Generalized inverse of fuzzy neutrosophic soft matrix

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Generalized inverse of fuzzy neutrosophic soft matrix R.Uma∗, P. Murugadas†and S.Sriram‥ *†Department of Mathematics, Annamalai University, Annamalainagar-608002, India. ‥ Department of Mathematics, Mathematics wing, D.D.E, Annamalai University, Annamalainagar-608002, India.

Abstract Aim of this article is to ďŹ nd the maximum and minimum solution of the fuzzy neutrosophic soft relational equation đ?‘Ľđ??´ = đ?‘? and đ??´đ?‘Ľ = đ?‘?, where đ?‘Ľ and đ?‘? are fuzzy neutrosophic soft vector and đ??´ is a fuzzy neutrosophic soft matrix. Whenever đ??´ is singular we can not ďŹ nd đ??´âˆ’1 . In that case we can use ginverse to get the solution of the above relational equation. Further, using this concept maximum and minimum g-inverse of fuzzy neutrosophic soft matrix are obtained.

AMS Subject classiďŹ cation code: 03E72, 15B15. Keywords: Fuzzy Neutrosophic Soft Set (FNSS), Fuzzy Neutrosophic Soft Matrix (FNSM), g-inverse.

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Introduction

Most of our real life problems in Medical Science, Engineering, Management, Environment and Social Sciences often involve data which are not necessarily crisp, precise and deterministic in character due to various uncertainties associated with these problems. Such uncertainties are usually being handled with the help of the ∗

uma83bala@gmail.com bodi muruga@yahoo.com ‥ ssm 3096@yahoo.co.in â€

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topics like probability, fuzzy sets, interval Mathematics and rough sets etc., Intuitionistic fuzzy sets introduced by Atanassov [3] is appropriate for such a situation. The intuitionistic fuzzy sets can only handle the incomplete information considering both the truth membership and falsity membership. It does not handle the indeterminate and inconsistent information which exists in belief system. Smarandache [13] announced and evinced the concept of neutrosophic set which is a Mathematical tool for handling problems involving imprecise, indeterminacy and inconsistent data. The neutrosophic components T,I, F which represents the membership, indeterminancy, and non-membership values respectively, where ]− 0, 1+ [ is the non-standard unit interval, and thus one deďŹ nes the neutrosophic set. For example the Schrodinger’s cat theory says that basically the quantum state of a photon can basically be in more than one place at the same time, which translated to the neutrosophic set which means an element (quantum state) belongs and does not belong to a set (one place) at the same time; or an element (quantum state) belongs to two dierent sets(two dierent places) in the same time. Diletheism is the view that some statements can be both true and false simultaneously. More precisely, it is belief that there can be true statement whose negation is also true. Such statement are called true contradiction, diletheia or nondualism. â€? All statements are trueâ€? is a false statement. The above example of true contradictions that dialetheists accept. Neutrosophic set, like dialetheism, can describe paradoxist elements, Neutrosophic set (paradoxist element)=(1,1,1), while intuitionistic fuzzy logic can not describe a paradox because the sum of components should be 1 in intuitionistic fuzzy set. In neutrosophic set there is no restriction on T,I,F other than they are subsets of − ] 0, 1+ [, thus − 0 ≤ đ?‘–đ?‘›đ?‘“ đ?‘‡ + đ?‘–đ?‘›đ?‘“ đ??ź + đ?‘–đ?‘›đ?‘“ đ??š ≤ đ?‘ đ?‘˘đ?‘?đ?‘‡ + đ?‘ đ?‘˘đ?‘?đ??ź + đ?‘ đ?‘˘đ?‘?đ??š ≤ 3+ Neutrosophic sets and logic are the foundations for many theories which are more general than their classical counterparts in fuzzy, intuitionistic fuzzy, paraconsistent set, dialetheist set, paradoxist set and tautological set. In 1999, Molodtsov [9] initiated the novel concept of soft set theory which is a completely new approach for modeling vagueness and uncertainty. In [7] Maji et al., initiated the concept of fuzzy soft sets with some properties regarding fuzzy soft union, intersection, complement of fuzzy soft set. Moreover in [8, 11] Maji et al., extended soft sets to intuitionistic fuzzy soft sets and neutrosophic soft sets. One of the important theory of Mathematics which has a vast application in Science and Engineering is the theory of matrices. Let đ??´ be a square matrix of full rank. Then, there exists a matrix đ?‘‹ such that đ??´đ?‘‹ = đ?‘‹đ??´ = đ??ź. This đ?‘‹ is called the 2

inverse of đ??´ and is denoted by đ??´âˆ’1 . Suppose đ??´ is not a matrix of full rank or it is a rectangular matrix, in such a case inverse does not exists. Need felt in numerous areas of applied Mathematics for some kind of partial inverse of a matrix which is singular or even rectangular, such inverse are called generalized inverse. Solving fuzzy matrix equation of the type đ?‘Ľđ??´ = đ?‘? where đ?‘Ľ = (đ?‘Ľ11 , đ?‘Ľ12 , đ?‘Ľ1đ?‘š ), đ?‘? = (đ?‘?11 , đ?‘?12 , đ?‘?1đ?‘› ) and đ??´ is a fuzzy matrix of order đ?‘š Ă— đ?‘› is of great interest in various ďŹ elds. We say đ?‘Ľđ??´ = đ?‘? is comptiable, if there exists a solution for đ?‘Ľđ??´ = đ?‘? and in this case we write max đ?‘šđ?‘–đ?‘›(đ?‘Ľ1đ?‘— , đ?‘Žđ?‘—đ?‘˜ ) = đ?‘?1đ?‘˜ for all đ?‘— ∈ đ??źđ?‘š and đ?‘˜ ∈ đ??źđ?‘› , where đ??źđ?‘› is an index set, đ?‘—

đ?‘– = 1, 2, ..., đ?‘›. Ί1 (đ??´, đ?‘?) represents the set of all solutions of đ?‘Ľđ??´ = đ?‘?. The authors extend this concept into fuzzy neutrosophic soft matrix. The fuzzy neutrosophic soft matrix equation is of the form đ?‘Ľđ??´ = đ?‘?,.....(1) where đ?‘Ľ = â&#x;¨đ?‘Ľđ?‘‡11 , đ?‘Ľđ??ź11 , đ?‘Ľđ??š11 â&#x;Š...â&#x;¨đ?‘Ľđ?‘‡1đ?‘š , đ?‘Ľđ??ź1đ?‘š , đ?‘Ľđ??š1đ?‘š â&#x;Š, đ?‘? = â&#x;¨đ?‘?đ?‘‡11 , đ?‘?đ??ź11 , đ?‘?đ??š11 â&#x;Š...â&#x;¨đ?‘?đ?‘‡1đ?‘› , đ?‘?đ??ź1đ?‘› , đ?‘?đ??š1đ?‘› â&#x;Š and đ??´ is a fuzzy neutrosophic soft matrix of order đ?‘š Ă— đ?‘›. The equation đ?‘Ľđ??´ = đ?‘? is compatible if there exist a solution for đ?‘Ľđ??´ = đ?‘? and in this case we write max đ?‘šđ?‘–đ?‘›â&#x;¨đ?‘Ľđ?‘‡1đ?‘— , đ?‘Ľđ??ź1đ?‘— , đ?‘Ľđ??š1đ?‘— â&#x;Š.â&#x;¨đ?‘Žđ?‘‡đ?‘—đ?‘˜ , đ?‘Žđ??źđ?‘—đ?‘˜ , đ?‘Žđ??šđ?‘—đ?‘˜ â&#x;Š = â&#x;¨đ?‘?đ?‘‡1đ?‘˜ , đ?‘?đ??ź1đ?‘˜ , đ?‘?đ??š1đ?‘˜ â&#x;Š for all đ?‘— ∈ đ??źđ?‘š and đ?‘˜ ∈ đ??źđ?‘› . Denote đ?‘—

Ί1 (đ??´, đ?‘?) = {đ?‘ĽâˆŁđ?‘Ľđ??´ = đ?‘?} represents the set of all solutions of đ?‘Ľđ??´ = đ?‘?. Several authors [4, 6, 12] have studied about the maximum solution đ?‘ĽË† and the minimum solution đ?‘ĽË‡ of đ?‘Ľđ??´ = đ?‘? for fuzzy matrix as well as IFMs. Li Jian-Xin [6] and Katarina Cechlarova [5] discussed the solvability of maxmin fuzzy equation đ?‘Ľđ??´ = đ?‘? and đ??´đ?‘Ľ = đ?‘?. In both the cases the maximum solution is unique and the minimum solution need not be unique. Let Ί2 (đ??´, đ?‘?) be the set of all solutions for đ??´đ?‘Ľ = đ?‘?. Murugadas [10] introduced a method to ďŹ nd maximum ginverse as well as minimum g-inverse of fuzzy matrix and intuitionistic fuzzy matrix. Let us restrict our further discussion in this section to fuzzy neutrosophic soft matrix equation of the form đ??´đ?‘Ľ = đ?‘? with đ?‘Ľ = [â&#x;¨đ?‘Ľđ?‘‡đ?‘–1 , đ?‘Ľđ??źđ?‘–1 , đ?‘Ľđ??šđ?‘–1 â&#x;ŠâˆŁđ?‘– ∈ đ??źđ?‘› ], đ?‘? = [â&#x;¨đ?‘?đ?‘‡đ?‘˜1 , đ?‘?đ??źđ?‘˜1 , đ?‘?đ??šđ?‘˜1 â&#x;Š ∈ đ??źđ?‘š ] where đ??´ ∈ đ??š đ?‘ đ?‘†đ?‘€đ?‘šđ?‘› . In this paper the authors extend the idea of ďŹ nding g-inverse to FNSM. And also ďŹ nds the maximum and minimum solution of the relational equation đ?‘Ľđ??´ = đ?‘? when đ??´ is a FNSM. Further this concept has been extended in ďŹ nding g-inverse of FNSM.

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preliminaries

DeďŹ nition 2.1. [13] A neutrosophic set đ??´ on the universe of discourse đ?‘‹ is deďŹ ned as đ??´ = {â&#x;¨đ?‘Ľ, đ?‘‡đ??´ (đ?‘Ľ), đ??źđ??´ (đ?‘Ľ), đ??šđ??´ (đ?‘Ľ)â&#x;Š, đ?‘Ľ ∈ đ?‘‹}, where đ?‘‡, đ??ź, đ??š : đ?‘‹ → ]− 0, 1+ [ and − 0 ≤ đ?‘‡đ??´ (đ?‘Ľ) + đ??źđ??´ (đ?‘Ľ) + đ??šđ??´ (đ?‘Ľ) ≤ 3+ ......(1). From philosophical point of view the neutrosophic set takes the value from real standard or non-standard subsets of ]− 0, 1+ [ . But in real life application especially in 3

scientiďŹ c and Engineering problems it is diďŹƒcult 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 subset of [0, 1]. Therefore we can rewrite the equation (1) as 0 ≤ đ?‘‡đ??´ (đ?‘Ľ) + đ??źđ??´ (đ?‘Ľ) + đ??šđ??´ (đ?‘Ľ) ≤ 3. In short an element đ?‘Ž Ëœ in the neutrosophic set đ??´, can be written as đ?‘Ž Ëœ = â&#x;¨đ?‘Žđ?‘‡ , đ?‘Žđ??ź , đ?‘Žđ??š â&#x;Š, where đ?‘Žđ?‘‡ denotes degree of truth, đ?‘Žđ??ź denotes degree of indeterminacy, đ?‘Žđ??š denotes degree of falsity such that 0 ≤ đ?‘Žđ?‘‡ + đ?‘Žđ??ź + đ?‘Žđ??š ≤ 3. Example 2.2. Assume that the universe of discourse đ?‘‹ = {đ?‘Ľ1 , đ?‘Ľ2 , đ?‘Ľ3 }, where đ?‘Ľ1 , đ?‘Ľ2 , and đ?‘Ľ3 characterises the quality, relaibility, and the price of the objects. It may be further assumed that the values of {đ?‘Ľ1 , đ?‘Ľ2 , đ?‘Ľ3 } are in [0, 1] and they are obtained from some investigations of some experts. The experts may impose their opinion in three components viz; the degree of goodness, the degree of indeterminacy and the degree of poorness to explain the characteristics of the objects. Suppose đ??´ is a Neutrosophic Set (NS) of đ?‘‹, such that đ??´ = {â&#x;¨đ?‘Ľ1 , 0.4, 0.5, 0.3â&#x;Š, â&#x;¨đ?‘Ľ2 , 0.7, 0.2, 0.4â&#x;Š, â&#x;¨đ?‘Ľ3 , 0.8, 0.3, 0.4â&#x;Š}, where for đ?‘Ľ1 the degree of goodness of quality is 0.4 , degree of indeterminacy of quality is 0.5 and degree of falsity of quality is 0.3 etc,. DeďŹ nition 2.3. [9] Let U be an initial universe set and đ??¸ be a set of parameters. Let P(U) denotes the power set of U. Consider a nonempty set đ??´, đ??´ ⊂ đ??¸. A pair (F,A) is called a soft set over U, where F is a mapping given by đ??š : đ??´ → đ?‘ƒ (đ?‘ˆ ). DeďŹ nition 2.4. [1] Let đ?‘ˆ be an initial universe set and đ??¸ be a set of parameters. Consider a non empty set đ??´, đ??´ ⊂ đ??¸. Let đ?‘ƒ (đ?‘ˆ ) denotes the set of all fuzzy neutrosophic sets of đ?‘ˆ . The collection (đ??š, đ??´) is termed to be the Fuzzy Neutrosophic Soft Set (FNSS) over đ?‘ˆ, where F is a mapping given by đ??š : đ??´ → đ?‘ƒ (đ?‘ˆ ). Hereafter we simply consider đ??´ as FNSS over đ?‘ˆ instead of (đ??š, đ??´). DeďŹ nition 2.5. [2] Let đ?‘ˆ = {đ?‘?1 , đ?‘?2 , ...đ?‘?đ?‘š } be the universal set and đ??¸ be the set of parameters given by đ??¸ = {đ?‘’1 , đ?‘’2 , ...đ?‘’đ?‘› }. Let đ??´ ⊆ đ??¸ . A pair (đ??š, đ??´) be a FNSS over đ?‘ˆ . Then the subset of đ?‘ˆ Ă— đ??¸ is deďŹ ned by đ?‘…đ??´ = {(đ?‘˘, đ?‘’); đ?‘’ ∈ đ??´, đ?‘˘ ∈ đ??šđ??´ (đ?‘’)} which is called a relation form of (đ??šđ??´ , đ??¸). The membership function, indeterminacy membership function and non membership function are written by đ?‘‡đ?‘…đ??´ : đ?‘ˆ Ă— đ??¸ → [0, 1], đ??źđ?‘…đ??´ : đ?‘ˆ Ă— đ??¸ → [0, 1] and đ??šđ?‘…đ??´ : đ?‘ˆ Ă— đ??¸ → [0, 1] where đ?‘‡đ?‘…đ??´ (đ?‘˘, đ?‘’) ∈ [0, 1], đ??źđ?‘…đ??´ (đ?‘˘, đ?‘’) ∈ [0, 1] and đ??šđ?‘…đ??´ (đ?‘˘, đ?‘’) ∈ [0, 1] are the membership value, indeterminacy value and non membership value respectively of đ?‘˘ ∈ đ?‘ˆ for each đ?‘’ ∈ đ??¸. If [(đ?‘‡đ?‘–đ?‘— , đ??źđ?‘–đ?‘— , đ??šđ?‘–đ?‘— )] = [(đ?‘‡đ?‘–đ?‘— (đ?‘˘đ?‘– , đ?‘’đ?‘— ), đ??źđ?‘–đ?‘— (đ?‘˘đ?‘– , đ?‘’đ?‘— ), đ??šđ?‘–đ?‘— (đ?‘˘đ?‘– , đ?‘’đ?‘— )] we deďŹ ne a matrix

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â&#x;¨đ?‘‡11 , đ??ź11 , đ??š11 â&#x;Š â&#x;¨đ?‘‡12 , đ??ź12 , đ??š12 â&#x;Š â‹… â‹… â‹… â&#x;¨đ?‘‡22 , đ??ź22 , đ??š22 â&#x;Š â‹… â‹… â‹… ⎢ â&#x;¨đ?‘‡21 , đ??ź21 , đ??š21 â&#x;Š =⎣ .. .. .. . . . â&#x;¨đ?‘‡đ?‘š1 , đ??źđ?‘š1 , đ??šđ?‘š1 â&#x;Š â&#x;¨đ?‘‡đ?‘š2 , đ??źđ?‘š2 , đ??šđ?‘š2 â&#x;Š â‹… â‹… â‹…

⎤ â&#x;¨đ?‘‡1đ?‘› , đ??ź1đ?‘› , đ??š1đ?‘› â&#x;Š â&#x;¨đ?‘‡2đ?‘› , đ??ź2đ?‘› , đ??š2đ?‘› â&#x;Š ⎼ ⎌

⎥

[â&#x;¨đ?‘‡đ?‘–đ?‘— , đ??źđ?‘–đ?‘— , đ??šđ?‘–đ?‘— â&#x;Š]đ?‘šĂ—đ?‘›

â&#x;¨đ?‘‡đ?‘šđ?‘› , đ??źđ?‘šđ?‘› , đ??šđ?‘šđ?‘› â&#x;Š

which is called an đ?‘š Ă— đ?‘› FNSM of the FNSS (đ??šđ??´ , đ??¸) over đ?‘ˆ. DeďŹ nition 2.6. Let đ?‘ˆ = {đ?‘?1 , đ?‘?2 ...đ?‘?đ?‘š } be the universal set and đ??¸ be the set of parameters given by đ??¸ = {đ?‘’1 , đ?‘’2 , ...đ?‘’đ?‘› }. Let đ??´ ⊆ đ??¸. A pair (đ??š, đ??´) be a fuzzy neutrosophic soft set. Then fuzzy neutrosophic soft set (đ??š, đ??´) in a matrix form as đ??´đ?‘šĂ—đ?‘› = (đ?‘Žđ?‘–đ?‘— )đ?‘šĂ—đ?‘› or đ??´ = { (đ?‘Žđ?‘–đ?‘— ), đ?‘– = 1, 2, ...đ?‘š, đ?‘— = 1, 2, ...đ?‘› where (đ?‘‡ (đ?‘?đ?‘– , đ?‘’đ?‘— ), đ??ź(đ?‘?đ?‘– , đ?‘’đ?‘— ), đ??š (đ?‘?đ?‘– , đ?‘’đ?‘— )) if đ?‘’đ?‘— ∈ đ??´ (đ?‘Žđ?‘–đ?‘— ) = â&#x;¨0, 0, 1â&#x;Š if đ?‘’đ?‘— ∈ /đ??´ where đ?‘‡đ?‘— (đ?‘?đ?‘– ) represent the membership of đ?‘?đ?‘– , đ??źđ?‘— (đ?‘?đ?‘– ) represent the indeterminacy of đ?‘?đ?‘– and đ??šđ?‘— (đ?‘?đ?‘– ) represent the non-membership of đ?‘?đ?‘– in the FNSS (đ??š, đ??´). If we replace the identity element â&#x;¨0, 0, 1â&#x;Š by â&#x;¨0, 1, 1â&#x;Š in the above form we get FNSM of type-II. FNSM of Type-I[14] Let đ?’Šđ?‘šĂ—đ?‘› denotes FNSM of order đ?‘š Ă— đ?‘› and đ?’Šđ?‘› denotes FNSM of order đ?‘› Ă— đ?‘›. âŒŞ âŒŞ ⌊ ⌊ DeďŹ nition 2.7. Let đ??´ = ( đ?‘Žđ?‘‡đ?‘–đ?‘— , đ?‘Žđ??źđ?‘–đ?‘— , đ?‘Žđ??šđ?‘–đ?‘— ), đ??ľ = ( đ?‘?đ?‘‡đ?‘–đ?‘— , đ?‘?đ??źđ?‘–đ?‘— , đ?‘?đ??šđ?‘–đ?‘— ) ∈ đ?’Šđ?‘šĂ—đ?‘› the componentwise addition and { componentwise } { multiplication } {isđ??šdeďŹ ned } as đ?‘‡ đ?‘‡ đ??ź đ??ź đ??š đ??´ ⊕ đ??ľ = (đ?‘ đ?‘˘đ?‘? đ?‘Žđ?‘–đ?‘— , đ?‘?đ?‘–đ?‘— , đ?‘ đ?‘˘đ?‘? đ?‘Žđ?‘–đ?‘— , đ?‘?đ?‘–đ?‘— , đ?‘–đ?‘›đ?‘“ đ?‘Žđ?‘–đ?‘— , đ?‘?đ?‘–đ?‘— ). đ??´ ⊙ đ??ľ = (đ?‘–đ?‘›đ?‘“ {đ?‘Žđ?‘‡đ?‘–đ?‘— , đ?‘?đ?‘‡đ?‘–đ?‘— }, đ?‘–đ?‘›đ?‘“ {đ?‘Žđ??źđ?‘–đ?‘— , đ?‘?đ??źđ?‘–đ?‘— }, đ?‘ đ?‘˘đ?‘?{đ?‘Žđ??šđ?‘–đ?‘— , đ?‘?đ??šđ?‘–đ?‘— }). DeďŹ nition 2.8. Let đ??´ ∈ đ?’Šđ?‘šĂ—đ?‘› , đ??ľ ∈ đ?’Šđ?‘›Ă—đ?‘? , the composition of đ??´ and đ??ľ is deďŹ ned as ( đ?‘› ) đ?‘› đ?‘› ∑ ∑ âˆ? đ?‘‡ đ?‘‡ đ??ź đ??ź đ??š đ??š đ??´âˆ˜đ??ľ = (đ?‘Žđ?‘–đ?‘˜ ∧ đ?‘?đ?‘˜đ?‘— ), (đ?‘Žđ?‘–đ?‘˜ ∧ đ?‘?đ?‘˜đ?‘— ), (đ?‘Žđ?‘–đ?‘˜ ∨ đ?‘?đ?‘˜đ?‘— đ?‘˜=1

( =

đ?‘˜=1

đ?‘˜=1

equivalently we can write the same as ) đ?‘› đ?‘› đ?‘› â‹€ â‹ â‹ (đ?‘Žđ??šđ?‘–đ?‘˜ ∨ đ?‘?đ??šđ?‘˜đ?‘— ) . (đ?‘Žđ??źđ?‘–đ?‘˜ ∧ đ?‘?đ??źđ?‘˜đ?‘— ), (đ?‘Žđ?‘‡đ?‘–đ?‘˜ ∧ đ?‘?đ?‘‡đ?‘˜đ?‘— ), đ?‘˜=1

đ?‘˜=1

đ?‘˜=1

The product đ??´ ∘ đ??ľ is deďŹ ned if and only if the number of columns of đ??´ is same as the number of rows of đ??ľ. đ??´ and đ??ľ are said to be conformable for multiplication. We shall use đ??´đ??ľ instead of đ??´ ∘ đ??ľ. FNSM of Type-II[14]

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DeďŹ nition 2.9. Let đ??´ = (â&#x;¨đ?‘Žđ?‘‡đ?‘–đ?‘— , đ?‘Žđ??źđ?‘–đ?‘— , đ?‘Žđ??šđ?‘–đ?‘— â&#x;Š), đ??ľ = (â&#x;¨đ?‘?đ?‘‡đ?‘–đ?‘— , đ?‘?đ??źđ?‘–đ?‘— , đ?‘?đ??šđ?‘–đ?‘— â&#x;Š) ∈ đ?’Šđ?‘šĂ—đ?‘› , the component wise addition and{ component wise } as } {is deďŹ ned } { multiplication đ??´ ⊕ đ??ľ = (â&#x;¨đ?‘ đ?‘˘đ?‘? đ?‘Žđ?‘‡đ?‘–đ?‘— , đ?‘?đ?‘‡đ?‘–đ?‘— , đ?‘–đ?‘›đ?‘“ đ?‘Žđ??źđ?‘–đ?‘— , đ?‘?đ??źđ?‘–đ?‘— , đ?‘–đ?‘›đ?‘“ đ?‘Žđ??šđ?‘–đ?‘— , đ?‘?đ??šđ?‘–đ?‘— â&#x;Š). đ??´ ⊙ đ??ľ = (â&#x;¨đ?‘–đ?‘›đ?‘“ {đ?‘Žđ?‘‡đ?‘–đ?‘— , đ?‘?đ?‘‡đ?‘–đ?‘— }, đ?‘ đ?‘˘đ?‘?{đ?‘Žđ??źđ?‘–đ?‘— , đ?‘?đ??źđ?‘–đ?‘— }, đ?‘ đ?‘˘đ?‘?{đ?‘Žđ??šđ?‘–đ?‘— , đ?‘?đ??šđ?‘–đ?‘— }â&#x;Š). Analogous to FNSM of type-I, we can deďŹ ne FNSM of type -II in the following way DeďŹ nition 2.10. Let đ??´ = (â&#x;¨đ?‘Žđ?‘‡đ?‘–đ?‘— , đ?‘Žđ??źđ?‘–đ?‘— , đ?‘Žđ??šđ?‘–đ?‘— â&#x;Š) = (đ?‘Žđ?‘–đ?‘— ) ∈ đ?’Šđ?‘šĂ—đ?‘› and đ??ľ = (â&#x;¨đ?‘?đ?‘‡đ?‘–đ?‘— , đ?‘?đ??źđ?‘–đ?‘— , đ?‘?đ??šđ?‘–đ?‘— â&#x;Š) = (đ?‘?đ?‘–đ?‘— ) ∈ â„ąđ?‘›Ă—đ?‘? the product of đ??´ and đ??ľ is deďŹ ned as ) ( đ?‘› đ?‘› đ?‘› ∑⌊ âˆ? âˆ? âŒŞ ⌊ âŒŞ ⌊ âŒŞ đ?‘Žđ??źđ?‘–đ?‘˜ ∨ đ?‘?đ??źđ?‘˜đ?‘— , đ?‘Žđ??šđ?‘–đ?‘˜ ∨ đ?‘?đ??šđ?‘˜đ?‘— đ??´âˆ—đ??ľ = đ?‘Žđ?‘‡đ?‘–đ?‘˜ ∧ đ?‘?đ?‘‡đ?‘˜đ?‘— , đ?‘˜=1

đ?‘˜=1

( =

đ?‘˜=1

equivalently we can write the same as ) đ?‘› đ?‘› đ?‘› â‹ â‹€ â‹€ ⌊ đ?‘‡ âŒŞ ⌊ âŒŞ ⌊ âŒŞ đ?‘Žđ?‘–đ?‘˜ ∧ đ?‘?đ?‘‡đ?‘˜đ?‘— , đ?‘Žđ??źđ?‘–đ?‘˜ ∨ đ?‘?đ??źđ?‘˜đ?‘— , đ?‘Žđ??šđ?‘–đ?‘˜ ∨ đ?‘?đ??šđ?‘˜đ?‘— . đ?‘˜=1

đ?‘˜=1

đ?‘˜=1

the product đ??´ ∗ đ??ľ is deďŹ ned if and only if the number of columns of A is same as the number of rows of B. A and B are said to be conformable for multiplication.

3

Main results

DeďŹ nition 3.1. đ??´ ∈ đ?‘ đ?‘šĂ—đ?‘› is said to be regular if there exists đ?‘‹ ∈ đ?‘ đ?‘›Ă—đ?‘š such that đ??´đ?‘‹đ??´ = đ??´. DeďŹ nition 3.2. If đ??´ and đ?‘‹ are two FNSM of order đ?‘š Ă— đ?‘› satisďŹ es the relation đ??´đ?‘‹đ??´ = đ??´, then đ?‘‹ is called a generalized inverse (g-inverse) of đ??´ which is denoted by đ??´âˆ’ . The g-inverse of an FNSM is not necessarily unique. We denote the set of all g-inverse by đ??´{1}. DeďŹ nition 3.3. Any element đ?‘ĽË† ∈ Ί1 (đ??´, đ?‘?) is called a maximal solution if for all đ?‘Ľ ∈ Ί1 (đ??´, đ?‘?), đ?‘Ľ ≼ đ?‘ĽË† implies đ?‘Ľ = đ?‘ĽË†. That is elements đ?‘Ľ, đ?‘ĽË† are component wise equal. DeďŹ nition 3.4. Any elementˇ đ?‘Ľ ∈ Ί1 (đ??´, đ?‘?) is called a minimal solution if for all đ?‘Ľ ∈ Ί1 (đ??´, đ?‘?), đ?‘Ľ ≤ đ?‘ĽË‡ implies đ?‘Ľ = đ?‘ĽË‡. That is elements đ?‘Ľ, đ?‘ĽË‡ are component wise equal. Lemma 3.5. Let đ?‘Ľđ??´ = đ?‘? as deďŹ ned in eqn (1) . If â&#x;¨max đ?‘Žđ?‘‡đ?‘—đ?‘˜ , max đ?‘Žđ??źđ?‘—đ?‘˜ , min đ?‘Žđ??šđ?‘—đ?‘˜ â&#x;Š < đ?‘—

â&#x;¨đ?‘?đ?‘‡1đ?‘˜ , đ?‘?đ??ź1đ?‘˜ , đ?‘?đ??š1đ?‘˜ â&#x;Š for some đ?‘˜ ∈ đ??źđ?‘› , then Ί1 (đ??´, đ?‘?) = đ?œ™. 6

đ?‘—

đ?‘—

Proof: If ⟨max 𝑎𝑇𝑗𝑘 , max 𝑎𝐼𝑗𝑘 , min 𝑎𝐹𝑗𝑘 ⟩ < ⟨𝑏𝑇1𝑘 , 𝑏𝐼1𝑘 , 𝑏𝐹1𝑘 ⟩ for some k, then 𝑗

𝑗

𝑗

𝑚𝑖𝑛{⟨𝑥𝑇1𝑗 , 𝑥𝐼1𝑗 , 𝑥𝐹1𝑗 ⟩, ⟨𝑎𝑇𝑗𝑘 , 𝑎𝐼𝑗𝑘 , 𝑎𝐹𝑗𝑘 ⟩} ≤ ⟨𝑎𝑇𝑗𝑘 , 𝑎𝐼𝑗𝑘 , 𝑎𝐹𝑗𝑘 ⟩ ≤ ⟨max 𝑎𝑇𝑗𝑘 , max 𝑎𝐼𝑗𝑘 , min 𝑎𝐹𝑗𝑘 ⟩ Hence max 𝑚𝑖𝑛{⟨𝑥𝑇1𝑗 , 𝑥𝐼1𝑗 , 𝑥𝐹1𝑗 ⟩, ⟨𝑎𝑇𝑗𝑘 , 𝑎𝐼𝑗𝑘 , 𝑎𝐹𝑗𝑘 ⟩} < 𝑗

𝑗 𝑗 𝐹 𝐼 𝑇 < ⟨𝑏1𝑘 , 𝑏1𝑘 , 𝑏1𝑘 ⟩ ⟨𝑏𝑇1𝑘 , 𝑏𝐼1𝑘 , 𝑏𝐹1𝑘 ⟩.

𝑗

Therefore, no values of 𝑥 satisfy the equation 𝑥𝐴 = 𝑏. Hence Ω(𝐴, 𝑏) = 𝜙. Theorem 3.6. For the equation 𝑥𝐴 = 𝑏, Ω1 (𝐴, 𝑏) ∕= 𝜑 if and only if 𝑥ˆ = [⟨ˆ 𝑥𝑇1𝑗 , 𝑥ˆ𝐼1𝑗 , 𝑥ˆ𝐹1𝑗 ⟩∣𝑗 ∈ ′ ′′ 𝐼𝑚 ] defined as ⟨ˆ 𝑥𝑇1𝑗 , 𝑥ˆ𝐼1𝑗 , 𝑥ˆ𝐹1𝑗 ⟩ = ⟨𝑚𝑖𝑛𝜎(𝑎𝑇𝑗𝑘 , 𝑏𝑇1𝑘 ), 𝑚𝑖𝑛𝜎 (𝑎𝐼𝑗𝑘 , 𝑏𝐼1𝑘 ), 𝑚𝑎𝑥𝜎 (𝑎𝐹𝑗𝑘 , 𝑏𝐹1𝑘 )⟩, where { 𝑏𝑇1𝑘 if 𝑎𝑇𝑗𝑘 > 𝑏𝑇1𝑘 𝑇 𝑇 𝜎(𝑎𝑗𝑘 , 𝑏1𝑘 ) = 1 otherwise { 𝑏𝐼 if 𝑎𝐼𝑗𝑘 > 𝑏𝐼1𝑘 ′ 𝜎 (𝑎𝐼𝑗𝑘 , 𝑏𝐼𝑗𝑘 ) = 1𝑘 1 otherwise { 𝑏𝐹 if 𝑎𝐹𝑗𝑘 < 𝑏𝐹1𝑘 ′′ 𝜎 (𝑎𝐹𝑗𝑘 , 𝑏𝐹1𝑘 ) = 1𝑘 0 otherwise is the maximum solution of 𝑥𝐴 = 𝑏 Proof: If Ω1 (𝐴, 𝑏) ∕= 𝜙, then 𝑥ˆ is a solution of 𝑥𝐴 = 𝑏. For if 𝑥ˆ is not a solution, then 𝑥ˆ𝐴 ∕= 𝑏 and therefore max 𝑚𝑖𝑛⟨ˆ 𝑥𝑇1𝑗 , 𝑥ˆ𝐼1𝑗 , 𝑥ˆ𝐹1𝑗 ⟩⟨𝑎𝑇𝑗𝑘 , 𝑎𝐼𝑗𝑘 , 𝑎𝐹𝑗𝑘 ⟩ ∕= ⟨𝑏𝑇1𝑘0 , 𝑏𝐼1𝑘0 , 𝑏𝐹1𝑘0 ⟩ for atleast one 𝑘0 ∈ 𝐼𝑛 . By the 𝑗

𝑥𝑇1𝑗 , 𝑥ˆ𝐼1𝑗 , 𝑥ˆ𝐹1𝑗 ⟩ ≤ ⟨𝑏𝑇1𝑘 , 𝑏𝐼1𝑘 , 𝑏𝐹1𝑘 ⟩ for each 𝑘 and so Definition of ⟨ˆ 𝑥𝑇1𝑗 , 𝑥ˆ𝐼1𝑗 , 𝑥ˆ𝐹1𝑗 ⟩, ⟨ˆ ⟨ˆ 𝑥𝑇1𝑗 , 𝑥ˆ𝐼1𝑗 , 𝑥ˆ𝐹1𝑗 ⟩ ≤ ⟨𝑏𝑇1𝑘0 , 𝑏𝐼1𝑘0 , 𝑏𝐹1𝑘0 ⟩. Therefore, ⟨ˆ 𝑥𝑇1𝑗 , 𝑥ˆ𝐼1𝑗 , 𝑥ˆ𝐹1𝑗 ⟩⟨𝑎𝑇𝑗𝑘 , 𝑎𝐼𝑗𝑘 , 𝑎𝐹𝑗𝑘 ⟩ < ⟨𝑏𝑇1𝑘0 , 𝑏𝐼1𝑘0 , 𝑏𝐹1𝑘0 ⟩ ⟨max 𝑎𝑇𝑗𝑘 , max 𝑎𝐼𝑗𝑘 , min 𝑎𝐹𝑗𝑘 ⟩ < ⟨𝑏𝑇1𝑘0 , 𝑏𝐼1𝑘0 , 𝑏𝐹1𝑘0 ⟩ for some 𝑘0 by our assumption. 𝑗

𝑗

𝑗

Hence by Lemma 3.5 Ω1 (𝐴, 𝑏) = 𝜙. which is a contradiction. Hence 𝑥ˆ is a solution. Let us prove that 𝑥ˆ is the maximum one. If possible let us assume that 𝑦ˆ is another solution such that 𝑦ˆ ≥ 𝑥ˆ that is 𝑇 𝐼 𝐹 ⟨𝑦1𝑗 , 𝑦1𝑗 , 𝑦1𝑗 ⟩ > ⟨ˆ 𝑥𝑇1𝑗0 , 𝑥ˆ𝐼1𝑗0 , 𝑥ˆ𝐹1𝑗0 ⟩ for atleast one 𝑗0 . 0 0 0 Therefore, by the definition of ⟨ˆ 𝑥𝑇1𝑗0 , 𝑥ˆ𝐼1𝑗0 , 𝑥ˆ𝐹1𝑗0 ⟩, ′ ′′ 𝑇 𝐼 𝐹 ⟨𝑦1𝑗 , 𝑦1𝑗 , 𝑦1𝑗 ⟩ > ⟨𝑚𝑖𝑛𝜎(𝑎𝑇𝑗0 𝑘 , 𝑏𝑇1𝑘 ), 𝑚𝑖𝑛𝜎 (𝑎𝐼𝑗0 𝑘 , 𝑏𝐼1𝑘 ), 𝑚𝑎𝑥𝜎 (𝑎𝐹𝑗0 𝑘 , 𝑏𝐹1𝑘 )⟩ 0 0 0 Since Ω(𝐴, 𝑏) ∕= 𝜙, by the Lemma 3.5 ⟨max 𝑎𝑇𝑗𝑘 , max 𝑎𝐼𝑗𝑘 , min 𝑎𝐹𝑗𝑘 ⟩ ≥ ⟨𝑏𝑇1𝑘0 , 𝑏𝐼1𝑘0 , 𝑏𝐹1𝑘0 ⟩ for each 𝑘0 . 𝑗

Hence

⟨𝑏𝑇1𝑘0 , 𝑏𝐼1𝑘0 , 𝑏𝐹1𝑘0 ⟩

= ∕

𝑗

𝑗 𝑇 ⟨max 𝑚𝑖𝑛(𝑦𝑗 , 𝑎𝑇𝑗𝑘0 ), max 𝑚𝑖𝑛(𝑦𝑗𝐼 , 𝑎𝐼𝑗𝑘0 ), min 𝑚𝑎𝑥(𝑦𝑗𝐹 , 𝑎𝐹𝑗𝑘0 )⟩ 𝑗 𝑗 𝑗

which is a contradiction to our assumption that 𝑦 ∈ Ω1 (𝐴, 𝑏). 7

Therefore đ?‘ĽË† is the maximum solution. The converse part is trivial. If the relational equation is the form đ??´đ?‘Ľ = đ?‘?.....(2) where đ??´ is an fuzzy neutrosophic soft matrix of order đ?‘š Ă— đ?‘›, đ?‘Ľ = (â&#x;¨đ?‘Ľđ?‘‡11 , đ?‘Ľđ??ź11 , đ?‘Ľđ??š11 â&#x;Š, ..., â&#x;¨đ?‘Ľđ?‘‡1đ?‘› , đ?‘Ľđ??ź1đ?‘› , đ?‘Ľđ??š1đ?‘› â&#x;Š)đ?‘‡ , đ?‘? = (â&#x;¨đ?‘?đ?‘‡11 , đ?‘?đ??ź11 , đ?‘?đ??š11 â&#x;Š, ..., â&#x;¨đ?‘?đ?‘‡1đ?‘š , đ?‘?đ??ź1đ?‘š , đ?‘?đ??š1đ?‘š â&#x;Š)đ?‘‡ we can prove the following Lemma and Theorem in similar fashion. Let Ί2 (đ??´, đ?‘?) be the set all solution of the relational equation đ??´đ?‘Ľ = đ?‘?. DeďŹ nition 3.7. Any element đ?‘ĽË† ∈ Ί2 (đ??´, đ?‘?) is called a maximal solution if for all đ?‘Ľ ∈ Ί2 (đ??´, đ?‘?), đ?‘Ľ ≼ đ?‘ĽË† implies đ?‘Ľ = đ?‘ĽË†. That is elements đ?‘Ľ, đ?‘ĽË† are component wise equal. DeďŹ nition 3.8. Any element đ?‘ĽË‡ ∈ Ί2 (đ??´, đ?‘?) is called a minimal solution if for all đ?‘Ľ ∈ Ί2 (đ??´, đ?‘?), đ?‘Ľ ≤ đ?‘ĽË‡ implies đ?‘Ľ = đ?‘ĽË‡. That is elements đ?‘Ľ, đ?‘ĽË‡ are component wise equal. Lemma 3.9. Let đ??´đ?‘Ľ = đ?‘? as deďŹ ned in (2). If â&#x;¨max đ?‘Žđ?‘‡đ?‘˜đ?‘– , max đ?‘Žđ??źđ?‘˜đ?‘– , min đ?‘Žđ??šđ?‘˜đ?‘– â&#x;Š < â&#x;¨đ?‘?đ?‘‡đ?‘˜1 , đ?‘?đ??źđ?‘˜1 , đ?‘?đ??šđ?‘˜1 â&#x;Š for some đ?‘˜ ∈ đ??źđ?‘š , then Ί2 (đ??´, đ?‘?) = đ?œ™. đ?‘–

đ?‘–

đ?‘–

Theorem 3.10. For the equation đ??´đ?‘Ľ = đ?‘?, Ί2 (đ??´, đ?‘?) ∕= đ?œ™ if only if đ?‘ĽË† = [â&#x;¨Ë† đ?‘Ľđ?‘‡đ?‘—1 , đ?‘ĽË†đ??źđ?‘—1 , đ?‘ĽË†đ??šđ?‘—1 â&#x;ŠâˆŁđ?‘— ∈ đ??źđ?‘› ]deďŹ ned as ′ ′′ â&#x;¨Ë† đ?‘Ľđ?‘‡đ?‘—1 , đ?‘ĽË†đ??źđ?‘—1 , đ?‘ĽË†đ??šđ?‘—1 â&#x;Š = â&#x;¨đ?‘šđ?‘–đ?‘›đ?œŽ(đ?‘Žđ?‘‡đ?‘˜đ?‘– , đ?‘?đ?‘‡đ?‘˜1 ), đ?‘šđ?‘–đ?‘›đ?œŽ (đ?‘Žđ??źđ?‘˜đ?‘– , đ?‘?đ??źđ?‘˜1 ), đ?‘šđ?‘Žđ?‘Ľđ?œŽ (đ?‘Žđ??šđ?‘˜đ?‘– , đ?‘?đ??šđ?‘˜1 )â&#x;Š, where { đ?‘?đ?‘‡ if đ?‘Žđ?‘‡đ?‘˜đ?‘– > đ?‘?đ?‘‡đ?‘˜1 đ?œŽ(đ?‘Žđ?‘‡đ?‘˜đ?‘– , đ?‘?đ?‘‡đ?‘˜1 ) = đ?‘˜1 1 otherwise { đ?‘?đ??ź if đ?‘Žđ??źđ?‘˜đ?‘– > đ?‘?đ??źđ?‘˜1 ′ đ?œŽ (đ?‘Žđ??źđ?‘˜đ?‘– , đ?‘?đ??źđ?‘˜1 ) = đ?‘˜1 1 otherwise { đ?‘?đ??š if đ?‘Žđ??šđ?‘˜đ?‘– < đ?‘?đ??šđ?‘˜1 ′′ đ?œŽ (đ?‘Žđ??šđ?‘˜đ?‘– , đ?‘?đ??šđ?‘˜1 ) = 1đ?‘˜ 0 otherwise is the maximum solution of đ??´đ?‘Ľ = đ?‘?. [ ] â&#x;¨0.5 0.6 0.2â&#x;Š â&#x;¨0.7, 0.5, 0.1â&#x;Š Example 3.11. Let đ??´ = and đ?‘? = [â&#x;¨0.2, 0.3, 0.5â&#x;Š â&#x;¨0.5, 0.3, 0.1â&#x;Š] â&#x;¨0.2 0.3 0.5â&#x;Š â&#x;¨0.6, 0.4, 0â&#x;Š then we can ďŹ nd đ?‘ĽË† = [â&#x;¨Ë† đ?‘Ľđ?‘‡11 , đ?‘ĽË†đ??ź11 , đ?‘ĽË†đ??š11 â&#x;Š, â&#x;¨Ë† đ?‘Ľđ?‘‡12 , đ?‘ĽË†đ??ź12 , đ?‘ĽË†đ??š12 â&#x;Š] in đ?‘Ľđ??´ = đ?‘?

8

′

′′

â&#x;¨Ë† đ?‘Ľđ?‘‡11 , đ?‘ĽË†đ??ź11 , đ?‘ĽË†đ??š11 â&#x;Š = â&#x;¨min đ?œŽ(đ?‘Žđ?‘‡1đ?‘˜ , đ?‘?đ?‘‡1đ?‘˜ ), min đ?œŽ (đ?‘Žđ??ź1đ?‘˜ , đ?‘?đ??ź1đ?‘˜ ), max đ?œŽ (đ?‘Žđ??š1đ?‘˜ , đ?‘?đ??š1đ?‘˜ )â&#x;Š đ?‘˜

đ?‘˜

đ?‘˜

= â&#x;¨min(0.2, 0.5), min(0.3, 0.3), max(0.5, 0)â&#x;Š đ?‘˜

đ?‘˜

đ?‘˜

= â&#x;¨0.2, 0.3, 0.5â&#x;Š â&#x;¨Ë† đ?‘Ľđ?‘‡12 , đ?‘ĽË†đ??ź12 , đ?‘ĽË†đ??š12 â&#x;Š

′′

′

= â&#x;¨min đ?œŽ(đ?‘Žđ?‘‡2đ?‘˜ , đ?‘?đ?‘‡1đ?‘˜ ), min đ?œŽ (đ?‘Žđ??ź2đ?‘˜ , đ?‘?đ??ź1đ?‘˜ ), max đ?œŽ (đ?‘Žđ??š2đ?‘˜ , đ?‘?đ??š1đ?‘˜ )â&#x;Š đ?‘˜

đ?‘˜

đ?‘˜

= â&#x;¨min(1, 0.5), min(1, 0.3), max(0, 0.1)â&#x;Š đ?‘˜

đ?‘˜

đ?‘˜

= â&#x;¨0.5, 0.3, 0.1â&#x;Š Then clearly [

] â&#x;¨0.5 0.6 0.2â&#x;Šâ&#x;¨0.7, 0.5, 0.1â&#x;Š (â&#x;¨0.2, 0.3, 0.5â&#x;Š â&#x;¨0.5, 0.3, 0.1â&#x;Š) = (â&#x;¨0.2, 0.3, 0.5â&#x;Š â&#x;¨0.5, 0.3, 0.1â&#x;Š) â&#x;¨0.2 0.3 0.5â&#x;Šâ&#x;¨0.6, 0.4, 0â&#x;Š To get the minimal solution đ?‘ĽË‡ of đ?‘Ľđ??´ = đ?‘? we follow the procedure as followed for fuzzy neutrosophic soft matrix equation. Step.1 Determine the sets đ??˝đ?‘˜ (ˆ đ?‘Ľ) = {đ?‘— ∈ đ??źđ?‘š âˆŁđ?‘šđ?‘–đ?‘›(â&#x;¨Ë† đ?‘Ľđ?‘‡1đ?‘— , đ?‘ĽË†đ??ź1đ?‘— , đ?‘ĽË†đ??š1đ?‘— â&#x;Š, â&#x;¨đ?‘Žđ?‘‡đ?‘—đ?‘˜ , đ?‘Žđ??źđ?‘—đ?‘˜ , đ?‘Žđ??šđ?‘—đ?‘˜ â&#x;Š) = đ?‘?đ?‘‡1đ?‘˜ } for all đ?‘˜ ∈ đ??źđ?‘› . Construct their cartesian product đ??˝(ˆ đ?‘Ľ) = đ??˝1 (ˆ đ?‘Ľ) Ă— đ??˝2 (ˆ đ?‘Ľ) Ă— ... Ă— đ??˝đ?‘› (ˆ đ?‘Ľ). Step.2 Denote the elements of đ??˝(ˆ đ?‘Ľ), by đ?›˝ = [đ?›˝đ?‘˜ /đ?‘˜ ∈ đ??źđ?‘› ]. For each đ?›˝ ∈ đ??˝(ˆ đ?‘Ľ) and each đ?‘— ∈ đ??źđ?‘š , determine the set đ?‘˜(đ?›˝, đ?‘—) = {đ?‘˜ ∈ đ??źđ?‘š âˆŁđ?›˝đ?‘˜ = đ?‘—}. Step.3 For each đ?›˝ ∈ đ??˝(ˆ đ?‘Ľ) generate the n-tuple đ?‘”(đ?›˝) = đ?‘”đ?‘— (đ?›˝)âˆŁđ?‘— ∈ đ??źđ?‘š }, where ⎧ ⎨ max â&#x;¨đ?‘?đ?‘‡1đ?‘˜ , đ?‘?đ??ź1đ?‘˜ , đ?‘?đ??š1đ?‘˜ â&#x;Š if đ?‘˜(đ?›˝, đ?‘—) ∕= 0 đ?‘”đ?‘— (đ?›˝) = đ?‘˜(đ?›˝,đ?‘—) ⎊â&#x;¨0, 0, 1â&#x;Š otherwise Step.4 From all the m-tuples đ?‘”(đ?›˝) generated in step.3, select only the minimal one by pairwise comparison. The resulting set of n-tuples is the minimal solution of the reduced form of equation đ?‘Ľđ??´ = đ?‘?. Example 3.12. Let us ďŹ nd the minimal solution to the linear equation given in Example 3.11 using the maximal solution đ?‘ĽË† Step 1. To determine đ??˝đ?‘˜ (ˆ đ?‘Ľ) for đ?‘˜ = 1, 2.

9

đ??˝1 (ˆ đ?‘Ľ) = {đ?‘— = 1, 2âˆŁđ?‘šđ?‘–đ?‘›(â&#x;¨đ?‘Ľđ?‘‡1đ?‘— , đ?‘Ľđ??ź1đ?‘— , đ?‘Ľđ??š1đ?‘— â&#x;Š, â&#x;¨đ?‘Žđ?‘‡đ?‘—đ?‘˜ , đ?‘Žđ??źđ?‘—đ?‘˜ , đ?‘Žđ??šđ?‘—đ?‘˜ â&#x;Š) = â&#x;¨đ?‘?đ?‘‡1đ?‘˜ , đ?‘?đ??ź1đ?‘˜ , đ?‘?đ??š1đ?‘˜ â&#x;Š} = {đ?‘šđ?‘–đ?‘›{â&#x;¨0.2, 0.3, 0.5â&#x;Šâ&#x;¨0.5, 0.6, 0.2â&#x;Š}, đ?‘šđ?‘–đ?‘›{â&#x;¨0.5, 0.3, 0.1â&#x;Šâ&#x;¨0.2, 0.3, 0.5â&#x;Š}} = â&#x;¨0.2, 0.3, 0.5â&#x;Š = {1, 2} đ??˝2 (ˆ đ?‘Ľ) = {đ?‘šđ?‘–đ?‘›{â&#x;¨0.2, 0.3, 0.5â&#x;Šâ&#x;¨0.7, 0.5, 0.1â&#x;Š}, đ?‘šđ?‘–đ?‘›{â&#x;¨0.5, 0.3, 0.1â&#x;Šâ&#x;¨0.6, 0.4, 0â&#x;Š}} = â&#x;¨0.5, 0.3, 0.1â&#x;Š = {2} Therefore đ??˝đ?‘˜ (ˆ đ?‘Ľ) = đ??˝1 (ˆ đ?‘Ľ) Ă— đ??˝2 (ˆ đ?‘Ľ) = {1, 2} Ă— {2} = {(1, 2, (2, 2))} = đ?›˝ Step 2: To determine the sets đ??ž(đ?›˝, đ?‘—) for each đ?›˝ = đ??˝đ?‘˜ (ˆ đ?‘Ľ) and for each j=1,2. For đ?›˝ = (1, 2) đ??ž(đ?›˝, 1) = {đ?‘˜ = 1, 2âˆŁđ?›˝đ?‘˜ = 1} = {1} đ??ž(đ?›˝, 2) = {đ?‘˜ = 1, 2âˆŁđ?›˝đ?‘˜ = 2} = {2} For đ?›˝ = (2, 2) đ??ž(đ?›˝, 1) = {đ?‘˜ = 1, 2âˆŁđ?›˝đ?‘˜ = 1} = {đ?œ‘} đ??ž(đ?›˝, 2) = {đ?‘˜ = 1, 2âˆŁđ?›˝đ?‘˜ = 2} = {1, 2} Thus the sets đ??ž(đ?›˝, đ?‘—) for each đ?›˝ ∈ đ??˝(ˆ đ?‘Ľ) and đ?‘— = 1, 2 are listed in the following table. đ??ž{đ?›˝, đ?‘—} 1 đ??ž (1, 2) {1} {2} (2, 2) {đ?œ™} {1, 2} Step 3. For each đ?›˝ ∈ đ??˝(ˆ đ?‘Ľ) we generate the tuples đ?‘”(đ?›˝) For đ?›˝ = ⎧ (1, 2) ⎨ max â&#x;¨0.2, 0.3, 0.5â&#x;Š if đ?‘˜(đ?›˝, 1) ∕= đ?œ™ đ?‘”1 (đ?›˝) = đ?‘˜âˆˆđ?‘˜(đ?›˝,1) ⎊â&#x;¨0, 0, 1â&#x;Š otherwise = â&#x;¨0.2, 0.3, 0.5â&#x;Š đ?‘”2 (đ?›˝) = â&#x;¨0.5, 0.3, 0.1â&#x;Š For đ?›˝ = (2, 2) đ?‘”1 (đ?›˝) = â&#x;¨0, 0, 1â&#x;Š đ?‘”2 (đ?›˝) = â&#x;¨0.5, 0.3, 0.1â&#x;Š Therefore we can get the following table for đ?›˝ đ?›˝ đ?‘”(đ?›˝) (1, 2) â&#x;¨0.2, 0.3, 0.5â&#x;Š, â&#x;¨0.5, 0.3, 0.1â&#x;Š (2, 2) â&#x;¨0, 0, 1â&#x;Š, â&#x;¨0.5, 0.3, 0.1â&#x;Š Out of which (â&#x;¨0, 0, 1â&#x;Š, â&#x;¨0.5, 0.3, 0.1â&#x;Š) is the minimal one. And also it satisfy đ?‘Ľđ??´ = đ?‘? that is đ?‘ĽË† = (â&#x;¨0, 0, 1â&#x;Š, â&#x;¨0.5, 0.3, 0.1â&#x;Š) 10

Using the same method we have followed, one can find the g-inverse of a fuzzy neutrosophic soft matrix if it exits. [ ] ⟨1, 1, 0⟩ ⟨1, 1, 0⟩ Example 3.13. Let 𝐴 = . To find the g-inverse, set 𝐴𝑋𝐴 = 𝐴 ⟨1, 1, 0⟩ ⟨0, 0, 1⟩ and 𝐴𝑋 = 𝐴, where [ 𝑇= 𝐵𝐼 so 𝐹that 𝐵𝐴 ] [ 𝑇 𝐼 𝐹 ] 𝑇 ⟨𝑥11 , 𝑥11 , 𝑥11 ⟩ ⟨𝑥12 , 𝑥𝐼12 , 𝑥𝐹12 ⟩ ⟨𝑏11 , 𝑏11 , 𝑏11 ⟩ ⟨𝑏𝑇12 , 𝑏𝐼12 , 𝑏𝐹12 ⟩ 𝑋= and 𝐵 = . ⟨𝑥𝑇21 , 𝑥𝐼21 , 𝑥𝐹21 ⟩ ⟨𝑥𝑇22 , 𝑥𝐼22 , 𝑥𝐹22 ⟩ ⟨𝑏𝑇21 , 𝑏𝐼21 , 𝑏𝐹21 ⟩ ⟨𝑏𝑇22 , 𝑏𝐼22 , 𝑏𝐹22 ⟩ To find 𝐵 and 𝑋: [ ] ⟨1, 1, 0⟩ ⟨1, 1, 0⟩ 𝐹 𝐼 𝑇 𝐹 𝐼 𝑇 = (⟨1, 1, 0⟩, ⟨1, 1, 0⟩) (⟨𝑏11 , 𝑏11 , 𝑏11 ⟩, ⟨𝑏12 , 𝑏12 , 𝑏12 ⟩) ⟨1, 1, 0⟩ ⟨0, 0, 1⟩ ′ ′ ′ ′′ ′′ ′′ ⟨𝑏𝑇11 , 𝑏𝐼11 , 𝑏𝐹11 ⟩ = ⟨min 𝜎(𝑎1𝑘 , 𝑏1𝑘 ), min 𝜎 (𝑎1𝑘 , 𝑏1𝑘 ), max 𝜎 (𝑎1𝑘 , 𝑏1𝑘 )⟩ = ⟨1, 1, 0⟩ 𝑘

𝑘

′

′

𝑘

′

′′

′′

′′

⟨𝑏𝑇12 , 𝑏𝐼12 , 𝑏𝐹12 ⟩ = ⟨min 𝜎(𝑎2𝑘 , 𝑏2𝑘 ), min 𝜎 (𝑎2𝑘 , 𝑏2𝑘 ), max 𝜎 (𝑎2𝑘 , 𝑏2𝑘 )⟩ = ⟨1, 1, 0⟩ 𝑘 𝑘 𝑘 [ ] ⟨1, 1, 0⟩ ⟨1, 1, 0⟩ 𝐹 𝐼 𝑇 𝐹 𝐼 𝑇 = (⟨1, 1, 0⟩, ⟨0, 0, 1⟩) Take (⟨𝑏21 , 𝑏21 , 𝑏21 ⟩, ⟨𝑏22 , 𝑏22 , 𝑏22 ⟩) ⟨1, 1, 0⟩ ⟨0, 0, 1⟩ ′ ′ ′′ ′′ ⟨𝑏𝑇21 , 𝑏𝐼21 , 𝑏𝐹21 ⟩ = ⟨min 𝜎1𝑘 , 𝑏2𝑘 ), min(𝜎1𝑘 , 𝑏2𝑘 ), max(𝜎1𝑘 , 𝑏2𝑘 )⟩ = ⟨0, 0, 1⟩ 𝑘

𝑘

′

𝑘

′

′′

′′

⟨𝑏𝑇22 , 𝑏𝐼22 , 𝑏𝐹22 ⟩ = ⟨min(𝜎2𝑘 , 𝑏2𝑘 ), min(𝜎2𝑘 , 𝑏2𝑘 ), max(𝜎2𝑘 , 𝑏2𝑘 )⟩ = ⟨1, 1, 0⟩ 𝑘 ] 𝑘 [𝑘 ⟨1, 1, 0⟩ ⟨1, 1, 0⟩ ˆ= Therefore 𝐵 which satisfy 𝐵𝐴 = 𝐴 ⟨0, 0, 1⟩ ⟨1, 1, 0⟩ The 𝐴𝑋 = 𝐵 becomes ] [ ] [ ][ ⟨1, 1, 0⟩ ⟨1, 1, 0⟩ ⟨1, 1, 0⟩ ⟨1, 1, 0⟩ ⟨𝑥𝑇11 , 𝑥𝐼11 , 𝑥𝐹11 ⟩ ⟨𝑥𝑇12 , 𝑥𝐼12 , 𝑥𝐹12 ⟩ = ⟨0, 0, 1⟩ ⟨1, 1, 0⟩ ⟨0, 0, 1⟩ ⟨1, 1, 0⟩ ⟨𝑥𝑇21 , 𝑥𝐼21 , 𝑥𝐹21 ⟩ ⟨𝑥𝑇22 , 𝑥𝐼22 , 𝑥𝐹22 ⟩ ′ ′ ′′ ′′ 𝐹 𝐼 𝑇 ⟨𝑥11 , 𝑥11 , 𝑥11 ⟩ = ⟨min(𝜎𝑘1 , 𝑏𝑘1 ), min(𝜎𝑘1 , 𝑏𝑘1 ), max(𝜎𝑘1 , 𝑏𝑘1 )⟩ = ⟨0, 0, 1⟩ 𝑘

𝑘

′

′

′

′

′

′

𝑘

′′

′′

′′

′′

′′

′′

⟨𝑥𝑇12 , 𝑥𝐼12 , 𝑥𝐹12 ⟩ = ⟨min(𝜎𝑘1 , 𝑏𝑘2 ), min(𝜎𝑘1 , 𝑏𝑘2 ), max(𝜎𝑘1 , 𝑏𝑘2 )⟩ = ⟨1, 1, 0⟩ 𝑘

𝑘

𝑘

⟨𝑥𝑇21 , 𝑥𝐼21 , 𝑥𝐹21 ⟩ = ⟨min(𝜎𝑘2 , 𝑏𝑘1 ), min(𝜎𝑘2 , 𝑏𝑘1 ), max(𝜎𝑘2 , 𝑏𝑘1 )⟩ = ⟨1, 1, 0⟩ 𝑘

𝑘

𝑘

⟨𝑥𝑇22 , 𝑥𝐼22 , 𝑥𝐹22 ⟩ = ⟨min(𝜎𝑘2 , 𝑏𝑘2 ), min(𝜎𝑘2 , 𝑏𝑘2 ), max(𝜎𝑘2 , 𝑏𝑘2 )⟩ = ⟨1, 1, 0⟩ 𝑘] 𝑘 [ 𝑘 ⟨0, 0, 1⟩ ⟨1, 1, 0⟩ ˆ= ˆ =𝐴 Therefore 𝑋 . Clearly 𝐴𝑋𝐴 ⟨1, 1, 0⟩ ⟨1, 1, 0⟩ ˆ is the maximum g-inverse of 𝐴𝑋𝐴 ˆ = 𝐴. Hence 𝑋 ˆ from 𝐵𝐴 ˆ = 𝐴 and using To get the minimal solution: Let us find the minimum 𝐵 ˆ ˆ the minimum we can find the minimum [ 𝐵 in 𝐴𝑋 = 𝐵] [ ] [ 𝑋 ] ⟨1, 1, 0⟩ ⟨1, 1, 0⟩ ⟨1, 1, 0⟩ ⟨1, 1, 0⟩ ⟨1, 1, 0⟩ ⟨1, 1, 0⟩ Consider = . ⟨0, 0, 1⟩ ⟨1, 1, 0⟩ ⟨1, 1, 0⟩ ⟨0, 0, 1⟩ ⟨1, 1, 0⟩ ⟨0, 0, 1⟩ ˆ Step 1. Determine the set 𝐽𝑖𝑗 (𝐵) 11

ˆ = {𝑚𝑖𝑛{⟨1, 1, 0⟩, ⟨1, 1, 0⟩}, 𝑚𝑖𝑛{⟨1, 1, 0⟩, ⟨1, 1, 0⟩}} = ⟨1, 1, 0⟩ 𝐽11 (𝐵) = {⟨1, 1, 0⟩, ⟨1, 1, 0⟩} = {1, 2} ˆ = {𝑚𝑖𝑛{⟨1, 1, 0⟩, ⟨1, 1, 0⟩}, 𝑚𝑖𝑛{⟨1, 1, 0⟩, ⟨0, 0, 1⟩}} = ⟨1, 1, 0⟩ 𝐽12 (𝐵) = {⟨1, 1, 0⟩, ⟨0, 0, 1⟩} = {1} ˆ = {𝑚𝑖𝑛{⟨0, 0, 1⟩, ⟨1, 1, 0⟩}, 𝑚𝑖𝑛{⟨1, 1, 0⟩, ⟨1, 1, 0⟩}} = ⟨1, 1, 0⟩ 𝐽21 (𝐵) = {⟨0, 0, 1⟩, ⟨1, 1, 0⟩} = {2} ˆ = {𝑚𝑖𝑛{⟨0, 0, 1⟩, ⟨1, 1, 0⟩}, 𝑚𝑖𝑛{⟨1, 1, 0⟩, ⟨0, 0, 1⟩}} = ⟨0, 0, 1⟩ 𝐽22 (𝐵) = {⟨0, 0, 1⟩, ⟨0, 0, 1⟩} = {1, 2} ˆ × 𝐽12 (𝐵) ˆ = {1, 2} × {1} = {(1, 2), (2, 1)} Let 𝛽1 = 𝐽11 (𝐵) ˆ × 𝐽22 (𝐵) ˆ = {2} × {1, 2} = {(2, 1), (2, 2)} 𝛽2 = 𝐽21 (𝐵) Step 2. Determine the set 𝐾(𝛽𝑘 , 𝑗) for 𝑘 = 1, 2 and 𝑗 = 1, 2 For 𝛽1 = (1, 1)𝐾(𝛽1 , 1) = {1, 2} 𝐾(𝛽1 , 2) = {𝜙} For 𝛽1 = (2, 1)𝐾(𝛽1 , 1) = {2} 𝐾(𝛽1 , 2) = {1} For 𝛽2 = (2, 1)𝐾(𝛽2 , 1) = {2} 𝐾(𝛽2 , 2) = {1} For 𝛽2 = (2, 2)𝐾(𝛽2 , 1) = {𝜙} 𝐾(𝛽2 , 2) = {1, 2} Writing the values in tabular form we get (𝛽1 , 𝑗) (1, 1) (2, 1)

1 2 {1, 2} 𝜙 {2} 1

(𝛽2 , 𝑗) 1 (2, 1) {2} (2, 2) {𝜙}

2 {1} {1, 2}

Step 3. For each 𝛽𝑘 let us generate the 𝑔(𝛽𝑘 ) tuples For 𝛽1 = (1, 1) 𝑔1 (𝛽1 ) = (1, 1) 𝑔1 (𝛽1 ) = max {⟨1, 1, 0⟩, ⟨1, 1, 0⟩} = ⟨1, 1, 0⟩ 𝑘∈𝐾(𝛽,1)

𝑔2 (𝛽1 ) = ⟨0, 0, 1⟩ For (𝛽1 ) = (2, 1) 12

𝑔1 (𝛽1 ) = ⟨1, 1, 0⟩ 𝑔2 (𝛽1 ) = ⟨1, 1, 0⟩ For 𝑔1 (𝛽1 ) = (2, 1) 𝑔1 (𝛽1 ) = ⟨1, 1, 0⟩ 𝑔2 (𝛽1 ) = ⟨1, 1, 0⟩ For 𝑔1 (𝛽1 ) = (2, 2) 𝑔1 (𝛽2 ) = ⟨0, 0, 1⟩ 𝑔2 (𝛽2 ) = 𝑚𝑎𝑥{⟨1, 1, 0⟩, ⟨0, 0, 1⟩} = ⟨1, 1, 0⟩ The corresponding tabular forms are given by (𝛽1 , 𝑗) (1, 1) (2, 1)

𝑔(𝛽1 ) (⟨1, 1, 0⟩, ⟨0, 0, 1⟩) (⟨1, 1, 0⟩, ⟨1, 1, 0⟩) (𝛽2 , 𝑗) (2, 1) (2, 1)

𝑔(𝛽2 ) (⟨0, 0, 1⟩, ⟨1, 1, 0⟩) (⟨0, 0, 1⟩, ⟨1, 1, 0⟩)

By pairwise comparison we can find out the minimum in each of the above table,[we get ] ⟨1, 1, 0⟩ ⟨0, 0, 1⟩ ˆ 𝐵= ⟨0, 0, 1⟩ ⟨1, 1, 0⟩ ˆ in 𝐴𝑋 = 𝐵 we can find the minimum 𝑋 ˆ Using the minimum 𝐵 ˆ is ] [ Now 𝐴𝑋 = 𝐵 [ ] [ ] ⟨1, 1, 0⟩ ⟨1, 1, 0⟩ ⟨0, 0, 1⟩ ⟨1, 1, 0⟩ ⟨1, 1, 0⟩ ⟨0, 0, 1⟩ = ⟨1, 1, 0⟩ ⟨0, 0, 1⟩ ⟨1, 1, 0⟩ ⟨1, 1, 0⟩ ⟨0, 0, 1⟩ ⟨1, 1, 0⟩ ˆ Step. 4 Determine the set 𝐽𝑖𝑗 (𝐵) ˆ = {𝑚𝑖𝑛⟨{⟨1, 1, 0⟩, ⟨0, 0, 1⟩}, 𝑚𝑖𝑛⟨{⟨1, 1, 0⟩, ⟨1, 1, 0⟩}} = ⟨1, 1, 0⟩ 𝐽11 (𝑋) = {⟨0, 0, 1⟩⟨1, 1, 0⟩ = {2} ˆ = {𝑚𝑖𝑛⟨{⟨1, 1, 0⟩, ⟨1, 1, 0⟩}, 𝑚𝑖𝑛⟨{⟨1, 1, 0⟩, ⟨1, 1, 0⟩}} = ⟨0, 0, 1⟩ 𝐽12 (𝑋) = {⟨1, 1, 0⟩⟨1, 1, 0⟩ = {𝜙} ˆ = {𝑚𝑖𝑛⟨{⟨1, 1, 0⟩, ⟨0, 0, 1⟩}, 𝑚𝑖𝑛⟨{⟨0, 0, 1⟩, ⟨1, 1, 0⟩}} = ⟨0, 0, 1⟩ 𝐽21 (𝑋) = {⟨0, 0, 1⟩⟨0, 0, 1⟩ = {1, 2} ˆ = {𝑚𝑖𝑛⟨{⟨1, 1, 0⟩, ⟨1, 1, 0⟩}, 𝑚𝑖𝑛⟨{⟨0, 0, 1⟩, ⟨1, 1, 0⟩}} = ⟨1, 1, 0⟩ 𝐽22 (𝑋) = {⟨1, 1, 0⟩⟨0, 0, 1⟩ = {1} ˆ × 𝐽12 𝐵 ˆ = {2} × 𝜙 Let 𝛽1 = 𝐽11 𝐵 13

ˆ Ă— đ??˝22 đ??ľ ˆ = {1, 2} Ă— {1} = {(1, 1)(2, 1)} đ?›˝2 = đ??˝21 đ??ľ Step 5. Determine the set đ??ž(đ?›˝đ?‘˜ , đ?‘—) for k=1,2 and j=1,2 For đ?›˝1 = {2} đ??ž(đ?›˝1 , 1) = đ?œ™ đ??ž(đ?›˝1 , 2) = {1} For đ?›˝2 = {2, 1}đ??ž(đ?›˝2 , 1) = {2} đ??ž(đ?›˝2 , 2) = {1} đ?‘˜(đ?›˝1 , đ?‘—) 1 2 {2} Ă— đ?œ™ đ?œ™ {1} đ?‘˜(đ?›˝2 , đ?‘—) (1, 1) (2, 1)

{1, 2} {1, 2} {2}

đ?œ™ đ?œ™ {1}

Step 6: For each đ?›˝đ?‘˜ Let as generate the đ?‘”(đ?›˝đ?‘˜ ) tuples. For đ?›˝1 = {2} Ă— đ?œ™ đ?‘”1 (đ?›˝1 ) = â&#x;¨0, 0, 1â&#x;Š đ?‘”2 (đ?›˝2 ) = â&#x;¨1, 1, 0â&#x;Š đ??š đ?‘œđ?‘&#x;đ?›˝2 = (1, 1) đ?‘”1 (đ?›˝2 ) = â&#x;¨1, 1, 0â&#x;Š đ?‘”2 (đ?›˝2 ) = â&#x;¨0, 0, 1â&#x;Š đ??š đ?‘œđ?‘&#x;đ?›˝2 = (2, 1) đ?‘”1 (đ?›˝2 ) = â&#x;¨1, 1, 0â&#x;Š đ?‘”2 (đ?›˝2 ) = â&#x;¨0, 0, 1â&#x;Š The corresponding tabular forms are given by đ?›˝1 đ?‘”(đ?›˝1 ) {2} Ă— đ?œ™ â&#x;¨0, 0, 1â&#x;Š, â&#x;¨1, 1, 0â&#x;Š đ?›˝2 đ?‘”(đ?›˝2 ) (1, 1) â&#x;¨1, 1, 0â&#x;Š, â&#x;¨0, 0, 1â&#x;Š (2, 1) â&#x;¨1, 1, 0â&#x;Š, â&#x;¨0, 0, 1â&#x;Š ˇ select a minimum row from each table, that is To get the đ?‘‹ [ ] â&#x;¨0, 0, 1â&#x;Š â&#x;¨1, 1, 0â&#x;Š ˇ= đ?‘‹ â&#x;¨1, 1, 0â&#x;Š â&#x;¨0, 0, 1â&#x;Š ˇ will satisfy đ??´đ?‘‹đ??´ = đ??´ and we observe that Clearly this [đ?‘‹ ] â&#x;¨0, 0, 1â&#x;Š â&#x;¨1, 1, 0â&#x;Š ′ ′′ ˇ đ?‘‹] ˆ = { [đ?‘‹, âˆŁ0 ≤ đ?›ź ≤ 1, 0 ≤ đ?›ź ≤ 1 and 0 ≤ đ?›ź ≤ 1 with ′ ′′ â&#x;¨1, 1, 0â&#x;Š â&#x;¨đ?›ź, đ?›ź , đ?›ź â&#x;Š ′ ′′ ˇ đ?‘‹]. ˆ đ?›ź + đ?›ź + đ?›ź ≤ 3} is the set of all g-inverse in [đ?‘‹,

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Conclusion: The maximum and minimum solution of the relational equation đ?‘Ľđ??´ = đ?‘? and đ??´đ?‘Ľ = đ?‘? has been obtained. Using this relational equation maximum and minimum g-inverse of a fuzzy neutrosophic soft matrix are also found.

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[13] F. Smarandache, Neutrosophic set, a generalisation of the intuitionistic fuzzy set, Inter. J.Pure ApplMath. (24), 2005, 287-297. [14] R. Uma, P. Murugadas and S. Sriram,Fuzzy Neutrosophic Soft Matrices of TypeI and Type-II, Communicated.

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