aa r X i v : . [ m a t h . R A ] J a n ON FUZZIFICATION OF n -LIE ALGEBRAS SHADI SHAQAQHA
Abstract.
The aim of this paper is to introduce the notion of intuitionistic fuzzyLie subalgebras and intutionistic fuzzy Lie ideals of n -Lie algebras. It is a generaliza-tion of intuitionistic fuzzy Lie algebras. Then, we investigate some of characteristicsof intutionistic fuzzy Lie ideals (resp. subalgebras) of n -Lie algeras. Finally, we de-fine the image and the inverse image of intuitionistic fuzzy Lie subalgebra under n -Lie algebra homomorphism. The properties of intuitionistic fuzzy n -Lie subal-gebras and intuitionistic fuzzy Lie ideals under homomorphisms of n -Lie algebrasare studied. Finally, we define the intuitionistic fuzzy quotient n -Lie algebra by anintuitionistic fuzzy ideal of n-Lie algebra and prove that it is a n -Lie algebra. Introduction
The concept of n -Lie algebras was introduced by Filippov [15] in 1985. If n = 2,then we get Lie algebra structure which was introduced by Sophus Lie (1842-1899)while he was attempting to classify certain smooth subgroups of general linear groupsthat are now called Lie groups. The case where n = 3 was initially appeared in Nan-mbu’s work [26] while he was attempting to describe simultaneous classical dynamicsof three particles. Takhtajan [29] investigated the geometrical and algebraic aspectsof the generalized Nambu mechanics, and established the connection between theNambu mechanics and Filippov’s theory of n -Lie algebras. More applictions to the n -Lie algebras can be found in [7, 16, 17, 23, 24, 25, 28, 34]. So, our results areexpected to be useful in various applications. Mathematics Subject Classification.
Key words and phrases. n -Lie algebras; n -Lie homomorphism; intutionistic fuzzy set; intuition-istic fuzzy n -Lie subalgebra; intuitionistic fuzzy n -Lie ideal. The notion of fuzzy sets was firstly introduced by Zadeh [37]. The fuzzy set theorystates that there are propositions with an infinite number of truth values, assumingtwo extreme values, 1 (totally true), 0 (totally false) and a continuum in between,that justify the term fuzzy. Applications of this theory can be found, for example,in artificial intelligence, computer science, control engineering, decision theory, logicand management science.After introducing of fuzzy sets by Zadeh, many researches were conducted on thegeneralizations of this fundamental concept. Among these generalizations was theconcept of intuitionistic fuzzy sets, which was introduced by Attanasov [8] in 1986,is the most important and interesting one. The elements of the intuitionistic fuzzysets are distinguished by an additional degree called the degree of uncertainty. Thereare numerous applications to this new concept include computer science, mathemat-ics, medicine, chemistry, economics, astronomy etc.. Many mathematicians haveinvolved in extending the concept of intuitionistic fuzzy sets to border of abstract al-gebra. Biswas applied the concepts of intuitionistic fuzzy sets to the theory of groupsand studied intuitionistic fuzzy subgroups in [10]. Also, Akram studied Lie algebrain intuitionistic fuzzy sets and obtained some results in [3].The fuzzy Lie subalgebras and fuzzy Lie ideals are considered in [21] by Kim and Lee,and in [35, 36] by Yehia. They established the analogues of most of the fundamentalground results involving Lie algebras in the fuzzy setting. The study of fuzzy subal-gebras (resp. ideals) of n -Lie algebras was initiated by B. Davvaz and WA. Dudek[13]. Recently, the complex (intuitionistic) fuzzy Lie algebras is studied in [32, 33] asa generalization of (intutionistic) fuzzy Lie algebras.In this paper we describe intuitionistic fuzzy n-Lie algebras. We will introduce n-Liealgebras into intuitionistic fuzzy set. Our work will generalize the theory of (intu-itionstic) fuzzy Lie algebras ([3, 4, 5, 6, 33, 35, 36]). N FUZZIFICATION OF n -LIE ALGEBRAS 3 Preliminaries n -Lie algebras were originally introduced by Filippov [15] in 1985. They generalizeLie algebras. In this article the ground field F is arbitrary. Definition 2.1.
Let n ∈ N , n ≥
2. An n -Lie algebra is a pair ( L, [ ]) where L is avector space and [ ] : L n → L ; ( x , . . . , x n ) [ x , . . . , x n ]is an n -linear map, called n -Lie bracket, that satisfies the following identities for all σ in the symmetric group S n and x , . . . , x n , y , . . . , y n ∈ L :(i) Skew symmetry:[ x σ (1) , . . . , x σ ( n ) ] = sign( σ )[ x , . . . , x n ] . (ii) The generalized Jacobi identity (called also the Filippov identity):[[ x , . . . , x n ] , y , . . . , y n ] = n X i =1 [ x , . . . , x i − , [ x i , y , . . . , y n ] , x i , . . . , x n ] . Subalgebras of n -Lie algebras and homomorphisms (or isomorphisms) between n -Lie algebras are defined as usual ([15]).Throughout this paper, L is a n -Lie algebra over field F .The concept of intuitionistic fuzzy set was introduced by Atanassov [8], where headded a new component (which determines the degree of non-membership) in thedefinition of fuzzy set (FS) that was given by Zadeh. Let X be a non-empty set,and let A = ( µ A , λ A ) = { ( x, µ A ( x ) , λ A ( x )) : x ∈ X } where µ A : X → [0 ,
1] and λ A : X → [0 ,
1] be mappings such that µ A ( x ) + λ A ( x ) ≤
1. Then A is called anintuitionistic fuzzy set. In this case the mappings µ A and λ A denote the degree ofmembership and the degree of non-membership to A respectively, for each element x ∈ X . The value π A ( x ) = 1 − µ A ( x ) − λ A ( x ) is called uncertainty or intuitionisticindex of the element x ∈ X to the intuitionistic fuzzy set A . It is obvious that each SHADI SHAQAQHA fuzzy set A = µ A = { ( x, µ A ( x )) : x ∈ X } can be represented as an intuitionisticfuzzy set where A = { ( x, µ A ( x ) , − µ A ( x )) : x ∈ X } . Definition 2.2.
Let A = ( µ A , λ A ) and B = ( µ B , λ B ) be two intuitionistic fuzzy setsof a set X . Then(i) The complement of A is ¯ A = ( λ A , µ A ),(ii) A ⊆ B if and only if µ A ( x ) ≤ µ B ( x ) and λ A ( x ) ≥ λ B ( x ) for all x ∈ X ,(iii) the intersection of A and B is A ∩ B = { ( x, min { µ A ( x ) , µ B ( x ) } , max { λ A ( x ) , λ B ( x ) } | x ∈ X ,(iv) the union of A and B is A ∪ B = { ( x, max { µ A ( x ) , µ B ( x ) } , min { λ A ( x ) , λ B ( x ) } | x ∈ X ,(v) ✷ A = { ( x, µ A ( x ) , µ ∁ A ( x )) : x ∈ X } where µ ∁ A ( x ) = (1 − µ A ( x )) for all x ∈ X ,(vi) ♦ A = { ( x, λ ∁ A ( x ) , λ A ( x )) : x ∈ X } where λ ∁ A ( x ) = (1 − λ A ( x )) for all x ∈ X .Let L be a n -Lie algebra over F . A fuzzy set A = µ A = { ( x, µ A ( x )) : x ∈ L } on L is a fuzzy Lie subalgebra if the following conditions are satisfied:(i) µ A ( x + y ) ≥ min { µ A ( x ) , µ A ( y ) } for all x, y ∈ L ,(ii) µ A ( αx ) ≥ µ A ( x ) for all x ∈ L and α ∈ F ,(iii) µ A ([ x , x , . . . , x n ]) ≥ min { µ A ( x ) , µ A ( x ) , · · · , µ A ( x n ) } for all x , x , . . . , x n ∈ L .It is called a fuzzy Lie ideal if the condition µ A ([ x , x , . . . , x n ]) ≥ min { µ A ( x ) , µ A ( x ) , · · · , µ A ( x n ) } is replaced by µ A ([ x , x , . . . , x n ]) ≥ max { µ A ( x ) , µ A ( x ) , · · · , µ A ( x n ) } ([13]). N FUZZIFICATION OF n -LIE ALGEBRAS 5 Intuitionistic Fuzzy n -Lie Algebras For the sake of simplicity, we shall use the symbols a ∧ b = min { a, b } and a ∨ b =max { a, b } . Definition 3.1.
Let A = ( µ A , λ A ) = { ( x, µ A ( x ) , λ A ( x )) : x ∈ L } be an intuitionisticfuzzy set of L . Then A is called an intuitionistic fuzzy Lie subalgebra of L if thefollowing conditions are satisfied:(i) µ A ( x + y ) ≥ µ A ( x ) ∧ µ A ( y ) and λ A ( x + y ) ≤ λ A ( x ) ∨ λ A ( y ) for all x, y ∈ L ,(ii) µ A ( αx ) ≥ µ A ( x ) and λ A ( αx ) ≤ λ A ( x ) for all x ∈ L and α ∈ F ,(iii) µ A ([ x , x , . . . , x n ]) ≥ µ A ( x ) ∧ µ A ( x ) ∧· · ·∧ µ A ( x n ) and λ A ([ x , x , . . . , x n ]) ≤ λ A ( x ) ∨ λ A ( x ) ∨ · · · ∨ λ A ( x n ) for all x , x , . . . , x n ∈ L .An intuitionistic fuzzy set A on L is called an intuitionistic fuzzy Lie ideal if theconditions ( i ) and ( ii ) are satisfied together with the following addition condition:( iii ) ′ µ A ([ x , x , . . . , x n ]) ≥ µ A ( x ) ∨ µ A ( x ) ∨ · · ·∨ µ A ( x n ) and λ A ([ x , x , . . . , x n ]) ≤ λ A ( x ) ∧ λ A ( x ) ∧ · · · ∧ λ A ( x n ) for all x , x , . . . , x n ∈ L .In the special case that n = 2, we obtain intuitionistic fuzzy Lie algebras ([3]).When first two conditions hold, we say that A is an intuitionistic fuzzy vector subspaceof L . The second condition implies µ A (0) ≥ µ A ( x ), µ A ( − x ) ≥ µ A ( x ), λ A (0) ≤ λ A ( x ),and λ A ( − x ) ≤ λ A ( x ) for all x ∈ L . Also, every intuitionistic fuzzy Lie ideal of an n -Lie algebra is an intuitionistic fuzzy Lie subalgebra, but the converse is not neces-sary true ( see [32, Example 3.1]).Let { A i = ( µ A i , λ A i ) | i ∈ I } be a collection of intuitionistic fuzzy sets on anonempty set X . Then \ i ∈ I A i = { ( x, µ T i ∈ I A i ( x ) , λ T i ∈ I A i ( x )) : x ∈ X } , SHADI SHAQAQHA where µ T i ∈ I A i ( x ) = inf i ∈ I { µ A i ( x ) } and λ T i ∈ I A i ( x ) = sup i ∈ I { λ A i ( x ) } , is an intuitionistic fuzzy set too (see [33]). We shall give the proof of the followingtheorem, established in [32] to the case of intuitionistic fuzzy Lie algebras, whichproves that the arbitrary intersection of intuitionistic fuzzy Lie subalgebras (resp.ideals) of an n -Lie algebra L is an intuitionistic fuzzy Lie subalgebras (resp. ideal) of L too. However it was proved in the case that L is a Lie (super)algebras and wherethe family is finite (see [11]). Theorem 3.1.
Let { A i } i ∈ I ( A i = ( µ A i , λ A i ) , i ∈ I ) be a collection of intuitionisticfuzzy Lie subalgebras (resp. ideals) on L . Then T i ∈ I A i is an intuitionistic fuzzy Liesubalgebra (resp. ideal) of L .Proof. Here we will prove the case of intuitionistic fuzzy Lie subalgera. For x, y ∈ L and α ∈ F , we have µ T i ∈ I A i ( x + y ) = inf i ∈ I { µ A i ( x + y ) }≥ inf i ∈ I { µ A i ( x ) ∧ µ A i ( y ) } = inf i ∈ I { µ A i ( x ) } ∧ inf i ∈ I { µ A i ( y ) } = µ T i ∈ I A i ( x ) ∧ µ T i ∈ I A i ( y ) . Also, µ T i ∈ I A i ( αx ) = inf i ∈ I { µ A i ( αx ) }≥ inf i ∈ I { µ A i ( x ) } = µ T i ∈ I A i ( x ) . N FUZZIFICATION OF n -LIE ALGEBRAS 7 Similarly, we can prove that λ T i ∈ I A i ( x + y ) ≤ λ T i ∈ I A i ( x ) ∨ λ T i ∈ I A i ( y ) and λ T i ∈ I A i ( αx ) ≤ λ T i ∈ I A i ( x ). Next, if x , . . . , x n ∈ L , then µ T i ∈ I A i ([ x , . . . , x n ]) = inf i ∈ I { µ A i ([ x , . . . , x n ]) }≥ inf i ∈ I { µ A i ( x ) ∧ · · · ∧ µ A i ( x n ) } = inf i ∈ I { µ A i ( x ) } ∧ · · · ∧ inf i ∈ I { µ A i ( x n ) } = µ T i ∈ I A i ( x ) ∧ · · · ∧ µ T i ∈ I A i ( x n ) . In a similar way, one can show λ T i ∈ I A i ([ x , . . . , x n ]) ≤ λ T i ∈ I A i ( x ) ∨ · · · ∨ λ T i ∈ I A i ( x n ).Therefore, T i ∈ I A i is an intuitionistic fuzzy Lie subalgebra. The proof of the case ofintuitionistic fuzzy Lie ideal is same, so we omit it. ✷ Let A = { ( x, µ A ( x ) , λ A ( x )) : x ∈ X } be an intuitionistic fuzzy set. For s, t ∈ [0 , A ts = { x ∈ X : µ A ( x ) ≥ s, λ A ( x ) ≤ t } is called the upper level subset of theintuitionistic fuzzy subset A . In particular if t = 1, then we get the upper s -level cut A s = U ( µ A ; s ) = { x ∈ X : µ A ( x ) ≥ s } . Also, if s = 0, then we get the lower t -levelcut A t = L ( λ A ; t ) = { x ∈ X : λ A ( x ) ≤ t } .The following two theorems will show relations between intuitionistic fuzzy Lie subal-gebras of L and Lie subalgebras of L . They are very similar to the case that suggestedby Kondo and Dudek in [22]. Theorem 3.2.
Let A = ( µ A , λ A ) be an intuitionistic fuzzy set of an n -Lie algebra L . Then A is an intuitionistic fuzzy Lie subalgebra of L if and only if the non-emptyset A ts is Lie subalgebra for all s, t ∈ [0 , .Proof. Let A = { ( x, µ A ( x ) , λ A ( x )) : x ∈ L } be an intuitionistic fuzzy Liesubalgebra. For x, y ∈ A st and γ ∈ F (i) µ A ( x + y ) ≥ µ A ( x ) ∧ µ A ( y ) ≥ s and λ A ( x + y ) ≤ λ A ( x ) ∨ λ A ( y ) ≤ t ,(ii) µ A ( γx ) ≥ µ A ( x ) ≥ s and λ A ( γx ) ≤ λ A ( x ) ≤ t . SHADI SHAQAQHA
Thus x + y, γx ∈ A ts . Also for x , . . . , x n ∈ L , we have µ A ([ x , , . . . , x n ]) ≥ µ A ( x ) ∧· · ·∧ µ A ( x n ≥ s and λ A ([ x , . . . , x n ]) ≤ λ A ( x ) ∨· · ·∨ λ A ( x n ) ≤ t . That is [ x , . . . , x n ] ∈ A ts . Therefore A ts is Lie subalgebra of L . Conversely, suppose that A st = ∅ is aLie subalgebra of L for every s, t ∈ [0 , x, y ∈ L and α ∈ F . Fix s = µ A ( x ) ∧ µ A ( x ) and t = λ A ( x ) ∨ λ A ( y ), so that x, y ∈ A t s . Since A t s is a subspaceof L , we have x + y and αx are in A t s , and so µ A ( x + y ) ≥ s = µ A ( x ) ∧ µ A ( y ), λ A ( x + y ) ≤ t = λ A ( x ) ∨ λ A ( y ). Also for x ∈ L , set s = µ A ( x ) and t = λ A ( x ). Then x ∈ A t s , and so γx ∈ A t s . Hence µ A ( γx ) ≥ µ A ( x ) and λ A ( γx ) ≤ λ A ( x ). Finally let x , x , . . . , x n ∈ L . Fix t = µ A ( x ) ∧ · · · ∧ µ A ( x n ) and s = λ A ( x ) ∨ · · · ∨ λ A ( x n ). Thus x i ∈ A ts for all i = 1 , . . . , n . Since A ts is a subalgebra of L , we have [ x , . . . , x n ] ∈ A ts ,so that µ A ([ x , . . . , x n ]) ≥ s = µ A ( x ) ∧ · · · ∧ µ A ( x ) and λ A ([ x , . . . , x n ]) ≤ t = λ A ( x ) ∨ · · · ∨ λ A ( x n ). The case of intuitionistic fuzzy ideals is almost same. ✷ Let A = { ( x, µ A ( x ) , λ A ( x )) : x ∈ X } be an intuitionistic fuzzy set. For s, t ∈ [0 , A t < s > = { x ∈ X : µ A ( x ) > s, λ A ( x ) < t } is called the strong upper level subsetof the intuitionistic fuzzy subset A . Theorem 3.3.
Let A = ( µ A , λ A ) be an intuitionistic fuzzy subset of L . Then A isan intuitionistic fuzzy Lie subalgebra of L if and only if the non empty set A t < s > is Liesubalgebra, for all s, t ∈ [0 , .Proof. The proof of the forward direction is almost identical to the proof in Theorem3.2. Conversely, suppose that the non-empty set A t < s > is a Lie subalgebra for all s, t ∈ [0 , x, y ∈ L . If µ A ( x ) = 0 or µ A ( y ) = 0, then it is clear that µ A ( x + y ) ≥ µ A ( x ) ∧ µ A ( y ),so we may assume that µ A ( x ) = 0 and µ A ( y ) = 0. Let s be the largest numberon the interval [0 ,
1] such that s < µ A ( x ) ∧ µ A ( y ) and there is no a ∈ L satisfying s < µ A ( a ) < µ A ( x ) ∧ µ A ( y ). Having x, y ∈ A t < s > where 1 > t > λ A ( x ) ∨ λ A ( y ) (such t exists because µ A ( x ) and µ A ( y ) are greater than 0 in addition to µ A ( x ) + λ A ( x ) N FUZZIFICATION OF n -LIE ALGEBRAS 9 and µ A ( y ) + λ A ( y ) are less than or equals to 1) implies that x + y ∈ A ts , and hence µ A ( x + y ) > s . Since there exist no a ∈ L with s < µ A ( a ) < µ A ( x ) ∧ µ A ( y ), itfollows that µ A ( x + y ) ≥ µ A ( x ) ∧ µ A ( y ). If λ A ( x ) = 1 or λ A ( y ) = 1, then it is obviousthat λ A ( x + y ) ≤ λ A ( x ) ∨ λ A ( y ). Assume now that λ A ( x ) = 1 = λ A ( y ). Let t be thesmallest number on the interval [0 ,
1] such that t > λ A ( x ) ∨ λ A ( y ) and there is no a ∈ L with t > λ A ( a ) > λ A ( x ) ∨ λ A ( y ). Hence x, y ∈ A t < s > where s < µ A ( x ) ∧ µ A ( y ),and so x + y ∈ A t < s > . Therefore λ A ( x + y ) < t . Since there is no a ∈ L such that t > λ A ( a ) > λ A ( x ) ∨ λ A ( y ), it follows that λ A ( x + y ) ≤ λ A ( x ) ∨ λ A ( y ) as desired.In a similar way we can show that µ A ( γx ) ≥ µ A ( x ) and λ A ( γx ) ≤ λ A ( x ) for all x ∈ L and γ ∈ F . Next let x , . . . , x n ∈ L . Again we may assume that neitherof µ A ( x ) , . . . , µ A ( x n ) is 0. Let s be the greatest number on the interval [0 ,
1] suchthat s < µ A ( x ) ∧ · · · ∧ µ A ( x n ) and there is no a ∈ L such that s < µ A ( a ) <µ A ( x ) ∧ · · · ∧ µ A ( x n ). Then x i ∈ A t < s > , where 1 > t > λ A ( x ) ∨ · · · ∨ µ A ( x n ), foreach i = 1 , . . . , n . Since A t < s > is subalgebra of L , we have [ x , . . . , x n ] ∈ A t < s > . Hence µ A ( x , . . . , x n ) ≥ µ A ( x ) ∧ · · · ∧ µ A ( x n ). In a similar fashion, we can prove that λ A ([ x , . . . , x n ]) ≤ λ A ( x ) ∨ · · · ∨ λ A ( x n ) for all x , . . . , x n ∈ L . ✷ From the proofs of Theorem 3.2 and Theorem 3.3, we can immediately obtain thefollowing result.
Corollary 3.1.
Let A = ( µ A , λ A ) be an intuitionistic fuzzy subset of an n -Lie algebra L . The following statements are equivalent for every s, t ∈ [0 , : (i) A is an intuitionistic fuzzy Lie subalgebra (resp. ideal) of L , (ii) The nonempty subsets A ts are n -Lie subalgebras (resp. ideals) of L , (iii) The nonempty subsets A t < s = { x ∈ L | µ A ( x ) ≥ s and λ A ( x ) < t } are n -Liesubalgebras (resp. ideals) of L , (iv) The nonempty subsets A ts > = { x ∈ L | µ A ( x ) > s and λ A ( x ) ≤ t } are n -Liesubalgebras (resp. ideals) of L , (v) The The nonempty subsets A t < s > are n -Lie subalgebras (resp. ideals) of L . Operations On Intuitionistic Fuzzy n -Lie ideals We omit the proofs for the following two results because they are straightforward.
Theorem 4.1.
Let A = ( µ A , λ A ) be an intuitionistic fuzzy set in an n -Lie algebra L . Then A is an intuitionistic fuzzy Lie subalgebra (resp. ideal) if and only if ✷ A and ♦ A are intuitionistic fuzzy Lie subalgebras (resp. ideals). Let A = ( µ A , λ A ) and B = ( µ B , λ B ) be two intuitionistic fuzzy subsets on a n -Liealgebra L . Let us introduce the following sum of A and B , which was defined byChen and Zhang [11] in the case of intuitionistic fuzzy Lie superalgebras: A + B = ( µ A + B , λ A + B ) , where µ A + B ( x ) = sup x = a + b { µ A ( a ) ∧ µ B ( b ) } , and λ A + B ( x ) = inf x = a + b { λ A ( a ) ∨ λ B ( b ) } . If A = ( µ A , λ A ) and B = ( µ B , λ B ) are intuitionistic fuzzy sets on L , then A + B is an intuitionistic fuzzy set of L . Indeed if x ∈ L with µ A + B ( x ) + λ A + B ( x ) > x = a + b { µ A ( a ) ∧ µ B ( b ) } + inf x = a + b { λ A ( a ) ∨ λ B ( b ) } >
1, and so there exist a , b ∈ L such that x = a + b with µ A ( a ) ∧ µ B ( b ) + inf x = a + b { λ A ( a ) ∨ λ B ( b ) } > µ A ( a ) ∧ µ B ( b ) + λ A ( a ) ∨ λ B ( b ) >
1. Without loss of generality wemay assume µ A ( a ) ≤ µ B ( b ). If λ A ( a ) ≥ λ B ( b ), then it is a clear contradiction. If λ A ( a ) ≤ λ B ( b ), then µ A ( a ) + λ B ( b ) >
1. Hence µ A ( a ) + 1 − µ B ( b ) >
1, and so µ A ( a ) > µ B ( b ). Contradiction. Therefore, A + B is an intuitionistic fuzzy set of L .It is well known that if A and B are ideals of L , then A + B is an ideal of L too. We are going to obtain an analogous result in the case of intuitionistic fuzzy Lie N FUZZIFICATION OF n -LIE ALGEBRAS 11 ideals. Similar result was obtained for complex fuzzy Lie subalgebra in [32], and forintuitionistic fuzzy Lie sub-superalgebras in [11]. Theorem 4.2.
Let A = ( µ A , λ A ) and B = ( µ B , λ B ) be two intuitionistic fuzzy n -Lieideals on L . Then A + B is an intuitionistic fuzzy n -Lie ideal of L too.Proof. We proceed as in the proof of corresponding result on complex fuzzy Lie alge-bras [33]. The only difference appears in the proof is to show that µ A ([ x , . . . , x n ]) ≥ µ A ( x ) ∨ · · · ∨ µ A ( x n ) and λ A ([ x , . . . , x n ]) ≥ λ A ( x ) ∧ µ A ( x n ) for each x , . . . , x n ∈ L .Let x , . . . , x n ∈ L . Suppose that µ A + B ([ x , . . . , x n ]) < µ A + B ( x ) ∨ · · · ∨ µ A + B ( x n ).Then there exists t such that µ A + B ([ x , . . . , x n ]) < t < µ A + B ( x ) ∨ . . . ∨ µ A + B ( x n ).Without loss of generality we may assume µ A + B ( x ) = µ A + B ( x ) ∨ · · · ∨ µ A + B ( x n ).Thus µ A + B ([ x , . . . , x n ]) < t < sup x = a + b { µ A ( a ) ∧ µ B ( b ) } , and so there exist a , b ∈ L such that µ A + B ([ x, y ]) < t < µ A ( a ) ∧ µ B ( b ). Therefore µ A ( a ) > t and µ B ( b ) > t .Hence µ A + B ([ x , . . . , x n ]) = µ A + B ([ a + b , . . . , x n ]) ≥ sup [ x ,...,x n ]=[ a,...,x n ]+[ b,...,x n ] { µ A ([ a, . . . , x n ]) ∧ µ B ([ b, . . . , x n ]) }≥ µ A ([ a , . . . , x n ]) ∧ µ B ([ b , . . . , x n ]) ≥ ( µ A ( a ) ∨ · · · ∨ µ A ( x n )) ∧ ( µ B ( b ) ∨ · · · ∨ µ B ( x n )) > t> µ A + B ([ x , . . . , x n ]) . Contradiction. Thus µ A + B ([ x , . . . , x n ] ≥ µ A + B ( x ) ∨· · ·∨ µ A + B ( x n ). Finally, suppose λ A + B ([ x , . . . , x n ]) > λ A + B ( x ) ∧ · · · ∧ λ A + B ( x n ). Then there exists s such that λ A + B ([ x , . . . , x n ]) > s > λ A + B ( x ) ∧ · · · ∧ λ A + B ( x n ). Without loss of generality wemay assume λ A + B ( x ) = λ A + B ( x ) ∧ · · · ∧ µ A + B ( x n ). Thus λ A + B ([ x , . . . , x n ]) > s > inf x = a + b { λ A ( a ) ∨ λ B ( b ) } , and so there exist a , b ∈ L such that λ A + B ([ x , . . . , x n ]) > s > λ A ( a ) ∨ λ B ( b ). Therefore λ A ( a ) < s and λ B ( b ) < s . Now λ A + B ([ x , . . . , x n ]) = λ A + B ([ a + b , . . . , x n ]) ≤ inf [ x, y ]=[ a,...,x n ]+[ b,...,x n ] { λ A ([ a, . . . , x n ]) ∨ λ B ([ b, . . . , x n ]) }≤ λ A ([ a , . . . , x n ]) ∨ λ B ([ b , . . . , x n ]) ≤ ( λ A ( a ) ∧ · · · ∧ λ A ( x n )) ∨ ( λ B ( b ) ∧ · · · ∧ λ B ( x n )) < s< λ A + B ([ x , . . . , x n ]) . Contradiction. Thus A + B is a complex intuitionistic fuzzy n -Lie ideal of L . ✷ In particular, if A and B are fuzzy Lie ideals of a Lie algebra L , then we obtain aspecial case of Shaqaqha’s result [32, Theorem 3.5].5. Direct Product of Intuitionistic fuzzy Lie n -subalgebras Let A = ( µ A , λ A ) and B = ( µ B , λ B ) be an intuitionistic fuzzy subsets of L and L , respectively. Then the generalized cartesian product A × B is defined to be A × B = ( µ A × µ B , λ A , λ B ) where µ A × µ B : L × L → [0 , x, y ) µ A ( x ) ∧ µ B ( y ) , and λ A × λ B : L × L → [0 , x, y ) λ A ( x ) ∨ λ B ( y ) . We have the following result.
Theorem 5.1.
Let A = ( µ A , λ A ) and B = ( µ B , λ B ) be an intuitionistic fuzzy n -Liesubalgebras of L and L , respectively. Then A × B is an intuitionistic fuzzy n -Liesubalgebra of L × L . N FUZZIFICATION OF n -LIE ALGEBRAS 13 Proof.
Note that A × B is an intuitionistic fuzzy subset of L × L . Indeed if x ∈ L and y ∈ L such that µ A ( x ) ≤ µ B ( y ) and λ B ( y ) ≥ λ B ( x ), then( µ A × µ B )( x, y ) + ( λ A × λ B )( x, y ) = µ A ( x ) + λ B ( y ) ≤ µ B ( y ) + λ B ( y ) ≤ . Similarly, if µ B ( y ) ≤ µ A ( x ) and λ A ( x ) ≥ λ B ( y ). Let ( x , y ) , ( x , y ) ∈ L × L .Then the proofs for ( µ A × µ B )(( x , y ) + ( x , y )) ≥ ( µ A × µ B )(( x , y )) ∧ ( µ A × µ B )(( x , y )), ( λ A × λ B )(( x , y )+( x , y )) ≤ λ A × λ B (( x , y )) ∨ λ A × λ B (( x , y )), ( µ A × µ B )( c ( x , y )) ≥ ( µ A × µ B )(( x , y )), and ( λ A × λ B )( c ( x , y )) ≤ ( λ A × λ B )(( x , y ))are similar to the proof of [11, Lemma 2.1].For ( x , y ) , ( x , y ) , . . . , ( x n , y n ) ∈ L × L , we have µ A ([( x , y ) , ( x , y ) , . . . , ( x n , y n )]) = ( µ A × µ B )([ x , x , . . . , x n ] , [ y , y , . . . , y n ])= µ A ([ x , x , . . . , x n ]) ∧ µ B ([ y , y , . . . , y n ]) ≥ ( µ A ( x ) ∧ · · · ∧ µ A ( x n )) ∧ ( µ B ( y ) ∧ · · · ∧ µ B ( y n ))= ( µ A × µ B )(( x , y )) ∧ · · · ∧ ( µ A × µ B )(( x n , y )) , and also λ A ([( x , y ) , ( x , y ) , . . . , ( x n , y n )]) = ( λ A × λ B )([ x , x , . . . , x n ] , [ y , y , . . . , y n ])= λ A ([ x , x , . . . , x n ]) ∨ λ B ([ y , y , . . . , y n ]) ≤ ( λ A ( x ) ∨ · · · ∨ λ A ( x n )) ∨ ( λ B ( y ) ∨ · · · ∨ λ B ( y n ))= ( λ A × λ B )(( x , y )) ∨ · · · ∨ ( λ A × λ B )(( x n , y )) . ✷ However the direct product of two intuitionistic fuzzy Lie ideals of n -Lie algebras L and L is not nesaccary to be an intuitionistic fuzzy Lie ideal of the n -Lie algebra L × L . On Lie Algebra Homomorphism of Intuitionistic Fuzzy n -LieAlgebras Let L and L be n -Lie algebras, A = ( µ A , λ A ) be an intuitionistic subset of L ,and f : L → L be a function. Then the intuitionistic fuzzy subset f ( A ) of f ( L )defined by f ( A ) = ( µ f ( A ) , λ f ( A ) ) where µ f ( A ) ( y ) = sup x ∈ f − ( y ) { µ A ( x ) } and λ f ( A ) ( y ) =inf x ∈ f − ( y ) { λ A ( x ) } for y ∈ f ( L ) is called the image of A under f . Similarly, if B = ( µ B , λ B ) is an intuitionistic fuzzy subset of L , then the intuitionistic fuzzy seton L is f − ( B ) = ( µ f − ( B ) , λ f − ( B ) ) where µ f − ( B ) ( x ) = µ B ( f ( x )) and λ f − ( B ) ( x ) = λ B ( f ( x ))(see for example [32]). The proofs of the following two results are omitted becausethey are routine and parallel to the corresponding results on intuitionistic fuzzy Liealgebras ([3, 33]). Theorem 6.1.
Let ϕ : L → L be a n -Lie algebra homomorphism. If B = ( µ B , λ B ) is an intuitionistic fuzzy n -Lie subalgebra (resp. ideal) on L , then the intuitionisticfuzzy set ϕ − ( B ) is an intuitionistic fuzzy n -Lie subalgebra (rep. ideal) of L . Theorem 6.2.
Let ϕ : L → L be a n -Lie algebra homomorphism. If A = ( µ A , λ A ) is an intuitionistic fuzzy n -Lie subalgebra (resp. ideal) on L , then the intuitionisticfuzzy set ϕ ( A ) is an intuitionistic fuzzy n -Lie subalgebra (resp. ideal) of im( ϕ ) . Intuitionistic Fuzzy Quotient n -Lie Algebras Let L be an n -Lie algebra and I be an ideal of L . Then the factor space L/I = { x + I : x ∈ I } acquires an n -Lie algebra structure (called a quotient n -Lie algebra)by setting [ x + I, x + I, . . . , x n + I ] = [ x , x , . . . , x n ] + I N FUZZIFICATION OF n -LIE ALGEBRAS 15 for x , x , . . . , x n ∈ L . In this paper we define and study the intuitionistic Fuzzyquotient n -Lie algebra by an intuitionistic fuzzy n -Lie ideal. Definition 7.1.
Let A = ( µ A , λ A ) be an intuitionistic fuzzy n -Lie ideal of an n -Liealgebra L . Then for x ∈ L , the intuitionistic fuzzy subset x + A = ( x + µ A , x + λ A )where x + µ A : L → [0 , y µ A ( y − x )and x + λ A : L → [0 , y λ A ( y − x )is called a coset (determined by x, µ A , and λ A ) of the intuitionistic fuzzy n -Lie ideal A .The case where n = 2 was introduced and studied by Chen [12] in 2010. The set ofall cosets of an intuitionistic fuzzy n -Lie ideal will be denoted by L/A . The followinglemma proves that a coset may have many different labels.
Lemma 1.
Let A = ( µ A , λ A ) be an intuitionistic fuzzy n -Lie ideal of an n -Lie algebra L , and let x, y ∈ L . The following statements are equivalent: (i) x + A = y + A , (ii) µ A ( x − y ) = µ A (0) and λ A ( x − y ) = λ A (0) .Proof. If x, y ∈ L with x + A = y + A , then x + µ A = y + µ A and x + λ A = y + λ A . Consider x + µ A = y + µ A , then evaluating both sides for x implies that µ A (0) = µ A ( x − y ). Similarly λ A ( x − y ) = λ A (0). Conversely, let µ A ( x − y ) = µ A (0)and λ A ( x − y ) = λ A (0). Then for any z ∈ L , we have ( y + µ A )( z ) = µ A ( z − y ) ≥ µ A ( z − x ) ∧ µ A ( x − y ) = µ A ( z − x ) ∧ µ A (0) = µ A ( z − x ) = ( x + µ A )( z ). Thus y + µ A ≥ x + µ A . Also µ A ( y − x ) = µ A ( − ( x − y )) = µ A ( x − y ) = µ A (0). So we can prove that x + µ A ≥ y + µ A in the same way as above. Hence x + µ A = y + µ A . Byalmost the same aegument we can prove that x + λ A = y + λ A . ✷ Theorem 7.1.
Let A = ( µ A , λ A ) be an intuitionistic fuzzy n -Lie ideal of L and L/A be the set of all cosets of L on A . Then the set L/A is an n -Lie algebra under thefollowing operations: (i) ( x + A ) + ( y + A ) = ( x + y ) + A for all x, y ∈ L , (ii) α ( x + A ) = αx + A for all x ∈ L and α ∈ F , (iii) [( x + A ) , ( x + A ) , . . . , ( x n + A )] = [ x , x , . . . , x n ]+ A for all x , x , . . . , x n ∈ L .Proof. First, we show that the operations are well defined. Let x, y, u and v be elements in L such that x + A = y + A and u + A = v + A . Consequently µ A ( x + u − ( y + v )) = µ A (( x − y ) + ( u − v )) ≥ µ A ( x − y ) ∧ µ A ( u − v ) = µ A (0)(because µ A ( x − y ) = µ A (0) and µ A ( u − v ) = µ A (0)). As µ A (0) ≥ µ A ( x + u − ( y + v )),we have µ A ( x + u − y − v ) = µ A (0). Almost the same argument one can obtainthat λ A ( x + u − y − v ) = λ A (0) Therefore ( x + u ) + A = ( y + v ) + A . Also, µ A ( αx − αy ) ≥ µ A ( x − y ) = µ A (0) and λ A ( αx − αy ) ≤ λ A ( x − y ) = λ A (0). Hence α ( x + A ) = α ( y + A ). If x , x , . . . , x n ∈ L such that x i + A = y i + A ( i = 1 , . . . , n ),then µ A ([ x , x , . . . , x n ] − [ y , y , . . . , y n ]) = µ A ([ x − y , x , . . . , x n ]+[ y , x , . . . , x n ] − [ y , y , . . . , y n ]) ≥ µ A ([ x − y , x , . . . , x n ]) ∧ µ A ([ y , x , . . . , x n ] − [ y , y , . . . , y n ]) ≥ µ A (0) ∧ µ A ([ y , x , . . . , x n ] − [ y , y , . . . , y n ]) N FUZZIFICATION OF n -LIE ALGEBRAS 17 Now, µ A ([ y , x , . . . , x n ] − [ y , y , . . . , y n ]) = µ A ([ y , x − y , x , . . . , x n ]+[ y , y , x , . . . , x ] − [ y , y , y , . . . , y n ]) ≥ µ A ( y ) ∨ µ A ( x − y ) ∨ µ A ( x ) ∨ · · · µ A ( x n ) ∧ µ A ([ y , y , x , . . . , x ] − [ y , y , y , . . . , y n ]) ≥ µ A (0) ∧ µ A ([ y , y , x , . . . , x ] − [ y , y , y , . . . , y n ]) . So µ A ([ x , x , . . . , x n ] − [ y , y , . . . , y n ]) ≥ µ A (0) ∧ µ A (0) ∧ µ A ([ y , y , x , . . . , x ] − [ y , y , y , . . . , y n ]) . Continuing in the same way, we obtain µ A ([ x , x , . . . , x n ] − [ y , y , . . . , y n ]) ≥ µ A (0) ∧ µ A (0) ∧ · · · ∧ µ A (0) ( n times) . Hence µ A ([ x , x , . . . , x n ] − [ y , y , . . . , y n ]) = µ A (0). Using the same method one canprove that λ A ([ x , x , . . . , x n ] − [ y , y , . . . , y n ]) = λ A (0). Therefore [ x , x , . . . , x n ] + A = [ y , y , . . . , y n ] + A . Now it is straightforward to see that the product on L/A isan n -linear map satisfying the Filippov identity. ✷ The n -Lie algebra constructed in Theorem 7.1 is called intuitionistic fuzzy quotient n -Lie algebra of L by A . References [1] K.S. Abdukhalikov, M.S. Tulenbaev, U.U. Umirbaev, On fuzzy subalgebras,
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Inform. Control (1965), 338-358.(Shadi Shaqaqha) Department of Mathematics, Yarmouk University, Irbid, Jordan
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