [p.summation over (I=1)] [b*'.sub.i][T.sub.i], [R.summation over(j=1)] [c*'.sub.j][I.sub.j], [s.summation over(k=1)] [d*'.sub.k][F.sub.k]) and [bar.a]*' [member of] [bar.e] [symmetry] (a', [p.summation over (I=1)] [b'.sub.i][T.sub.i], [R.summation over(j=1)] [c'.sub.j][I.sub.j], [s.summation over(k=1)] [d'.sub.k][F.sub.k]).

Let a, b, e [member of] H and a [less than or equal to] b. Having 0 [less than or equal to] 0 implies that

(a, [p.summation over (I=1)] 0[T.sub.i], [r.summation over (j=1)] 0[I.sub.j], [s.summation over(k=1)] 0[F.sub.k]) [??] (b, [p.summation over (i=1)] 0[T.sub.i], [r.summation over (j=1)] 0[I.sub.j], [s.summation over(k=1)] 0[F.sub.k]).

Since (RNQ(H), [symmetry], [??]) is a refined neutrosophic quadruple po-hypergroup, it follows that for

[bar.e] =(e, [p.summation over (i=1)] 0[T.sub.i], [r.summation over (j=1)] 0[I.sub.j], 0[F.sub.k]),

[bar.e] [symmetry] (a, [p.summation over (i=1)] 0[T.sub.i], [r.summation over (j=1)] 0[I.sub.j], [s.summation over (k=1)] 0[F.sub.k]) [??] [bar.e] [symmetry] (b, [p.summation over (i=1)] 0[T.sub.i], [r.summation over (j=1)] 0[I.sub.j], ([pounds sterling]) 0[F.sub.k]).

It is clear that e + a [less than or equal to] e + b.

Corollary 4.3. Let (H, +, [less than or equal to]) be a po-hypergroup containing an identity element 0 with the property that 0 + 0 = 0. Then there is infinite number of refined neutrosophic quadruple po-hypergroups.

Proof. The proof is starightforward using Theorem 4.2.

Example 4.4. Let [H.sub.1] = {0,1} and define ([H.sub.1], [+.sub.1]) as in Example 3.9. It is clear that ([H.sub.1], [+.sub.1], [less than or equal to]) is a po-hypergroup. Here, the partial order relation "[less than or equal to]" is directed to the set {(0, 0), (1,1)}. By using Theorem 4.2, we get (RNQ([H.sub.1]), [symmetry], [??]) is a refined neutrosophic quadruple po-hypergroup.

Example 4.5. Let (H, [??]) be any poset and define (H, +) as the biset hypergroup, i.e. x + y = {x, y} for all x,y [member of] H. Then (RNQ(H), [symmetry], [??]) is a refined neutrosophic quadruple po-hypergroup.

Theorem 4.6. [16] Let (H, [omicron]) be a hypergroup such that there exists an element 0 [member of] H and the following conditions hold:

1. 0 [omicron] 0 = 0;

2. {0, x} [??] 0 [omicron] x for all x [member of] H;

3. If x [omicron] 0 = y [omicron] 0 then x = y for all x, y [member of] H.

Then there exist a relation "[less than or equal to]" on H such that (H, [omicron], [less than or equal to]) is a po-hypergroup.

Heidari et al. [16], in their proof of Theorem 4.6, defined the binary relation "[less than or equal to]" on H as follows:

x [less than or equal to] y [??] x E y [omicron] 0, for all x,y [member of] H.

Corollary 4.7. Let (H, +) be a hypergroup satisfying conditions of Theorem 4.6. Then there exist a relation "[??]" on RNQ(H) such that (RNQ(H), [symmetry], [??]) is a refined neutrosophic quadruple po-hypergroup.

Proof. The proof follows from Theorems 4.2 and 4.6.

Example 4.8. Let H = {0, x, y} and define "+" by the following table:
+    0        x       y
0    0       {0, x}   H
c   {0, x}    H       H
y    H        H       H


Then (RNQ(H), [symmetry], [??]) is a refined neutrosophic quadruple po-hypergroup. Here the partial order relation "[less than or equal to]" is directed to the set {(0, 0), (x, x), (y, y), (x, y), (0, x), (0, y)} and [??] is defined in the usual way on RNQ(H).

Definition 4.9. Let (RNQ(H), [[symmetry].sub.1], [??]) and (RNQ(J), [[symmetry].sub.2], [??]) be refined neutrosophic quadruple po-hypergroups. A function [phi] : RNQ(H) [right arrow] RNQ(J) is called an ordered refined neutosophic homomorphism if the following conditions hold:

1. [phi]([0.sub.H], [p.summation over (I=1)] [0.sub.H][T.sub.i], [R.summation over(j=1)] P[0.sub.H][I.sub.j], [s.summation over(k=1)] [0.sub.H][F.sub.k]) = ([0.sub.J], [p.summation over (I=1)] [0.sub.J][T.sub.i], [R.summation over(j=1)] [0.sub.J]I, [s.summation over(k=1)] [0.sub.J][F.sub.k]);

2. [phi](x [[symmetry].sub.1] y) = [phi](x) [[symmetry].sub.2] [phi](y) for all x, y e RNQ(H);

3. if x [??] y then [phi](x) [??] [phi](y) for all x,y [member of] RNQ(H).

If [phi] is an ordered refined neutrosophic homomorphism and is bijective then it is called an ordered refined neutrosophic isomorphism and we say RNQ(H) and RNQ(J) are isomorphic refined neutrosophic quadruple po-hypergroups.

Example 4.10. Let (H, +, [less than or equal to]) be a po-hypergroup. Then f : RNQ(H) [right arrow] RNQ(H) is an ordered refined neutrosophic isomorphism, where f (x) = x for all x [member of] RNQ(H).

Example 4.11. Let (H, +, [less than or equal to]) be a po-hypergroup, 0 [member of] H and f : RNQ(H) [right arrow] RNQ(H) be defined as follows:

f ((a, [p.summation over (I=1)] [b.sub.i][T.sub.i], [R.summation over(j=1)] [c.sub.j][I.sub.j], [s.summation over(k=1)] [d.sub.k] F)) = (a, [p.summation over (I=1)] 0[T.sub.i], [R.summation over(j=1)] 0[I.sub.j], [s.summation over (k=1)] 0F).

Then f is an ordered refined neutrosophic homomorphism.

Example 4.12. Let (H, [+.sub.1], [[less than or equal to].sub.1]) and (J, [+.sub.2], [[less than or equal to].sub.2]) be po-hypergroups, [0.sub.H] [member of] H, [0.sub.J] [member of] J and g : H [right arrow] J be an ordered homomorphism. Then f : RNQ(H) [right arrow] RNQ(J) is an ordered refined neutrosophic homomorphism. Here, f is defind as follows:

f ((a, [p.summation over (I=1)] [b.sub.i][T.sub.i], [R.summation over(j=1)] [c.sub.j][I.sub.j], [s.summation over(k=1)] [d.sub.k] F)) = (g(a), [p.summation over (I=1)] [0.sub.J][T.sub.i], [R.summation over(j=1)] [0.sub.J] [I.sub.j], [s.summation over(k=1)] [0.sub.J][F.sub.k]).

Proposition 4.13. Let (H, [+.sub.1], [[less than or equal to].sub.1]) and (J, [+.sub.2], [[less than or equal to].sub.2]) be po-hypergroups with [0.sub.H] [member of] H, [0.sub.J] [member of] J. If there exist an ordered homomorphism f : H [right arrow] J with f ([0.sub.H]) = [0.sub.J] then there exist an ordered refined neutrosophic homomorphism from (RNQ(H), [[symmetry].sub.1], [??]) to (RNQ(J), [[symmetry].sub.2], [??]).

Proof. Let [phi] : RNQ(H) [right arrow] RNQ(J) be defined as follows:

[phi]((a, [p.summation over (i=1)] [b.sub.i][T.sub.i], [r.summation over (j=1)] [c.sub.j][I.sub.j], [s.summation over (k=1)] [d.sub.k][F.sub.k])) = (f(a), f([b.sub.i])[T.sub.i], f([c.sub.j])[I.sub.j], f([d.sub.k])[F.sub.k]).

It is clear that [phi] is an ordered refined neutrosophic homomorphism.

Corollary 4.14. Let (H, [+.sub.1], [[less than or equal to].sub.1]) and (J, [+.sub.2], [[less than or equal to].sub.2]) be isomorphic po-hypergroup with [0.sub.H] [member of] H, [0.sub.J] [member of] J. Then (RNQ(H), [[symmetry].sub.1], [??]) and (RNQ(J), [[symmetry].sub.2], [??]) are isomorphic refined neutrosophic quadruple po-hypergroups.

Proof. The proof is straightforward by using Proposition 4.13.

5 Fundamental group of refined neutrosophic quadruple hypergroups

This section presents the study of fundamental relation on refined neutrosophic quadruple hypergroups and finds their fundamental refined quadruple neutrosophic groups.

Theorem 5.1. Let (G, +) be a groupoid. Then (RNQ(G), [symmetry]) is a group with identity element

[bar.0] = (0,[p.summation over (i=1)] 0[T.sub.i], [r.summation over (j=1)] 0[I.sub.j], [s.summation over (k=1)] 0[F.sub.k]) if and only if (G, +) is a group with identity element 0.

Proof. It is clear that [bar.0] = (0,[p.summation over (i=1)] 0[T.sub.i], [r.summation over (j-1)] 0[I.sub.j], [s.summation over (k=1)] 0[F.sub.k]) is the identity of (RNQ(G), [symmetry]) if and only if 0 is the

identity of G. Let [bar.x] = (a,[p.summation over (i=1)] [B.sub.I][T.sub.I],[r.summation over (j=1)] [c.sub.j][I.sub.j], [s.summation over (k=1)] [d.sub.k][F.sub.k]) [member of] RNQ(G). Then the inverse

-[bar.x] = (a,[p.summation over (i = 1)] (-[b.sub.i])[T.sub.I],[r.summation over (j = 1)] (-[c.sub.j])[I.sub.j],[s.summation over (k=1)](-[d.sub.k])[F.sub.k]) of [bar.x] exists if and only if the inverse -y of y exists in G for all

y [member of] G. The proof of (RNQ(G), [symmetry]) is binary closed if and only if (G, +) is binary closed is similar to that of Proposition 3.1. And the proof of (RNQ(G), [symmetry]) is associative if and only if (G, +) is associative is similar to that of Proposition 3.2.

NOTATION 2. Let (G, +) be a group with identity element "0". Then (RNQ(G), [symmetry]) is called refined neutrosophic quadruple group.

Proposition 5.2. Let G, G' be isomorphic groups. Then RNQ(G) and RNQ(G') are isomorphic neutrosophic quadruple groups.

Definition 5.3. [14] For all n> 1, we define the relation [[beta].sub.n] on a semihypergroup (H,o) as follows:

x[[beta].sub.n]y if there exist [a.sub.1],..., [a.sub.n] in H such that {x, y} [??] [??] [a.sub.i]

Here, [??] [a.sub.i] = [a.sub.1] o [a.sub.2]... o [a.sub.n]. And we set [beta] = [??] [[beta].sub.n], where [[beta].sub.1] = {(x,x) | x E H} is the diagonal relation on H.

Koskas [18] introduced this relation as an important tool to connect hypergroups with groups. And due to it's importance in connecting algebraic hyperstructures with algebraic structures, different researchers studied it on various hypergroups and some extended this definition to cover other types of hyperstructures. Clearly, the relation [beta] is reflexive and symmetric. Denote by [beta]* the transitive closure of [beta]. Then [beta]* is called the fundamental equivalence relation on H and it is the smallest strongly regular relation on H. If H is a hypergroup then [beta] = [beta] (*) and H/[beta] (*) is called the fundamental group.

Throughout this section, [beta] and [beta]* are the relation on H and [[beta].sub.N] and [[beta]*.sub.N] are the relations on RNQ(H).

Theorem 5.4. Let (H, +) be a hypergroup with identity element element "0" and 0 + 0 = 0 and let a, a', b[b.sub.i], [b'.sub.i],

[c.sub.j], [c'.sub.j], [d.sub.k], [d'.sub.k] [member of] H. Then

(a, [p.summation over (i = 1)] [B.sub.I][T.sub.I], [r.summation over (j = 1)] [c.sub.j][I.sub.j],[s.summation over (k=1)] [d.sub.k][F.sub.k])[[beta].sub.n] (a', [p.summation over (i = 1)] [B'.sub.I][T.sub.I],[r.summation over (j = 1)] [c'.sub.j][I.sub.j], [d'.sub.k][F.sub.k]) if and only if a[beta]a', [b.sub.i][beta][b'.sub.i], [c.sub.j] [beta][c'.sub.j] and [d.sub.k] [beta][d'.sub.k].

Proof. Let (a, [p.summation over (i=1)] [b.sub.i][T.sub.i], [r.summation over (j=1)] [c.sub.j][I.sub.j],[s.summation over (k-1)] [d.sub.k][F.sub.k])[[beta].sub.n] (a', [p.summation over (i=1)] [b'.sub.i][T.sub.i],[r.summation over (j=1)] [c'.sub.j][I.sub.j], [d'.sub.k][F.sub.k]). Then there exist (at, [p.summation over (i=1)] [b.sub.it][T.sub.i], [r.summation over (j=1)] [c.sub.jt][I.sub.j], [s.summation over (k-1)] [d.sub.kt][F.sub.k]) with t = 1,..., n such that

{(a, [p.summation over (i=1)] [b.sub.i][T.sub.i], [r.summation over (j-1)] [c.sub.j][I.sub.j], [s.summation over (k=1)] [d.sub.k][F.sub.k]), (a', [p.summation over (i=1)] [b'.sub.i][T.sub.i], [r.summation over (j-1)] [c'.sub.j][I.sub.j], [s.summation over (k=1)] [d'.sub.k][F.sub.k])}

is a subset of

(a1, [p.summation over (i=1)] [b.sub.i1][T.sub.i], [r.summation over (j=1)] [c.sub.j1][I.sub.j], [s.summation over (k=1)] [d.sub.k][F.sub.k]) [symmetry]... [symmetry] ([a.sub.n], [p.summation over (i=1)] [b.sub.in][T.sub.i], [r.summation over (j=1)] [c.sub.jn][I.sub.j], [s.summation over (k=1)] [d.sub.kn][F.sub.k]).

The latter implies that a, a' e [a.sub.1] + ... + [a.sub.n], [b.sub.i], [b'.sub.i] [member of] [b.sub.i1] + ... + [b.sub.in], [c.sub.j], [c'.sub.j] [member of] [c.sub.j1] + ... + [c.sub.jn] and [d.sub.k], [d'.sub.k] [member of]

[d.sub.k1] + ... + [d.sub.kn]. Thus, a[beta]a', [b.sub.i][beta][b'.sub.i], [c.sub.j][beta][c'.sub.j] and [d.sub.k][beta][d'.sub.k].

Conversely, let a/3a', [b.sub.i][beta][b'.sub.i], [c.sub.j][beta][c'.sub.j] and [d.sub.k][beta][d'.sub.k]. Then there exist [t.sub.1], [t.sub.2], [t.sub.3], [t.sub.4] e N and[mathematical expression not reproducible] [member of] H such that [mathematical expression not reproducible] and [mathematical expression not reproducible]. By setting t = max{[t.sub.1], [t.sub.2], [t.sub.3], [t.sub.4]} and [x.sub.m] = 0 for [t.sub.1] < m [less than or equal to] t, [y.sub.im] = 0 for [t.sub.2] < m [less than or equal to] t, [z.sub.jm] = 0 for [t.sub.3] < m [less than or equal to] t and [w.sub.km] = 0 for [t.sub.4] < m [less than or equal to] t and using the fact that e [member of] 0 + e [intersection] e + 0 for all e [intersection] H, we get [mathematical expression not reproducible] and [d.sub.k], [d'.sub.k] [member of] [W.sub.k1] + ... + [W.sub.kt]. The latter implies that {(a, [p.summation over (i=1)] [b.sub.i][T.sub.i], [r.summation over (j=1)] [c.sub.j][I.sub.j], [s.summation over (k=1)] [d.sub.k][F.sub.k]), (a', [p.summation over (i=1)] [b'.sub.i][T.sub.i], [r.summation over (j=1)] [c'.sub.j][I.sub.j], [s.summation over (k=1)] [d'.sub.k][F.sub.k])} is a subset of ([x.sub.1] [p.summation over (i=1)] [y.sub.i1][T.sub.i], [r.summation over (j=1)] [z.sub.j1][I.sub.j], [s.summation over (k-1)] [W.sub.k1][F.sub.k]) [symmetry] ... [symmetry] ([x.sub.t], [p.summation over (i=1)] [y.sub.it][T.sub.i], [r.summation over (j=1)] [z.sub.jt][I.sub.j], [s.summation over (k-1)] [w.sub.kt][F.sub.k]). Thus,

(a, [p.summation over (i=1)] [b.sub.i][T.sub.i], [r.summation over (j-1)] [c.sub.j][I.sub.j], [s.summation over (k=1)] [d.sub.k][F.sub.k]), (a', [p.summation over (i=1)] [b'.sub.i][T.sub.i], [r.summation over (j-1)] [c'.sub.j][I.sub.j], [s.summation over (k=1)] [d'.sub.k][F.sub.k])

Theorem 5.5. Let (H, +) be a hypergroup with identity "0" and 0 + 0 = 0. Then RNQ(H)/[[beta].sub.N] [congruent to] RNQ(H/[beta]).

Proof. Let [phi] : RNQ(H)/[[beta].sub.N] [right arrow] RNQ(H/[beta]) be defined as

[phi]([[beta].sub.N] ((a, [p.summation over (i=1)] [b.sub.i][T.sub.i], [r.summation over (j=1)] [c.sub.j][I.sub.j], [s.summation over (k=1)] [d.sub.k][F.sub.k]))) = ([beta](a), [p.summation over (i=1)] [beta] ([b.sub.i])[T.sub.i], [r.summation over (j=1)] [beta] ([c.sub.j])[I.sub.j], [s.summation over (k=1)] [beta]([d.sub.k])[F.sub.k]).

Theorem 5.4 asserts that [phi] is well-defined and one-to-one. Also, it is clear that [phi] is onto. We need to show that [phi] is a group homomorphism. Let

[bar.a] = (a, [p.summation over (i=1)] [b.sub.i][T.sub.i], [r.summation over (j=1)] [c.sub.j][I.sub.j], [s.summation over (k=1)] [d.sub.k][F.sub.k]) and [bar.a'] = (a', [p.summation over (i=1)] [b'.sub.i][T.sub.i], [r.summation over (j=1)] [c'.sub.j][I.sub.j], [s.summation over (k=1)] [d'.sub.k][F.sub.k]). Since [[beta].sub.N] ([bar.a]) [??]' [[beta].sub.N] ([bar.a']) = [[beta].sub.N] ([bar.x]) where [bar.x] = x [p.summation over (i=1)] [y.sub.i][T.sub.i], [r.summation over (j=1)] [z.sub.j][I.sub.j], [s.summation over (k-1)] [w.sub.k][F.sub.k]) [member of] (a, [p.summation over (i=1)] [b.sub.i][T.sub.i], [r.summation over (j=1)] [c.sub.j][I.sub.j], [s.summation over (k-1)] [d.sub.k][F.sub.k])[symmetry](a', [p.summation over (i=1)] [b'.sub.i][T.sub.i] [r.summation over (j=1)] [c'.sub.j][I.sub.j], [s.summation over (k-1)] [d'.sub.k][F.sub.k]) = (a + a', [p.summation over (i=1)] ([b.sub.i]+ [b'.sub.i])[T.sub.i], [r.summation over (j=1)] ([C.sub.J] + [c'.sub.j])[I.sub.j], [s.summation over (k=1)] ([d.sub.k] + [d'.sub.k])[F.sub.k]), it follows that

[phi]([[beta].sub.N]([bar.a]) [??] [[beta].sub.N]([bar.a'])) = [phi]([bar.x]) = ([beta](x), [p.summation over (i=1)][beta]([y.sub.i])[T.sub.i], [r.summation over (j=1)]([z.sub.j])[I.sub.j], [s.summation over (k=1)] [beta]([w.sub.k])[F.sub.k]).

Having [beta](x) = [beta](a) [symmetry] [beta](a'), [beta]([y.sub.i]) = [beta]([b.sub.i]) [symmetry] [beta]([b'.sub.i]), [beta]([z.sub.i]) = [beta]([c.sub.j]) [symmetry] [beta]([c.sub.j]) and [beta]([w.sub.k]) = [beta]([d.sub.k]) [??] [beta]([d'.sub.k]) imply that [phi]([[beta].sub.N]([bar.a]) [??] [[beta].sub.N] ([.bar.a'])) = [phi]([[beta].sub.N] ([bar.a])) [symmetry]' [phi]([[beta].sub.N][bar.a']).

Corollary 5.6. Let (H, +) be a hypergroup with identity element "0" and 0 + 0 = 0. If G is the fundamental group of H (up to isomorphism) then RNQ(G) is the fundamental group of RNQ(H) (up to isomorphism).

Proof. The proof follows from Proposition 5.2 and Theorem 5.5.

Corollary 5.7. Let (H, +) be a hypergroup with identity element "0" and 0 + 0 = 0. If H has a trivial fundamental group then RNQ(H) has a trivial fundamental group.

Proof. The proof is straightforward by applying Corollary 5.6.

Theorem 5.8. [8] Every single power cyclic hypergroup has a trivial fundamental group.

Corollary 5.9. Let (H, +) be a single power cyclic hypergroup with 0 [member of] H and 0 + 0 = 0. Then RNQ(H) has a trivial fundamental group.

Proof. The proof follows from Corollary 5.7 and Theorem 5.8.

6 Conclusion

This paper contributed to the study of neutrosophic hyperstructures by introducing refined neutrosophic quadruple hypergroups (po-hypergroups) and determining their fundamental refined neutrosophic quadruple groups. Several interesting results related to these new hypergroups were obtained. For future work, it will be interesting to study new properties of other types of refined neutrosophic quadruple hyperstructures.

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Received: April 26, 2019. Accepted: June 04, 2019.

M. Al-Tahan (1), B. Davvaz (2,*)

(1) Lebanese International University, Bekaa, 961, Lebanon.

E-mail: madeline.tahan@liu.edu.lb

(2) Yazd University, Yazd, 98, Iran.

E-mail: davvaz@yazd.ac.ir

(*) Correspondence: B. Davvaz (davvaz@yazd.ac.ir)
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