# On a Matrix over NC and Multiset NC Semigroups

Authors: Mohammed A. Saleem; Mohamed Abdalla; A. Elrawy
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we define a matrix over neutrosophic components (NCs), which was built using the four different intervals0,1,0,1,0,1, and 0,1. This definition was made clear by introducing some examples. Then, the study of the algebraic structure of matrices over NC under addition modulo 1, the usual product, and product by using addition modulo 1 was introduced, from which it was found that the matrix over NC built using interval 0,1 happens to be an abelian group under addition modulo 1. Furthermore, it is proved that the matrix over NC defined on the interval 0,1 is not a regular semiring. Also, we define a matrix over multiset NC semigroup using the interval 0,1. Moreover, we define a matrix over m-multiplicity multiset NC semigroup for finite m. Several interesting properties are discussed for the three structures. It was concluded that the last two structures are semigroups and semirings under addition modulo 1 and usual product, respectively.


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## Body

## 1. Introduction

Semigroups play main roles in algebraic structures [1–9]. Neutrosophic sets were introduced by Smarandache [10]. Every element in a neutrosophic set has three associated functions, namely, the membership function, the non-membership function, and the indeterminacy function, all of which are defined on the universe of discourse X. Moreover, these three functions are completely independent. Since then, this concept has become an interesting area of major research both in the area of algebraic structures [11–16] and analysis [17] and in applications ranging from medical diagnosis to sentiment analysis [18, 19]. Also, matrices always play a significant role in technology and science. However, in some situations, the classical matrix theory fails to solve problems involving uncertainties that emerge in an unpredictable world. Some researchers have studied and applied neutrosophic matrix [20–22]. For instance, they introduced some novel operations on neutrosophic matrices, that is, type-1 product operation of the SVN-matrices is a similar Hadamard product of matrices. Therefore, the type-1 product allows making some applications related to Hadamard product under the SVN environment. On the other hand, there is no study for neutrosophic matrix under the usual addition and product operations. Motivated by the previous study of Vasantha et al. [23], where they defined the NC over the intervals 0,1,0,1,0,1, and 0,1 under the usual addition, addition modulo 1, and classic product, here we venture to study matrix over NC on all the four mentioned intervals results in several interesting algebraic structures. Indeed, it is proven that there is no algebraic structure under addition modulo 1 and the new type of product for matrix over NC over the intervals 0,1,0,1, and 0,1. It is shown that the interval 0,1 provides an interesting algebraic structure. Thus, the aim of the present paper is to define a matrix over neutrosophic components and multiset neutrosophic components. Also, we define some operations between a matrix over neutrosophic components and multiset neutrosophic components to investigate some of their properties. The remaining part of the paper is organized as follows. Section 2 gives a brief summary of neutrosophic components, multiset neutrosophic components, and operations on these sets. In Section 3, we introduce and study a matrix over neutrosophic components. In Section 4, we define and study matrix multiset NC built using the interval 0,1 semigroups under semigroup under addition modulo 1 and product ×⊕. Section 5 is devoted to studying matrix over m-multiplicity multiset NC on the interval 0,1 and obtaining several interesting properties. In Section 6, we draw some conclusions.


## 2. Basic Concepts

This section introduces the fundamental principles needed to make this paper self-contained.Now, we recall the definition of semigroup in the following form.Definition 1 (see [6]).
A coupleS,× is said to be a semigroup if the operation × is closed and associative.Definition 2 (see [6]).
A coupleS,× is said to be a regular semigroup if for each element a∈S, there exists x∈S such that a=a⋅x⋅a.Definition 3.
A semigroupS,×is said to be a torsion-free semigroup if fora,b∈S,a≠b,an≠bnfor any1≤n<∞.Definition 4 (see [23]).
The neutrosophic component (NC) is a tripleta,b,c, whereais the truth membership function,bis the indeterminacy membership function, andcis the falsity membership function, and all of them are from the unit interval0,1.Definition 5 (see [10]).
A neutrosophic multiset is a neutrosophic set where one or more elements are repeated with same neutrosophic components or with different neutrosophic components.Now, we define four sets of neutrosophic components as follows.Definition 6 (see [23]).
LetSl,l=1,2,3,4. Then, we define four sets of NC as follows: S1=x,y,z:x,y,z∈0,1,S2=x,y,z:x,y,z∈0,1,S3=x,y,z:x,y,z∈0,1, and S4=x,y,z:x,y,z∈0,1.Definition 7 (see [23]).
LetS2,× be a multiset NC semigroup under ×; then, the elements of the form a,b,0, 0,0,c, and so on which are infinite in number with a,b,c∈S2 contribute to zero divisors. Hence, multisets using these types of elements contribute to zeros of the form n0,0,0,1<n<∞. As the zeros are of varying multiplicity, we call these zero divisors as special type of zero divisors.Remark 1.
We denote the set of the multisets of NC using elements ofSl,l=1,2,3,4 by MSl.Definition 8 (see [23]).
The collection of all multisets with entries fromSl,l=1,2,3,4, with at most multiplicity m, 2≤m<∞, is called m-multiset NC and we denote it by m-MSl.


## 3. Matrix over NC

In this section, we introduce matrices over a set of NC (MNC for short) and support them by presenting concrete examples. Also, we give the definition of usual addition modulo 1 and two types of multiplication of NC to obtain some algebraic structures for MNC. Furthermore, we give some examples and some results on MNC. LetSl,l=1,2,3,4, be the four sets of NC and Mm×nSl be the set of all m×n matrices defined on Sl.Definition 9.
A matrix over NC (MNC) of orderm×nis defined asA=AijwhereAij=aij,bij,cij∈Slare called truth-membership, indeterminacy-membership, and falsity-membership values of theithandjthelements inA. Letℳm×ndenote the set of all MNC of orderm×n. In particular, ℳnSldenotes the set of all square MNCs of ordern.Definition 10.
LetA=Aijbe anm×nMNC. If all of its entries areAij=0,0,0,∀i,j, thenAis called the zero MNC and denoted byO. Ifm=n,Aij=0,0,0,∀i≠j, andAij=1,1,1,∀i=j, thenAis called the identity MNC and denoted byℐ.Definition 11.
LetA=Aijandℬ=Bijbe twom×nMNC; then, addition modulo 1 is defined as follows:(1)A⊕1ℬ=Aij⊕1Bij.It is well known that for twon-square MMs A=Aij and ℬ=Bij, the usual product is defined as A×ℬ=∑j=1nAijBjk. Now, we introduce the new type of the product of two n-square MNCs by using addition modulo 1 as in the following definition.Definition 12.
LetA=Aijandℬ=Bijben-square MNCs; then, we define the product of MNC by using addition modulo 1 as follows:(2)A×⊕1ℬ=⊕j=1nAijBjk,where ⊕j=1n represents the summation using addition modulo 1.Next, we illustrate how addition modulo 1, usual product, and a new type of product are performed on any two MNCs.Example 1.
LetA,ℬ,C,D∈ℳ2Si,l=1,2,3,4, where(3)A=0.3,0.2,0.40.6,0.7,0.10.6,0.7,0.10.1,0.3,0.2,ℬ=0.1,0.2,0.50.8,0.4,0.50.1,0.3,0.60.7,0.4,0.5,C=0.7,0.8,0.60.4,0.3,0.90.4,0.3,0.90.9,0.7,0.8,D=0.9,0.3,0.40.8,0.9,0.60.1,0.2,0.10.2,0.4,0.5.
Then, we have, for example, thatA+C=1,1,11,1,11,1,11,1,1 and A⊕1C=O.
Also,(4)A+ℬ=0.4,0.4,0.91.4,1.1,0.60.7,1,0.70.8,0.7,0.7,A⊕1ℬ=0.4,0.4,0.90.4,0.1,0.60.7,0,0.70.8,0.7,0.7.
Since0,0,0∉Sl,l=1,3, then it is clear that the two operations+and⊕1are not closed inℳ2Sl,l=1,3. Furthermore, the two operations×and⊕j=1nare not closed.
In caseA,D∈ℳ2S4,(5)A+D=1.2,0.5,0.81.4,1.6,0.70.7,0.9,0.20.3,0.7,0.7∉ℳ2S4,and we cannot define addition modulo 1 since 1,1,1∈S4.Since the operations⊕1 and + are not closed, then the operations × and ⊕j=1n are not closed. The following example explains the new type of the product defined in Definition 12.Example 2.
LetA,ℬ∈ℳ2S2, whereA=0.9,0.3,0.40.9,0.9,0.90.1,0.2,0.10.2,0.1,0.3andℬ=0.8,0.8,0.20.2,0.3,0.10.7,0.2,0.10.2,0.4,0.1.Hence,A×ℬ=1.35,0.42,0.170.36,0.45,0.130.22,0.18,0.050.06,0.1,0.04∉ℳ2S1. But the new type of the product by using addition modulo 1 gives the following:(6)A×⊕1ℬ=0.35,0.42,0.170.36,0.45,0.130.22,0.18,0.050.06,0.1,0.04.Remark 2.
SinceS1andS3do not include the elements0,0,0and1,1,1, thenℳn×nS1andℳn×nS3do not include the matricesOandℐ.Lemma 1.
The above two examples explain thatℳnSl,l=1,2,3, are not algebraic structures under usual product and under addition modulo 1. Moreover, they are not semigroups under usual product and under addition modulo 1.Theorem 1.
LetS2=x,y,z:x,y,z∈0,1be the collection of NC andℳm×nS2be a set ofm×nmatrices overS2; then, ℳm×nS2,⊕1is a commutative group under addition modulo 1.Proof.
Assume thatA,ℬ∈ℳS2; then, A⊕1ℬ∈ℳS2 and associative law hold. O∈ℳS2 is a zero identity. Also, for every A∈ℳS2, there is a unique ℬ∈ℳS2 such that A⊕1ℬ=O. Further, we find A⊕1ℬ=ℬ⊕1A. Thus, ℳS2,⊕1 is a commutative group.Theorem 2.
ℳnS2,×⊕1 is only a semigroup and not a monoid. It has infinite number of zero divisors.Proof.
Suppose thatA=Aij,ℬ=Bjk∈ℳnS2; then, A×⊕1ℬ=⊕j=1nAijBjk, and since ℳnS2,⊕1 is a group, A×⊕1ℬ∈ℳnS2. Hence, ℳnS2,×⊕1 is a semigroup under the ×⊕1 product. It is obvious that ℐ∉ℳnS2, and we find ℳS2,×⊕1 is not a monoid. Finally, O∈ℳnS2, so ℳnS2,×⊕1 has infinite number of zero divisors, and hence the claim is satisfied.Theorem 3.
ℳnS2,⊕1,×⊕1 is a ring with infinite number of zero divisors and has no multiplicative identity ℐ.Proof.
SinceℳnS2,⊕1 is a group from Theorem 1 and ℳnS2,×⊕1 is a semigroup from Theorem 2, the distributive property is acquired from the number theoretic properties of modulo integers, hence the result.
Next result shows thatS2,⊕1,× is not regular semiring.Lemma 2.
S2,⊕1,× is not regular semiring.Proof.
To prove this result, we use the method contradiction. LetS2,⊕1,× be regular semiring, and then for all a,b,c∈S2, there exists e,f,g∈S2 such that aea,bfb,cgc=a,b,c, and we thus observe that ea=1,fb=1andgc=1. This completes the proof.Remark 3.
SinceS2,⊕1,× is not regular semiring, then MS2,⊕1,× is not regular.


## 4. Matrix over Multiset NC Semigroups

In this section, we introduce and study matrices over multiset NC semigroups usingS2. We are referring to the collection of such matrices by ℳnMS2, where MS2 is a collection of all multisets built using S2. Also, we prove that the algebraic structure ℳnℳS2,⊕1,×⊕1 is a semiring of infinite order.Now, we illustrate how the addition and the multiplication operations are performed on any two matrices over multiset NC inMS2.Definition 13 (see [23]).
LetA,B∈MS2, where A=na,b,c:a,b,c∈S2,n∈ℕ and B=mx,y,x:x,y,z∈S2,m∈ℕ. Then, we define the product and the addition modulo 1 as follows:(7)A×B=nmax,by,cz:a,b,c,x,y,z∈S2,n,m∈ℕ,A⊕1B=nma⊕1x,b⊕1y,c⊕1z:a,b,c,x,y,z∈S2,n,m∈ℕ.Now, we illustrate how the two operations⊕1 and ×⊕1 are performed in ℳnMS2. To define these operations, we give the following example.Example 3.
LetℳA1=A,ℬ,C and ℳℬ1=D,E in ℳnMS2. Then, the sum of ℳA1 with ℳℬ1 under addition modulo 1 is given by(8)ℳA1⊕1ℳℬ1=A⊕1D,A⊕1E,ℬ⊕1D,ℬ⊕1E,C⊕1D,C⊕1E,which is in ℳnMS2. Also, the product of ℳA1 and ℳℬ1 is given by(9)ℳA1×⊕1ℳℬ1=A×⊕1D,A×⊕1E,ℬ×⊕1D,ℬ×⊕1E,C×⊕1D,C×⊕1E,which is in ℳnMS2.Example 4.
LetA,ℬ∈ℳ2MS2, where(10)A=90.3,0.2,0.420.6,0.7,0.120.6,0.7,0.10.1,0.3,0.2,ℬ=50.1,0.2,0.5100.8,0.4,0.550.1,0.2,0.5100.8,0.4,0.5.
Then, we define the addition modulo 1 ofA and ℬ as follows:(11)A⊕1ℬ=450.4,0.4,0.9200.4,0.1,0.6100.7,0.9,0.6100.9,0.7,0.7.
Also, we define the new type of product×⊕1 of A and ℬ as follows:(12)A×⊕1ℬ=4500.09,0.18,0.2518000.72,0.36,0.25500.07,0.2,0.152000.56,0.4,0.15.Theorem 4.
LetℳnMS2 be the matrix over multiset NC built using S2; then, ℳnMS2,⊕1 is a semigroup under addition modulo 1.Proof.
SinceℳnMS2 is closed under the binary operation addition modulo 1, ℳnMS2,⊕1 is a semigroup.Theorem 5.
ℳnMS2,×⊕1 is an infinite non-commutative matrix over multiset NC semigroup, which is not a monoid and has special type of zero divisors.Proof.
SinceℳnMS2is closed under the binary operation product with associative, let ℳnMS2be noncommutative. Thus, ℳnMS2,×⊕1 is a non-commutative semigroup of infinite order. Furthermore, ℐ∉ℳnS2, so ℳnMS2,×⊕1 is not a monoid. From Definition 7 of special zero divisors, we find that ℳnMS2,×⊕1 has infinite number of special type of zero divisors.
Using Theorems4 and 5, we have the following.Theorem 6.
LetℳnMS2,⊕1,×⊕1 be an algebraic structure of matrices over multiset NC. Then, it is a non-commutative semiring of infinite order. Moreover, it has infinite numbers of special type of zero divisors.Definition 14.
Any subset ofc containing elements of the form m1a,0,0, m20,b,c,a,b,c∈S2, which are infinite numbers, contributes to zero divisor, 1<m1,m2<∞. Hence, any matrix over this subset contributes to zero matrix with entries of the form m0,0,0,1<m<∞. As the zeros are of varying multiplicity, we call these divisors as special types of zero divisors.Example 5.
LetA,ℬ∈ℳ2MS2, where(13)A=30.6,0,020,0.2,040,0.2,050,0.2,0.3,ℬ=20,0,0.120,0,0.250.1,0,020.3,0,0.
Then,(14)A×⊕1ℬ=600,0,0240,0,02000,0,0800,0,0is a special type of zero divisor of ℳ2MS2.Theorem 7.
ℳnMS2 has infinite number of special types of zero divisor and non-trivial idempotents.Proof.
SinceMS2 has infinite number of special types of zero divisor and non-trivial idempotents, we get the claim of the theorem.


## 5. Matrix over m-Multiplicity NC Semigroups

In this section, we define the matrix overm-multiplicity NC semigroups. Also, we explain that the matrix over m-multiplicity NC under ⊕1 and ×⊕1 is a semiring of infinite order.Definition 15.
Letm-MS2 be the collection of all multisets with entries in S2 of at most multiplicity m, where 2≤m<∞. Then, we denote the n-square matrix over m-MS2 by ℳnm−MS2.Before we build a strong structure using the interval0,1, it is useful to give this example.Example 6.
Let 3-MS2 be the collection of all multisets with entries in S2 where(15)A=30.5,0.7,0.430.1,0.9,0.720.1,0.2,0.30.8,0.8,0.8,30.1,0.2,0.730.1,0.1,0.120.1,0.4,0.50.2,0.3,0.4,ℬ=0.1,0.2,0.320.1,0.4,0.330.3,0.2,0.10.2,0.1,0.1,0.1,0.2,0.30.3,0.2,0.130.4,0.5,0.520.3,0.3,0.2.
whereA,ℬ∈ℳ23−MS2
Now, we arrive at(16)A⊕1ℬ=30.6,0.9,0.730.2,0.3,030.4,0.4,0.40,0.9,0.9,30.6,0.9,0.730.4,0.1,0.830.5,0.7,0.820.1,0.1,0,30.2,0.4,030.2,0.5,0.430.4,0.6,0.60.4,0.4,0.5,30.2,0.4,030.4,0.3,0.230.5,0.9,020.5,0.6,0.6,whereA⊕1ℬ∈ℳ23−MS2,A×⊕1ℬ=30.08,0.32,0.1930.07,0.37,0.1930.25,0.2,0.1730.17,0.16,0.17,30.09,0.59,0.4730.18,0.41,0.1830.33,0.44,0.4930.27,0.28,0.19,30.04,0.06,0.2230.03,0.09,0.2230.07,0.14,0.1930.05,0.19,0.19,30.05,0.09,0.2630.06,0.07,0.0930.09,0.23,0.3530.09,0.17,0.13,whereA×⊕1ℬ∈ℳ23−MS2.Theorem 8.
Letℳnm−MS2 be a set of n×n matrices over a collection of all m-multisets of S2, where 2≤m<∞. Then, the following axioms are satisfied:(1)
ℳnm−MS2,+ is not a semigroup under usual addition.(2)
ℳnm−MS2,× is not a semigroup under usual addition.(3)
ℳnm−MS2,⊕1 is a semigroup.(4)
ℳnm−MS2,×⊕1 is an infinite non-commutative semigroup and is not a monoid.Proof.
For (1) and (2), suppose thatA,ℬ∈ℳ23−MS2, where(17)A=30.5,0.7,0.430.1,0.9,0.720.1,0.2,0.30.8,0.8,0.8,ℬ=30.1,0.2,0.730.1,0.1,0.120.1,0.4,0.50.2,0.3,0.4.
Then,A+ℬ and A×ℬ are not in ℳ23−MS2. Thus, ℳ23−MS2,+ is not a semigroup under the usual addition and the usual product.
For (3) and (4), sinceℳnMS2,⊕1 and ℳnMS2,×⊕1 are closed under ⊕1 and ×⊕1, respectively, then ℳnm−MS2,⊕1 and ℳnm−MS2,×⊕1, replacing the numbers greater than m by m in the resultant product, are semigroups as claimed. Also, ℳnm−MS2,×⊕1 is non-commutative under ×⊕1. Thus, it is a non-commutative semigroup of infinite order. Furthermore, since ℐ∉ℳnS2, then it is not a monoid.Theorem 9.
ℳnm−MS2,⊕1,×⊕1 is a non-commutative semiring of infinite order and has no unit.Proof.
The claim follows from (3) and (4) of Theorem8.


## 6. Conclusions

In this paper, we study the matrices over NC which are built using the intervals0,1,0,1,0,1, and 0,1. We define the usual addition, addition modulo 1, multiplication, and a new type of product by using addition modulo 1 on each of these intervals that are different from the studies conducted so far.The main interesting properties for matrices were developed for the current structures which are also totally different from the usual ones. In addition, the interval [0; 1) gives the algebraic structure of the commutative group and the semigroup under addition modulo 1 and usual multiplication, respectively. This result leads to constructing a structure of matrices over NC which is a non-commutative ring under usual addition modulo 1 and the new type of the product. Moreover, it has an infinite number of zero divisors.Furthermore, the notion of the matrix over multiset of NC was introduced by using the interval0,1 under the new type of product and addition modulo 1. Finally, the matrix over m-multiplicity multiset of NC was also presented.



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