Aim of this article is to study the idea of minus-ordering on fuzzy neutrosophic soft matrix. It is shown that the minus ordering in the set of all fuzzy neutrosophic soft matrix is a partial ordering. Further some properties of minus ordering on fuzzy neutrosophic soft matrix are discussed

M.Kavitha

P. Murugadas

S.Sriram

English

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Minus Ordering on Fuzzy NeutrosophicSoft MatricesM.Kavitha∗, P. Murugadas†and S.Sriram‡* Department of Mathematics, Annamalai University, Annamalainagar-608002, India.† Department of Mathematics, Government Arts College (Autonomous), Karur.‡ Mathematics wing, Directorate of Distance Education, Annamalai University,Annamalainagar-608002, India.AbstractAim of this article is to study the idea of minus-ordering on fuzzyneutrosophic soft matrix. It is shown that the minus ordering in the set of allfuzzy neutrosophic soft matrix is a partial ordering. Further some propertiesof minus ordering on fuzzy neutrosophic soft matrix are discussed.Keywords: Fuzzy Neutrosophic Soft Set, Fuzzy Neutrosophic Soft Matrix, FuzzyNeutrosophic Soft Minus Ordering.1 IntroductionThe theory of Fuzzy set was presented by Zadeh [20] in 1965. Intuitionistic fuzzyset introduced by Atanassov [3] can managed the incomplete information consideringboth the truth membership and falsity membership values. It does not deal the in-determinate and inconsistent information which can be in faith system. The perceptof Neutrosophic set was started by Smarandache[16]. Neutrosophic sets and logic arethe foundations for many theories which are general than their classical counterpartsin fuzzy, intuitionistic fuzzy, paraconsistent sets, dialetheist sets, paradoxist set and∗kavithakathir3@gmail.com†bodi muruga@yahoo.com‡ssm 3096@yahoo.co.in1International Journal of Pure and Applied MathematicsVolume 120 No. 6 2018, 7665-7675ISSN: 1314-3395 (on-line version)url: http://www.acadpubl.eu/hub/Special Issue http://www.acadpubl.eu/hub/7665tautological set. The Neutrosophic set which is handling the problems as T-truthvalue, I-indeterminate value and F-falsity value. Thus T, I, F be a real standard(or)non-standard subset of ]−0, 1+[ withSup T = t−sup, inf T = t−inf,Sup I = i−sup, inf I = i−inf,Sup F = f−sup, inf F = f−inf. and n−sup = t−sup+ i−sup+ f−sup,n−inf = t−inf + i−inf + f−inf. T, I, F are called neutrosophic components. Thesuperior sums and inferior sums.nsup = sup T + sup I + sup F ∈ ]−0, 3+[,may be as high as 3 or 3+ while,ninf = inf T + inf I + inf F ∈ ]−0, 3+[,may be as low as 0 or 0−.The concept of soft set theory was formulated at first by Molodtsov[7] in 1999. InMaji et al., [8] initiated the idea of fuzzy soft sets with the operations of union,intersection, complement of fuzzy soft set. The fuzzy soft set is extended into intu-itionistic fuzzy soft set and fuzzy Neutrosophic soft set. In [19]Yong Yang and ChenliJ: put forth a matrix representation of fuzzy soft set and used it in decision makingproblems. Rajarajeswari and Dhanalakshmi [14] innovated the intuitionistic fuzzysoft matrices and started to use in the application of medical diagnosis. Sumathiand Arockiarani [1, 2] displayed new operation on fuzzy Neutrosophic soft matrices.In [11] P. Murugadas introduced maximum g-inverse as well as minimum g-inverseof fuzzy matrix and intuitionistic fuzzy matrices. Meenakshi.AR and Inbam [10]explicate the minus ordering for fuzzy matrices and established that the minus or-dering is a partial ordering in the set of all regular fuzzy matrices. Susanta, k.khanand anita pal [17] brought in the theory of generalized inverse for intuitionistic fuzzymatrices. In [18] R.Uma et al., introduced the concept of fuzzy neutrosophic soft ma-trices of Type-1 and Type-2. The section-2 recalls some basic definition. In section-3introduced Minus ordering of Fuzzy neutrosophic soft matrices (FNSMs).Notation:N -denotes by the set of all fuzzy Neutrosophic Soft Matrices(FNSMs).Nmn-The set of all FNSMs of order m× n.Nnn-The set of all FNSMs of order n× n.N+mn = {A ∈ Nmn|A has Moore-Penrose Inverse }.N−mn = {A ∈ Nmn|A has Generalized-inverse}.2International Journal of Pure and Applied Mathematics Special Issue76662 PreliminariesDefinition 2.1. [16] A neutrosophic set A on the universe of discourse X is definedas A = {〈x, TA(x), IA(x), FA(x)〉, x ∈ X}, where T, I, F : X → ]−0, 1+[ and−0 ≤ TA(x) + IA(x) + FA(x) ≤ 3+......(1). From philosophical point of view the neu-trosophic set takes the value from real standard or non-standard subsets of ]−0, 1+[.But in real life application especially in scientific and engineering problems it is diffi-cult to use neutrosophic set with value from real standard or non-standard subset of]−0, 1+[. Hence we consider the neutrosophic set which takes the value from the sub-set of [0, 1]. Therefore we can rewrite equation (1) as 0 ≤ TA(x)+IA(x)+FA(x) ≤ 3.In short an element a˜ in the neutrosophic set A, can be written as a˜ = 〈aT , aI , aF 〉,where aT denotes degree of truth, aI denotes degree of indeterminacy, aF denotesdegree of falsity such that 0 ≤ aT + aI + aF ≤ 3Example 2.2. Assume that the universe of discourse X = {x1, x2, x3} wherex1, x2 and x3 characterize the quality, reliability, and the price of the objects. Itmay be further assumed that the values of {x1, x2, x3} are in [0,1] and they areobtained from some investigations of some experts. The experts may impose theiropinion in three components viz; the degree of goodness, the degree of indeterminacyand the degree of poorness to explain the characteristics of the objects. Suppose Ais a Neutrosophic Set (NS) of X, such thatA = {〈x1, 0.4, 0.5, 0.3〉 〈x2, 0.7, 0.2, 0.4〉, 〈x3, 0.8, 0.3, 0.4〉} where for x1 thedegree of goodness of quality is 0.4, degree of indeterminacy of quality is 0.5 anddegree of falsity of quality is 0.3 etc,. .Definition 2.3. Let U be the initial universe set and E be a set of parameter.Consider a non-empty set A,A ⊂ E. Let P (U) denote the set of all fuzzy neutro-sophic sets of U . The collection (F,A) is termed to be the fuzzy neutrosophic softset (FNSS) over U , where F is a mapping given by F : A → P (U). Here after wesimply consider A as FNSS over U instead of (F,A).Definition 2.4. [1] Let U = {c1, c2, ...cm} be the universal set and E be the setof parameters given by E = {e1, e2, ...em}. Let A ⊂ E. A pair (F,A) be a FNSSover U . Then the subset of U × E is defined by RA = {(u, e); e ∈ A, u ∈ FA(e)}which is called a relation form of (FA E). The membership function, indeterminacymembership function and non membership function are written byTRA : U × E → [0, 1], IRA : U × E → [0, 1] and FRA : U × E → [0, 1] whereTRA(u, e) ∈ [0, 1], IRA(u, e) ∈ [0, 1] and FRA(u, e) ∈ [0, 1] are the membership value,indeterminacy value and non membership value respectively of u ∈ U for each e ∈ E.3International Journal of Pure and Applied Mathematics Special Issue7667If [(Tij, Iij, Fij)] = [Tij(ui, ej), Iij(ui, ej), Fij(ui, ej)] we define a matrix[〈Tij , Iij, Fij〉]m×n =〈T11, I11, F11〉 · · · 〈T1n, I1n, F1n〉〈T21, I21, F21〉 · · · 〈T2n, I2n, F2n〉.........〈Tm1, Im1, Fm1〉 · · · 〈Tmn, Imn, Fmn〉Which is called an m× n FNSM of the FNSS (FA, E) over U.FNSMs OF Type-I [18]Definition 2.5. Let A = (〈aTij, aIij, aFij〉), B = 〈(bTij, bIij, bFij〉) ∈ Nm×n. The compo-nent wise addition and component wise multiplication is defined asA⊕B = (sup{aTij, bTij}, sup{aIij, bIij}, inf{aFij, bFij})AB = (inf{aTij, bTij}, inf{aIij, bIij}, sup{aFij, bFij}).Definition 2.6. Let A ∈ Fm×n, B ∈ Nm×p, the composition of A and B is definedasA ◦B =(n∑k=1(aTik ∧ bTkj),n∑k=1(aIik ∧ bIkj),n∏k=1(aFik ∨ bFkj))equivalently we can write the same as=(n∨k=1(aTik ∧ bTkj),n∨k=1(aIik ∧ bIkj),n∧k=1(aFik ∨ bFkj)).The product A ◦ B is defined if and only if the number of columns of A is same asthe number of rows of B. A and B are said to be conformable for multiplication.We shall use AB instead of A ◦B.Where∑(aTik ∧ bTkj) means max-min operation and∏(aFik ∨ bFkj) means min-max operation.3 Minus Ordering of Fuzzy Neutrosophic SoftMatricesDefinition 3.1. If two FNSMs A and X of order m × n and n × m respectivelysatisfies the following A{1} = {X ∈ Nnm| AXA = A},A{2} = {X ∈ Nnm| XAX = X}, A{3} = {X ∈ Nnm| (AX)t = AX} andA{4} = {X ∈ Nnm|(XA)t = XA}, then A{1, 2, 3, 4} is called Moore-Penroseinverse of A which is denoted by A+, t- denote the transpose.4International Journal of Pure and Applied Mathematics Special Issue7668Definition 3.2. Let A ∈ Nmn and X ∈ Nnm satisfying AXA = A, then A has ag-inverse. The g-inverse of A is denoted as A− and A{1} is the set of all g-inverse ofA.Definition 3.3. Let A, B ∈ Nmn. The T-ordering AT≤ B is defined asAT≤ B ⇔ AtA = AtB and AAt = BAt.Definition 3.4. The row space R(A) is the subspace of the set of all FNSMs oforder m×n, generated by the rows of A. Similarly, column space of A is denoted byC(A) and is generated by the columns of A.Definition 3.5. For A ∈ N−mn and B ∈ Nmn, the minus ordering denoted as <¯ isdefined as A<¯B ⇔ A−A = A−B and AA− = BA− for some A− ∈ A{1}.To specify the minus ordering with respect to a particular g-inverse of A let us writeA<¯ B with respect to X ⇔ XA = XB and AX = BX for X ∈ A{1}.Example 3.6. A =( 〈1 1 0〉 〈1 1 0〉〈0.5 0.5 0.5〉 〈0 0 1〉), B =(〈1 1 0〉 〈1 1 0〉〈1 1 0〉 〈0 0 1〉)A− =(〈0 0 1〉 〈0.5 0.5 0.5〉〈1 1 0〉 〈1 1 0〉), where A− is the g-inverse of A.and A−A = A−B and AA− = BA−.Therefore A<¯B.Remark 3.7. For A ∈ N−mn and B ∈ Nmn, if A+ exists, then A+ is unique andA+ = At. We have, AtAT≤ B ⇔ A <¯ B with respect to A+ ⇔ AtA = AtB and AAt = BAt, which isprecisely the Definition 3.3 of T- ordering. Thus “T-ordering” is a special case ofminus ordering. However, the converse A <¯ B ⇒ A T≤ B need not be true. This isillustrated in the following example.Example 3.8. A =(〈0 0 1〉 〈0 0 1〉〈1 1 0〉 〈0 0 1〉), B =(〈1 1 0〉 〈0 0 1〉〈0 0 1〉 〈1 1 0〉).Since At is a g-inverse of A, A+ exists and A+ = At, also A isidempotent, A itself is a g-inverse of A, A = AB = BA ⇒ A <¯ B with respect toA. But AtA 6= AtB and AAt 6= BAt. Hence A <¯ B ; A T≤ B.Lemma 3.9. For A ∈ N−mn and B ∈ Nmn, the following are equivalent:(i). A <¯ B,(ii). A = AA−B = BA−A = BA−B.5International Journal of Pure and Applied Mathematics Special Issue7669Proof:(i)⇒ (ii): A <¯ B ⇒ AA− = BA− and A−A = A−Bfor some A− ∈ A{1}.Now, A = A(A−A) = AA−B A = (AA−)A = BA−A A = B(A−A) = BA−B.(ii)⇒(i):Let X = A−AA− AXA = A(A−AA−)A = (AA−A)A−A = A⇒ X ∈ A{1}.Now XA = (A−AA−)AA−B = A−(AA−A)A−B = (A−AA−)B = XB.Similarly, AX = BX.Hence A <¯ B with respect to X ∈ A{1}.Remark 3.10. In general, in the definition of minus ordering A <¯ B, B need notbe regular. This is illustrated in the following example.Example 3.11. A =(〈1 1 0〉 〈1 1 0〉〈1 1 0〉 〈1 1 0〉), B =(〈1 1 0〉 〈1 1 0〉〈0 0 1〉 〈1 1 0〉)Since A is idempotent, A is a regular and A itself is g-inverse of A. Therefore AXA =A, then A = X. Hence A <¯ B with respect to A = X ∈ A{1}.But B is not regular. Therefore BXB 6= B.Theorem 3.12. Let A, B ∈ N−mn. If A <¯ B, then B{1} ⊆ A{1}.Proof: A <¯ B ⇒ A = AA−B = BA−A. ( by Lemma 3.9).For B− ∈ B{1},AB−A = (AA−B)B−(BA−A)= AA−(BB−B)AA−= (AA−B)A−A = AA−A = A.Hence AB−A = A for each B− ∈ B{1}. Therefore, B{1} ⊆ A{1}.Remark 3.13. For complex matrices, the converse of the above Theorem 3.12 holdand this need not be the case for fuzzy neutrosophic soft matrices. This is illustratedin the following example.Example 3.14. A =( 〈0 0 1〉 〈0 0 1〉〈0.5 0.5 0.5〉 〈0 0 1〉), B =( 〈1 1 0〉 〈1 1 0〉〈0.5 0.5 0.5〉 〈0 0 1〉).The g-inverse of A and B are computed asA{1} = {X|X =(〈aT aI aF 〉 〈bT bI bF 〉〈cT cI cF 〉 〈dT dI dF 〉)}0 ≤ aT , aI ≤ 1, aF ∈ [0.1]6International Journal of Pure and Applied Mathematics Special Issue76700 ≤ bT , bI ≤ 1, bF ∈ [0, 1]0.5 ≤ cT , cI ≤ 1, cF ∈ [0, 0.5]0 ≤ dT , dI ≤ 0.5, dF ∈ [0, 1].We haveB{1} = {X|X =(〈aT aI aF 〉 〈bT bI bF 〉〈cT cI cF 〉 〈dT dI dF 〉)}0 ≤ bT , bI , bF ≤ 10.5 ≤ cT , cI ≤ 1, cF ∈ (0, 0.5)0 ≤ dT , dI , dF ≤ 1.But there is no A− ∈ A{1} satisfying AA− = BA− and A−A = A−B.Hence A¯B.Corollary 3.15. If A <¯ B with B idempotent, then B ∈ A{1}.Proof: Since B is idempotent, B is regular and B itself is a g-inverse of B. HenceB ∈ B{1}, by the Theorem 3.12, B{1} ⊆ A{1}. Hence B ∈ A{1}.Remark 3.16. In the above corollary (3.15), the condition on B to be idempotentis essential for B ∈ A{1}. This is illustrated in the following exampleExample 3.17. Let A =( 〈1 1 0〉 〈1 1 0〉〈0.5 0.5 0.5〉 〈0 0 1〉), and B =(〈1 1 0〉 〈1 1 0〉〈1 1 0〉 〈0 0 1〉)B is not idempotent.A{1} =X =(〈0 0 1〉 〈βt βI βF 〉〈γT γI γF 〉 〈αT αI αF 〉), 0 ≤ βT , βI , βF ≤ 1 and0.5 ≤ γT , γI , γF ≤ 1 and 0 ≤ αT , αI , αF ≤ 1.Here A<¯ B for A− =(〈0 0 1〉 〈1 1 0〉〈1 1 0〉 〈1 1 0〉)but B /∈ A{1}.Theorem 3.18. For A ∈ N−mn and B ∈ Nmn with A <¯ B, we have (i). If B = B2,then A = A2. (ii). If B2 = 0, then A2 = 0.Proof:(i).A2 = A. A = (AA−B)(BA−A) (by Lemma 3.9)= AA−B2A−A= (AA−B)A−A (by B = B2)= AA−A= A.7International Journal of Pure and Applied Mathematics Special Issue7671(ii). A2 = AA = (AA−B)(BA−A) = AA−(B2)A−A = 0.Remark 3.19. In the above Theorem 3.18, if A<¯B with A idempotent then B neednot be idempotent. ConsiderExample 3.20. A =(〈1 1 0〉 〈1 1 0〉〈1 1 0〉 〈1 1 0〉), B =(〈0 0 1〉 〈1 1 0〉〈1 1 0〉 〈0 0 1〉).Here B2 6= B, but A2 = A and A<¯B.Theorem 3.21. For A, B ∈ N+m,n the following conditions are equivalent:i. A<¯B with respect to A+(AT< B).ii. B ∈ A+{1, 3, 4}.iii. B+ ∈ A{1, 3, 4}.Proof(i)⇒(ii) A <¯ B with respect to A+ ⇒ A+A = A+B and AA+ = BA+.Now A+ = A+AA+ = A+BA+ ⇒ B ∈ A+{1}.(A+B)T = (A+A)T = A+A = A+B ⇒ B ∈ A+{3}.(BA+)T = (AA+)T = AA+ = BA+ ⇒ B ∈ A+{4}.(ii)⇒(iii): Since A+ = AT and B+ = BT , we haveB ∈ A+{1, 3, 4} ⇒ B+ ∈ A{1, 3, 4}.(iii)⇒ (i) B+ ∈ A{1, 3, 4} ⇒AB+A = A, (AB+)T= AB+= BA+and (B+A)T = B+A= A+B.A+A = A+A(B+A)= (A+AA+)B= A+B.AA+ = (AB+A)A+ = (AB+)AA+ = BA+AA+ = BA+.Hence A <¯ B with respect to A+.Theorem 3.22. Let A ∈ (N )+mn. ThenH <¯ A+ <¯ G for every G ∈ A{1, 3, 4} and every H ∈ A{2, 3, 4}.Proof.H ∈ A{2, 3, 4} ⇒ H{1, 3, 4} 6= Φ⇒ H+ exists.8International Journal of Pure and Applied Mathematics Special Issue7672Also, HAH = H, (HA)T = HA and (AH)T = AH.Now, HAH = H ⇒ HHTA+ = H⇒ (H+HH+)A+ = H+H⇒ H+A+ = H+H.Similarly, we have A+H+ = HH+.Hence H <¯ A+ with respect to H− = H+.Now G ∈ A{1, 3, 4} ⇒ A+ = GAG⇒ AA+ = (AGA)G = AG.Also A+A = GAGA = G(AGA) = GA.Hence A+ <¯ G with respect to A.4 ConclusionAs minus ordering has close relationship with g-inverse, it has some special rolein relational equation. Recently the hybrid structure neutrosophic soft set is widelyused in all areas such as Engineering, Medical Science, Decision Making etc,. Thusthe study of minus ordering on fuzzy neutrosophic soft matrices has some future.References[1] I. Arockiarani, I. R. Sumathi and Martina Jency, Fuzzy Neutrosophic Soft Topo-logical Spaces, International Journal of Mathematical Archive, 4 (10) (2013),225-238.[2] I.Arockiarani and I.R.sumathi, A Fuzzy Neutrosophic Soft Matrix Approach inDecision Making, Journal of Glabal Research in mathematical Archives, 2(2),(2014), 14-23.[3] K.Atanassov, Intuitionistic Fuzzy Sets, and Fuzzy Sets and System, 20, (1983),87-96.[4] Cen Jianmiao , Fuzzy Matrix Partial Orderings and Generalized Inverses, 105,(1999), 453-458.[5] K.H.Kim and F.W.Roush, Generalized Fuzzy Matrices, Fuzzy Sets and Systems,4(3), (1980), 293-315.[6] Young Bim Im, Eun Pyo Lee, The Determinant of Square Intuitionistic FuzzyMatrix, Far East Journal of Mathematical Science, 3(5), (2001), 789-796.9International Journal of Pure and Applied Mathematics Special Issue7673[7] D.Molodtsov, Soft Set Theory First Resuls, Computer and Mathematics withApplication, 37, (1999), 19-31.[8] P.K.Maji, R.Biswas, and A.R.Roy, Fuzzy Soft sets, The journal of Fuzzy Math-ematics, 9(3), (2001), 589-602.[9] AR.Meenakshi and Inbam, The Minus Partial Order in Fuzzy Matrices, TheJournal of Fuzzy Mathematics, 12(3), (2004), 695-700 .[10] AR.Meenakshi , Fuzzy Matrices Theory and Application, MJP Publishers[11] P.Murugadas , Contribution to a study on Generalized Fuzzy matrices, Ph.DThesis Depertmant of Mathematices.Annamalai university , July-(2011).[12] Mamonuni.Dhar, Said Broumi, Florentin Smarandache, A Note on Square Neu-trosophic Fuzzy Matrices, 13, (2014).[13] S.K. Mitra, The Minus Portial Order and Shorted Matrix, J.Lin. Alg. Appl,,83,(1986), 1-27.[14] P.Rajarajeswari and P.Dhanalakshmi, Intuitionistic Fuzzy Soft Matrix Theoryand it Application in Medical Diagnosis, Annals of Fuzzy Mathematics and In-formatics, 7(5), (2014) 765-772.[15] F.Smarandach, A Uniflying Field in Logics. Neutrosophy: Neutrosophic Proba-bility, Set and logic. Rehoboth: American Research press, (1998).[16] F.Smarandach, Neutrosophic Set a Generalisation of the Intuitionistic Fuzzy Set,International Journal of Pure and Applied Mathematics, (24), (2005) ,287-297.[17] K.Khan Susanta ,Anita pal, The Generalized Inverse of Intuitionistic FuzzyMatrices, Journal of Physical Science, 11, (2007), 62-67.[18] Uma, P.Murugadas and S.Sriram, Fuzzy Neutrosophic Soft Matrices of Type-Iand Type-II, Communicated.[19] Yong Yang and Chenli Ji., Fuzzy Soft Matrices and Their Application, Part I,LNAI, 7002; (2011), 618-627.[20] L.A.Zadeh, Fuzzy Sets, Information and Control, 8, (1965), 338-353.10International Journal of Pure and Applied Mathematics Special Issue767476757676

Minus Ordering on Fuzzy Neutrosophic Soft Matrices

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Minus Ordering on Fuzzy Neutrosophic Soft Matrices

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