A Study on Some Properties of Neutrosophic Multi Topological Group

: In this paper, we studied some properties of the neutrosophic multi topological group. For this, we introduced the deﬁnition of semi-open neutrosophic multiset, semi-closed neutrosophic multiset, neutrosophic multi regularly open set, neutrosophic multi regularly closed set, neutrosophic multi continuous mapping, and then studied the deﬁnition of a neutrosophic multi topological group and some of their properties. Moreover, since the concept of the almost topological group is very new, we introduced the deﬁnition of neutrosophic multi almost topological group. Finally, for the purpose of symmetry, we used the deﬁnition of neutrosophic multi almost continuous mapping to deﬁne neutrosophic multi almost topological group and study some of its properties.


Introduction
Following the introduction of the fuzzy set (FS) [1], a variety of studies on generalisations of FS concepts were performed. In the sense that the theory of sets should have been a particular case of the theory of FSs, the theory of FSs is a generalisation of the classical theory of sets. Following the generalisation of FSs, many scholars used the theory of generalised FSs in a variety of fields in science and technology. Fuzzy topology (FT) was first introduced by Chang [2], and Intuitionistic fuzzy topological space (FITS) was defined by Coker [3]. Many researchers studied topology based on neutrosophic sets (NS), such as Lupianez [4][5][6][7] and Salama et al. [8]. Kelly [9] defined the concept of bitopological space (BTS) in 1963. Kandil et al. [10] studied the topic of fuzzy bitopological space (FBTS). Some characteristics of Intuitionistic Fuzzy Bitopological Space (IFBTS) were addressed by Lee et al. [11]. Garg [12] investigated how to rank interval-valued Pythagorean FSs using a modified score function. A Pythagorean fuzzy method for order of preference by similarity to ideal solution (TOPSIS) method based on Pythagorean FSs was discussed, which took the experts' preferences in the form of interval-valued Pythagorean fuzzy decision matrices. Moreover, different explorations of the theory of Pythagorean FSs can be seen in [13][14][15][16][17][18][19]. Yager [20] proposed the q-rung orthopair FSs, in which the sum of the qth powers of the membership (MS) and non-MS degrees is restricted to one [21]. Peng and Liu [22] studied the systematic transformation for information measures for q-rung orthopair FSs. Pinar and Boran [23] applied a q-rung orthopair fuzzy multi-criteria group decision-making method for supplier selection based on a novel distance measure.
Cuong et al. [24] proposed a picture FS as an extension of FS and Intuitionistic fuzzy set (IFS) that contains the concept of an element's positive, negative, and neutral MS de-

Motivation
There is a lot of ambiguity information in the real world that crisp values cannot manage. The FS theory [1], proposed by Zadeh, is an age-old and excellent tool for dealing with uncertain information; however, it can only be used on random processes. As a result, Sebastian et al. [56] introduced FMSs, Atanassov [57] suggested the IFS theory, and Shinoj et al. [58] launched intuitionistic FMSs, all based on FS theory. The theories mentioned above have expanded in a variety of ways and have applications in a variety of fields, including algebraic structures. Some of the selected papers are those on FSs [59][60][61], FMSs [62][63][64], IFSs [65][66][67][68][69][70][71][72], and intuitionistic FMSs [73]. However, these theories are incapable of dealing with all forms of uncertainty, such as indeterminate and inconsistent data in various decision-making situations. To address this shortfall, Smarandache [74] proposed the NS theory, which makes Atanassov's [57] theory very practical and easy to apply. In this current decade, neutrosophic environments are mainly interested by different fields of researchers. In Mathematics, much theoretical research has also been observed in the sense of neutrosophic environment. A more theoretical study will be required to build a broad framework for decision-making and to define patterns for the conception and implementation of complex networks. Deli et al. [75] and Ye [76,77] proposed the notion of neutrosophic multiset (NMS) for modelling vagueness and uncertainty in order to improve the NS theory further. From the literature survey, it was noticed that precisely the properties of the neutrosophic multi topological group (NMTG) are not performed. Now, as an update for the research in NMS, we introduced the definition of a neutrosophic semi-open set, neutrosophic semi-closed set, neutrosophic regularly open set, neutrosophic regularly closed set, neutrosophic continuous mapping, neutrosophic open mapping, neutrosophic closed mapping, neutrosophic semi-continuous mapping, neutrosophic semiopen mapping, neutrosophic semi-closed mapping. Moreover, we tried to prove some of their properties and also cited some examples. We defined the neutrosophic multi almost topological group by using the definition of neutrosophic multi almost continuous mapping and investigate some properties and theorems of a neutrosophic multi almost topological group.

Materials and Methods
Definition 1 ([42]). Let X be a non-empty fixed set. A neutrosophic set (NS) A is an object with the form , and γ A (x) represents the degree of MS function, the degree indeterminacy, and the degree of non-MS function, respectively, of each element x ∈ X to set A.

Definition 2 ([78]).
A neutrosophic multiset (NMS) is a type of neutrosophic set (NS) in which one or more elements are repeated with the same or different neutrosophic components.  ([52]). The Empty NMS is defined as 0 NM = m ∈ X; < m (0,1,1) > , where m can be repeated.

Definition 5 ([52]
). Let X = φ, and a neutrosophic multiset (NMS) A on X can be expressed where m can be repeated depending on its multiplicity, and the T, , F values may or may not be equal.

Definition 7 ([78]
). Let X = φ, and a neutrosophic multiset topology (NMT) on X is a family τ X of neutrosophic multi subsets of X if the following conditions hold: Then (X, τ X ) is known as a neutrosophic multi topological space (NMTS), and any NMS in τ X is called a neutrosophic multi-open set (NMOS). The element of τ X are said to be NMOSs, an NMS F is neutrosophic multi closed set (NMCoS) if F c is NMOS.

Definition 8 ([52]
). Let X be a classical group and A be a neutrosophic multiset (NMS) on X. Then A is said to be neutrosophic multi groupoid over X if , ∀ m, n ∈ X and i = 1, 2, . . . , P.
Moreover, A is said to be neutrosophic multi-group (NMG) over X if the neutrosophic multi groupoid satisfies the following: , ∀ m ∈ X and i = 1, 2, . . . , P.

Definition 9 ([52]
). Let G be an NMG in a group X, and e be the identity of X. We define the NMS G e by We note for an NMG G in a group X, for every m ∈ X : Moreover, for the identity e ∈ X : T G (e) T G (m), G (e) G (m) and F G (e) F G (m).

Results
Definition 10. Let (X, τ X ) be NMTS. Then for an NMS A = < x, µ N i , σ N i , δ N i > : x ∈ X , the neutrosophic interior of A can be defined as N M Definition 11. Let (X, τ X ) be NMTS. Then for an NMS A = < x, µ N i , σ N i , δ N i > : x ∈ X , the neutrosophic closure of A can be defined as N M Definition 12. Let G be an NMG on a group X. Let τ X be a NMT on G, then (G , τ X ) is known as a neutrosophic multi topological group (NMTG) if it satisfies the given conditions: Then τ = {0 X , 1 X , B} is neutrosophic multi topological space. Then Cl(B) = 1 X , Int(Cl(B)) = 1 X . Hence, B is NMSOS.
Proof. Proof is straightforward.

Lemma 2.
Let A, B be NMSs of X and Y, then Proof. For each p ∈ X, we have , where a subfamily B of (X, τ X ) is said to be subbase for (X, τ X ) if the collection of all intersections of members of B forms a base for (X, τ X ).

Lemma 6. For an NMS
Proof. Proof is straightforward. Theorem 1. The statements below are equivalent:

Proof. (i) and (ii) are equivalent follows from Lemma 6, since for an NMS
(ii)⇔(iv) can similarly be proved.

Theorem 2. (i) Arbitrary union of NMSOSs is an NMSOS;
(ii) Arbitrary intersection of NMSCoSs is an NMSCoS.

Remark 1. It is clear that every NMOS (NMCoS) is an NMSOS (NMSCoS)
. The converse is not true.
are NMTSs, and X is a product related to Y. Then the product A × B of an NMSOS A of X and an NMSOS B of Y is an NMSOS of the neutrosophic multi-product space X × Y.
Hence, A is NMROS.
Proof. It follows from Lemma 3.

Remark 2.
It is obvious that every NMROS (NMRCoS) is an NMOS (NMCoS). The converse need not be true.

Remark 3.
The union (intersection) of any two NMROSs (NMRCoS) need not be an NMROS (NMRCoS). Here, Hence, A and B is NROS, but A B is not NROS.

Theorem 5. (i) The intersection of any two NMROSs is an NMROS;
(ii) The union of any two NMRCoSs is an NMRCoS.
Proof. (i) Let A 1 and A 2 be any two NMROSs of an NMTS (X, τ X ).
(ii) Let A 1 and A 2 be any two NMROSs of an NMTS (X, τ X ).  Then   Then τ X = {0 X , 1 X , A} and τ Y = {0 Y , 1 Y , B} are neutrosophic multi topological spaces. Let us define a mapping f : Thus, f is NMSCM, which is not an NMCM. Theorem 7. Let X 1 , X 2 , Y 1 and Y 2 be NMTSs such that X 1 is product related to X 2 . Then, the product φ 1 × φ 2 : where A α 's and B β 's are NMOSs of Y 1 and Y 2 , respectively, be an NMOS of Y 1 × Y 2 . By using Lemma 1(i) and Lemma 3, we have is an NMSOS follows from Theorem 3 and Theorem 2 (i).
Theorem 8. Let X, X 1 and X 2 be NMTSs and p i :

Proof. For an NMOS
Theorem 9. Let φ : X −→ Y be a mapping from an NMTS X to another NMTS Y. Then if the graph ψ : Since ψ is an NMSCM and 1 NM × A is an NMOS X × Y, φ −1 (A) is an NMSOS of X and hence φ is an NMSCM.
Then the below statements are equivalent: (a) φ is an NMACM; (b) ⇔ (d) similarly can be proved.

Remark 6.
Clearly, an NMCM is an NMACM. The converse need not be true.
Theorem 14. Let X, X 1 and X 2 be an NMTSs and p i : X 1 × X 2 → X i (i = 1, 2) be the projection of X 1 × X 2 onto X i . Then if φ : X → X 1 × X 2 is an NMACM, p i φ is also an NMACM.

Again, since (i) each p i is an NMOS, and (ii) for any NMS
Theorem 15. Let X and Y be NMTSs such that X is product related to Y and let φ : X → Y be a mapping. Then, the graph ψ : Proof. Consider that ψ is an NMACM and A is an NMOS of Y. Then, using Lemma 4 and Theorems 10 (c), we have Thus, by Theorem 10 (c), φ is NMACM. Conversely, let φ be an NMACM and B = B α × A β , where B α 's and A β 's are NMOSs of X and Y, respectively, be an NMOS of X × Y.
Since B α N M Int φ −1 N M Int N M Cl A β is an NMOSs of X contained in and hence, using Lemmas 1(i), 4 and 5, and Theorems 10 (c), we have Thus, by Theorem 10(c), ψ is NMACM.

Definition 25.
Let G be an NMG on a group X. Now, if τ X is an NMT on G, then (G , τ X ) is said to be a neutrosophic multi almost topological group (NMATG) if the given conditions are satisfied: Then (G , τ X ) is known as an NMATG.
Remark 7. (G , τ X ) is an NMATG if the below conditions hold good: (i) For g 1 , g 2 ∈ G and every NMROS P containing g 1 g 2 in G , ∃ open neighborhoods R and S of g 1 and g 2 in G such that R * S P; (ii) For g ∈ G and every N in G containing g −1 , ∃ open neighborhood R of g in G so that R −1 S.

Remark 8.
For any P, Q G, we denote P * Q by P Q and defined as P Q = {gh : g ∈ P, h ∈ Q} and P −1 = g −1 : g ∈ P . If P = {a} for each a ∈ G, we denote P * Q by aQ and Q * P by P a.
Theorem 16. Let (G , τ X ) be an NMATG and let a be any element of G. Then (a) µ a : Proof. (a) Let p ∈ G and let R be an NMROS containing ap in G. By Definition 25, ∃ open neighborhoods P, Q of a, p in G such that P Q R. Especially, aQ R, i.e., µ a (Q) R. This proves that µ a is NMACM at p, and hence, µ a is NMACM.
(b) Suppose p ∈ G and R ∈ N MRO(G) contain pa. Then ∃ open sets p ∈ P and a ∈ Q in G such that P Q R. This proves P a R. This shows that λ a is NMACM at p. Since arbitrary element p is in G, hence, λ a is NMACM. Proof. (a) We first show that mU ∈ τ X . Let p ∈ mU . Then by Definition 25 of NMATGs, ∃ NMOSs m −1 ∈ W 1 and p ∈ W 2 in G such that W 1 W 2 U . Especially, m −1 W 2 U . That is, equivalently, W 2 mU . This indicates that p ∈ N M Int(mU ) and thus, N M Int(mU ) = mU . That is mU ∈ τ X . Consequently, mU N M Int(N M Cl(mU )). Now, we have to prove that N M Int(N M Cl(mU )) mU . As U is NMOS, (b) Following the same steps as in part (1)  Proof. (a) Assume p ∈ N M Cl(U m) and consider q = pm −1 . Let q ∈ W be NMOS in G. Then ∃ NMOSs m −1 ∈ V 1 and p ∈ V 2 in G, such that V 1 V 2 N M Int(N M Cl(W)). By hypothesis, there is g ∈ U m Conversely, let q ∈ N M Cl(U )m. Then q = pg for some p ∈ N M Cl(U ). Conversely, let q ∈ N M Int(mQ) be an arbitrary element. Suppose q = mp, for some p ∈ Q. By hypothesis, this proves mQ is NMCoS, and that is N M Int(mQ) is NMROS in G. Assume that m ∈ U and p ∈ V be NMOSs in G, such that UV N M Int(mQ). Then mV mQ, which means that mV mN M Int(Q). Thus, N M Int(mQ) mN M Int(Q).
(b) Following the same steps as in part (1) above, we can prove that N M Int(Qm)

Conclusions
To deal with uncertainty, the NS uses the truth membership function, indeterminacy membership function, and falsity membership function. By discovering this concept, we were able to generalise the idea of an almost topological group to an NMATG. First, we developed the definitions of NMSOS, NMSCoS, NMROS, NMRCoS, NMCM, NMOM, NMCoM, NMSCM, NMSOM, NMSCoM to propose the definition of NMATG. Some properties of NMACM were demonstrated. Finally, we defined NMATG and demonstrated some of their properties using the definition of NMACM. In this study, an NMATG is conceptualised for the environments of the NS along with some of their elementary properties and theoretic operations. Novel numerical examples are given for definitions and remarks to study NMATG. We expect that our study may spark some new ideas for the construction of the NMATG. Future work may include the extension of this work for: (1) The development of the NMATG of the neutrosophic multi-vector spaces, etc.; (2) Dealing NMATG with multi-criteria decision-making techniques.