n-Dimensional Fuzzy Negations
aa r X i v : . [ c s . L O ] J u l n -Dimensional Fuzzy Negations Benjamin Bedregal a and Ivan Mezzomo b and Renata H.S. Reiser c a Department of Informatics and Applied Mathematics,Federal University of Rio Grande do Norte [email protected] b Center of Exact and Natural Sciences,Rural Federal University of SemiArid [email protected] c Center of Tecnological Development,Federal University of Pelotas [email protected]
Abstract n -Dimensional fuzzy sets is a fuzzy set extension where the membership values are n -tuples of real numbers in the unit interval [0 ,
1] orderly increased, called n -dimensionalintervals. The set of n -dimensional intervals is denoted by L n ([0 , ,
1] – n -representable fuzzy negations on L n ([0 , ,
1] are preserved by representable(strong) fuzzy negation on L n ([0 , n -dimensionalautomorphism on an n -dimensional fuzzy negation provides a method to obtain other n -dimensional fuzzy negation, in which properties such as representability, continuity andmonotonicity on L n ([0 , Keywords
Fuzzy negations, n -dimensional fuzzy sets, n -dimensional fuzzy negations, n -dimensional automorphisms. The notion of an n -dimensional fuzzy set on L n -fuzzy set theory was introduced in [25] asa special class of L -fuzzy set theory, generalizing the theories underlying the fuzzy logic andmany other multivalued fuzzy logics: the interval-valued fuzzy set, the intuitionistic fuzzyset, the interval-valued intuitionistic fuzzy set and the type-2 fuzzy logic [10].In accordance with the Zadeh’s Extension Principle, L n -fuzzy set theory provides addi-tional degrees of freedom that makes it possible to directly model uncertainties in computa-tional systems based on fuzzy logics. Such uncertainties are frequently associated to systemswhere time-varying, non stationary statistical attributes or knowledge of experts using ques-tionnaires including uncertain words from natural language. However, the correspondingmathematical description of such models is unknown or not totally consolidated yet.This paper considers the main properties of an n -dimensional fuzzy set A over a referenceset X , where each element x ∈ X = ∅ is related with an n -dimensional interval, characterizedby its ordered n -membership values: µ A ( x ) ≤ . . . ≤ µ A n ( x ). Thus, for i = 1 , . . . , n , each n -membership function µ A i : X → [0 , i -th membership degree of A , canprovide an interpretation to model the uncertainty of n -distinct parameters from evaluationprocesses or fuzzy measures in computational systems modelled by L n -fuzzy set theory.1 .1 Main contribution The main contribution of this paper is concerned with representability of fuzzy negations onthe set of n -dimensional intervals, denoted by L n ([0 , ⊆ -monotonicityand monotonicity by part of corresponding n -membership function. These topics are closelyconnected to degenerate elements and equilibrium points of n -dimensional fuzzy negations.By considering an n -dimensional fuzzy negation N and related n -projections, the n -representability of N is discussed and the notion of ⊆ i -monotonicity is formalized on L n ([0 , i = 0 , . . . , n .Additionally, it is shown that the partial order of fuzzy negation can be extended from[0 ,
1] to L n ([0 , ,
1] are preserved by representable strong n -dimensionalfuzzy negations on L n ([0 , n -dimensionalfuzzy negations as operators preserving degenerate elements on L n ([0 , ,
1] have a counterpart on L n ([0 , n -dimensionalfuzzy negations, which can be generated by action of n -dimensional automorphisms is studied.The paper also investigates the conditions under which equilibrium points and degenerateelements are preserved by conjugate fuzzy negations on L n ([0 , In [25], the definitions of cut set on an n-dimensional fuzzy set and its corresponding n-dimensional vector level cut set of Zadeh fuzzy set are presented in order to study not onlydecomposition but also representation theorems of the n -dimensional fuzzy sets. Thus, newdecomposition and representation theorems of the Zadeh fuzzy set are proposed.In [8], the authors consider the study of aggregation operators for these new conceptsof n -dimensional fuzzy sets, starting from the usual aggregation operator theory and alsoincluding a new class of aggregation operators containing an extension of the OWA operator,which is based on n -dimensional fuzzy connectives. The results presented in such contextallow to extend fuzzy sets to interval-valued Atanassov’s intuitionistic fuzzy sets and alsopreserve their main properties. In particular, in [8], it was introduced the notion of n -dimensional fuzzy negation and showed a way of building n -dimensional fuzzy negation from n comparable fuzzy negations. In [21], we were introduced and studied the n -dimensionalstrict fuzzy negations.In the context of lattice-valued fuzzy set theory [18], the notion of fuzzy connectives forlattice-valued fuzzy logics was generalized in [3, 6, 4, 23] by taking into account axiomaticdefinitions. In [24], it was extended the notion n -dimensional fuzzy set by considered arbitrarybounded lattice L and, in[23], it was introduced the notion of n -dimensional lattice-valuednegation.Following the results in the above cited works, this paper studies the possibility of dealingwith main properties of representable fuzzy negation on L n ([0 , n -dimensional automorphisms. In particular, we studied the n -dimensional strongfuzzy negations. In this section, we will briefly review some basic concepts which are necessary for the de-velopment of this paper. The previous main definitions and additional results concern-2ng the study of n -dimensional fuzzy negations presented in this work can be found in[1, 2, 5, 8, 9, 11, 14, 15, 17, 19]. According with [11, Definition 0], a function ρ : [0 , → [0 ,
1] is an automorphism if it iscontinuous, strictly increasing and verifies the boundary conditions ρ (0) = 0 and ρ (1) = 1,i.e., if it is an increasing bijection on U , meaning that for each x, y ∈ [0 , x ≤ y , then ρ ( x ) ≤ ρ ( y ).Automorphisms are closed under composition, i.e., denoting A ([0 , , ρ, ρ ′ ∈ A ([0 , ρ ◦ ρ ′ ( x ) = ρ ( ρ ′ ( x )) ∈ A ([0 , ρ − of an automorphism ρ is also an automorphism, meaning that ρ − ( x ) ∈ A ([0 , ρ on a function f : [0 , n → [0 , f ρ and named the ρ -conjugate of f is defined as, for all ( x , . . . , x n ) ∈ [0 , n : f ρ ( x , . . . , x n )= ρ − ( f ( ρ ( x ) , . . . , ρ ( x n ))) . (1)Let f , f : [0 , n → [0 ,
1] be functions. The functions f and f are conjugated to eachother , if there exists an automorphism ρ such that f ( x , . . . , x n )= ρ − ( f ( ρ ( x ) , . . . , ρ ( x n ))) , for all ( x , . . . , x n ) ∈ [0 , n . Notice that, if f = f ρ then f = f ρ − . Let n ∈ N such that n ≥
2. A function A : [0 , n → [0 ,
1] is an n -ary aggregation operator if, for each x , . . . , x n , y , . . . , y n ∈ [0 , A satisfies the following conditions:A1. A (0 , . . . ,
0) = 0 and A (1 , . . . ,
1) = 1;A2. If x i ≤ y i , for each i = 1 , . . . , n , then A ( x , . . . , x n ) ≤ A ( y , . . . , y n ). A function N : [0 , → [0 ,
1] is a fuzzy negation ifN1: N (0) = 1 and N (1) = 0;N2: If x ≤ y , then N ( x ) ≥ N ( y ), for all x, y ∈ [0 , N satisfying the involutive propertyN3: N ( N ( x )) = x , for all x ∈ [0 , strong fuzzy negation . And, a continuous fuzzy negation N is strict if it verifiesN4: N ( x ) < N ( y ) when y < x , for all x ∈ [0 , N S ( x ) = 1 − x .An equilibrium point of a fuzzy negation N is a value e ∈ [0 ,
1] such that N ( e ) = e .See [5, Remarks 2.1 and 2.2] and [5, Proposition 2.1] for additional studies related to mainproperties of equilibrium points. Example 1
The function C k : [0 , → [0 , given by C k ( x ) = n − k +1 p − x n − k +1 , ∀ k ∈ { , , . . . , n } (2) is a strong fuzzy negation. Since C k is strong, it is also a strict fuzzy negation. Moreover,based on [20, Theorem 3.4], every continuous fuzzy negation has a unique equilibrium point. o, C k has a unique equilibrium point. Notice that e = n − k +1 q is the equilibrium point of C k , i.e., n − k +1 √ − e n − k +1 = e . Proposition 1
The function C k : [0 , → [0 , given by C k ( x ) = 1 − x k , ∀ k ∈ { , , . . . , n } . (3) is a strict but not strong fuzzy negation. Proof:
Straightforward from [5]. (cid:3)
Fuzzy negations have at most one equilibrium point, as proved by Klir and Yuan, in [20,Theorem 3.2]. Therefore, if a fuzzy negation has an equilibrium point then it is unique.However, not all fuzzy negations have an equilibrium point [5], e.g. the fuzzy negations N ⊥ and N ⊤ , respectively given as: N ⊥ ( x ) = (cid:26) , if x > , if x = 0; N ⊤ ( x ) = (cid:26) , if x = 1;1 , if x < . Clearly, for all fuzzy negation N , it holds that N ⊥ ≤ N ≤ N ⊤ .In [22, Prop. 4.2], Navara introduced the notion of negation-preserving automorphisms,assuring that an N S -preserving automorphism ρ ∈ Aut ([0 , N S , meaning that ρ ( N S ( x )) = N S ( ρ ( x )), for all x ∈ [0 , N be a fuzzy negation.A function ρ ∈ Aut ([0 , N -preserving automorphism if and only if ρ verifies thecondition ρ ( N ( x )) = N ( ρ ( x )) , ∀ x ∈ [0 , . (4)The Navara’s characterization for negation-preserving automorphisms is generalized be-low. Proposition 2 [5, Proposition 2.6] Let N be a strong fuzzy negation which has e as theunique equilibrium point of N . When ρ ∈ Aut ([0 , e ]) then ρ N : [0 , → [0 , , defined by ρ N ( x ) = (cid:26) ρ ( x ) , if x ≤ e ;( N ◦ ρ ◦ N )( x ) , if x > e (5) is an N -preserving automorphism. Additionally, N -preserving automorphisms are given as Eq.(5). Proposition 3 [5, Proposition 2.7] Let N be a strong fuzzy negation which has e as theunique equilibrium point. When ρ ∈ Aut ([0 , e ]) then ρ N − is an N -preserving automorphism. n -Dimensional fuzzy sets Let X be a non empty set and n ∈ N + = N − { } . According to [25], an n -dimensionalfuzzy set A over X is given by A = { ( x, µ A ( x ) , . . . , µ A n ( x )) : x ∈ X } ,4here, for each i = 1 , . . . , n , µ A i : X → [0 ,
1] is called i -th membership degree of A , whichalso satisfies the condition: µ A ( x ) ≤ . . . ≤ µ A n ( x ), for x ∈ X .In [7], for n ≥
1, an n -dimensional upper simplex is given as L n ([0 , { ( x , . . . , x n ) ∈ [0 , n : x ≤ . . . ≤ x n } , (6)and its elements are called n -dimensional intervals .For each i = 1 , . . . , n , the i -th projection of L n ([0 , π i : L n ([0 , → [0 , π i ( x , . . . , x n ) = x i .Notice that L ([0 , ,
1] and L ([0 , , degenerate element x ∈ L n ([0 , π i ( x ) = π j ( x ) , ∀ i, j = 1 , . . . , n. (7)The degenerate element ( x, . . . , x ) of L n ([0 , x ∈ [0 , /x/ and the set of all degenerate elements of L n ([0 , D n .An m -ary function F : L n ([0 , m → L n ([0 , D n -preserve function or afunction preserving degenerate elements if the following condition holds( DP ) F ( D mn ) = F ( /x /, . . . , /x m / ) ∈ D n , ∀ x , . . . , x m ∈ [0 , L n ([0 , x , y ∈ L n ([0 , x ∨ y = (max( x , y ) , . . . , max( x n , y n )) , (8) x ∧ y = (min( x , y ) , . . . , min( x n , y n )) . (9)And, by considering the natural extension of the order ≤ on L ([0 , x , y ∈ L n ([0 , x ≤ y iff π i ( x ) ≤ π i ( y ) , ∀ i = 1 , . . . , n. (10) n -dimensional interval function In the following, the continuity of a function F : L n ([0 , → L n ([0 , n -dimensional function or an n -dimensional interval function , will be studied, based onthe continuity on L ([0 , n ). Proposition 4
Let ρ : [0 , n → L n ([0 , be the function defined by ρ ( x , . . . , x n ) = [ x (1) , . . . , x ( n ) ] ,when ( x (1) , . . . , x ( n ) ) is a fixed-permutation of a tuple ( x , . . . , x n ) such that x ( i ) ≤ x ( i +1) forall i = 1 , . . . , n − and let σ : L n ([0 , → [0 , n be the function defined by σ ([ x , . . . , x n ]) =( x , . . . , x n ) . For an n -dimensional interval function F : L n ([0 , → L n ([0 , , the corre-sponding operator F ρ : [0 , n → [0 , n given by F ρ ( x , . . . , x n ) = σ ( F ( ρ ( x , . . . , x n ))) , (11) for all ( x , . . . , x n ) ∈ L n ([0 , , is a non-injective function. Proof:
Straightforward. (cid:3)
Based on the above results, from Eq. (8) to Eq. (10), the lattice ( L n ([0 , , ≤ ) satisfiesthe notion of continuity given as follows: Definition 5 An n -dimensional function F : L n ([0 , → L n ([0 , is continuous if thefunction F ρ : [0 , n → [0 , n given by Eq. (11) is also continuous. Hence, the continuity of an n -dimensional function F : L n ([0 , → L n ([0 , F ρ = σ ◦ F ◦ ρ from the usual continuity notion on [0 , n .5 Fuzzy negations on L n ([0 , L n ([0 , , ≤ ) as conceivedby Bedregal in [8] and their relation with usual notion of fuzzy negation. Definition 6
A function N : L n ([0 , → L n ([0 , is an n -dimensional fuzzy negation if it satisfies the following properties: N1 : N ( / / ) = / / and N ( / / ) = / / ; N2 : If x ≤ y then N ( x ) ≥ N ( y ) , for all x , y ∈ L n ([0 , . Proposition 7 [8, Proposition 3.1] Let N , . . . , N n be fuzzy negations such that N ≤ . . . ≤ N n . Then ^ N . . . N n : L n ([0 , → L n ([0 , defined by ^ N . . . N n ( x ) = ( N ( π n ( x )) , . . . , N n ( π ( x ))) (12) is an n -dimensional fuzzy negation. Additionally, according to [7], when i = 1 , . . . , n −
1, the ⊆ i -relation with respect to the i -th component of x , y ∈ L n ([0 , x ⊆ i y when π i ( y ) ≤ π i ( x ) ≤ π i +1 ( x ) ≤ π i +1 ( y ) . (13) L n ([0 , In order to analyse properties related to equilibrium point, representable and monotone fuzzynegations on L n ([0 , n -dimensional fuzzy negation N is called n -representable if there exist fuzzy nega-tions N , . . . , N n such that N ≤ . . . ≤ N n and N = ^ N . . . N n . (14)By reducing notation, when N i = N for all i = 1 , . . . , n , an n -representable fuzzy negation ^ N . . . N will be denoted by e N . Proposition 8
Let C k ( x ) = 1 − x k , for k ∈ { , , . . . , n } . Then an n -dimensional function C : L n ([0 , → L n ([0 , given as the following C ( x ) = ^ C . . . C n ( x ) = ( C ( π n ( x )) , . . . , C n ( π ( x ))) (15) is an n -representable fuzzy negation on L n ([0 , . Proof:
For all 1 ≤ i ≤ j ≤ n and x ∈ [0 , − x i ≤ − x j , resulting thefollowing inequalities: C ≤ C i ≤ C j ≤ C n . Additionally, the following is verified: N1: ^ C . . .C n ( / / ) = ( C ( π n ( / / )) , . . ., C n ( π ( / / ))) = ( C (0) , . . ., C n (0)) = / / ; and ^ C . . .C n ( / / ) = ( C ( π n ( / / )) , . . ., C n ( π ( / / ))) = ( C (1) , . . ., C n (1)) = / / ; N2:
Based on monotonicity of projection-functions, if x ≥ y then( C ( π n ( x )) , . . . , C n ( π ( x ))) ≤ ( C ( π n ( y )) , . . . , C n ( π ( y ))) , therefore, ^ C . . . C n ( x ) ≤ ^ C . . . C n ( y ) meaning that C ( x ) ≤ C ( y ).Concluding, C = ^ C . . .C n is an n -representable fuzzy negation on L n ([0 , (cid:3) i ∈ { , . . . , n − } . An n -dimensional fuzzy negation N is called ⊆ i -monotone if, forany x , y ∈ L n ([0 , N ( x ) ⊆ i N ( y ) whenever x ⊆ n − i y . (16)Moreover, one can say that an n -dimensional fuzzy negation N is called(i) ⊆ -monotone if N is ⊆ i -monotone for all i = 1 , . . . , n −
1; and(ii) monotone by part when, for all i = 1 , . . . , n and x , y ∈ L n ([0 , π i ( N ( x )) ≤ π i ( N ( y )) whenever π n − i +1 ( x ) ≥ π n − i +1 ( y ) . (17) Remark 1
When we say that N is ⊆ i -monotone, for all i = 1 , . . . , n − , it does not meanthat we will only consider x , y ∈ L n ([0 , such that π i ( y ) ≤ π i ( x ) ≤ π i +1 ( x ) ≤ π i +1 ( y ) , forall i = 1 , . . . , n − . Instead, we consider all x , y ∈ L n ([0 , and if for some i , x ⊆ i y thenby Eq. (16) we have that N ( x ) ⊆ n − i N ( y ) . For example, consider a n -dimensional fuzzynegation N which is ⊆ -monotone, x = (0 . , . , . , . , , and y = (0 . , . , . , . , .Clearly, x ⊆ y , x ⊆ y and x ⊆ y but x y . Since N is ⊆ -monotone we can concludethat of N ( x ) ⊆ N ( y ) , N ( x ) ⊆ N ( y ) and N ( x ) ⊆ N ( y ) . Proposition 9
Let N be an n -dimensional fuzzy negation. Then, for all i = 1 , . . . , n , thefunction N i : [0 , → [0 , defined by N i ( x ) = π i ( N ( /x/ )) (18) is a fuzzy negation. Proof:
Trivially, N i (0) = π i ( N ( / / ) = π i ( / / ) = 1 and N i (1) = π i ( N ( / / ) = π i ( / / ) = 0.Let x, y ∈ [0 , x ≤ y ⇒ /x/ ≤ /y/ ⇒ N ( /x/ ) ≥ N ( /y/ ) by N2 ⇒ π i ( N ( /x/ )) ≥ π i ( N ( /y/ )) ⇒ N i ( x ) ≥ N i ( y ) by Eq. (10)Therefore, Proposition 9 holds. (cid:3) In the following, the necessary and sufficient conditions under which we can obtain n -representable fuzzy negation on L n ([0 , Theorem 10 An n -dimensional fuzzy negation N is n -representable iff N is ⊆ -monotone. Proof: ( ⇒ ) If N is n -representable, then there exist fuzzy negations N ≤ . . . ≤ N n suchthat N = ^ N . . . N n . For each i = 1 , . . . , n −
1, by the antitonicity of N i ′ s , it holds that x ⊆ n − i y ⇒ π n − i ( y ) ≤ π n − i ( x ) ≤ π n − i +1 ( x ) ≤ π n − i +1 ( y ) by Eq.(13) ⇒ N i ( π n − i +1 ( y )) ≤ N i ( π n − i +1 ( x )) ≤ N i +1 ( π n − i ( x )) ≤ N i +1 ( π n − i ( y ))by N N i ≤ N i +1 ⇒ π i ( N ( y )) ≤ π i ( N ( x )) ≤ π i +1 ( N ( x )) ≤ π i +1 ( N ( y )) by Eq.(12) ⇒ N ( x ) ⊆ i N ( y ) by Eq.(13)Hence, N is ⊆ -monotone. 7 ⇐ ) Firstly, for all x ∈ L n ([0 , i = 1 , . . . , n −
1, we have that /x n − i +1 / ⊆ n − i +1 x as well as /x n − i +1 / ⊆ n − i x . Since N is ⊆ -monotone then for each i = 2 , . . . , n − /x n − i +1 / ⊆ n − i x ⇒ N ( /x n − i +1 / ) ⊆ i N ( x ) by Eq.(16) ⇒ π i ( N ( x )) ≤ π i ( N ( /x n − i +1 / )) by Eq.(13)and /x n − i +1 / ⊆ n − i +1 x ⇒ N ( /x n − i +1 / ) ⊆ i − N ( x ) by Eq.(16) ⇒ π i ( N ( /x n − i +1 / )) ≤ π i ( N ( x )) by Eq.(13)So, for each i = 2 , . . . , n − N i ( x n − i +1 ) = π i ( N ( /x n − i +1 / )) = π i ( N ( x )).On the other hand, since N is ⊆ -monotone and decreasing, then /x n / ⊆ n − x ⇒ N ( /x n / ) ⊆ N ( x ) by Eq.(16) ⇒ π ( N ( x )) ≤ π ( N ( /x n / )) by Eq.(13)and x ≤ /x n / ⇒ N ( /x n / ) ≤ N ( x ) by N2 ⇒ π ( N ( /x n / )) ≤ π ( N ( x )) by Eq.(10)Therefore, N ( x n ) = π ( N ( /x n / )) = π ( N ( x )). Analogously, since N is ⊆ -monotone anddecreasing, then /x / ⊆ x ⇒ N ( /x / ) ⊆ n − N ( x ) by Eq.(16) ⇒ π n ( N ( /x / )) ≤ π n ( N ( x )) by Eq.(13)and /x / ≤ x ⇒ N ( x ) ≤ N ( /x / ) by N2 ⇒ π n ( N ( x )) ≤ π n ( N ( /x / )) by Eq.(10)Concluding, N n ( x ) = π n ( N ( /x / )) = π n ( N ( x )). So, N i ( x n − i +1 ) = π i ( N ( x )) for each i = 1 , . . . , n and consequently, N = ^ N . . . N n and by Proposition 9, the N i ′ s are fuzzynegations and then N is n-representable. (cid:3) Proposition 11
If an n -dimensional fuzzy negation N is n -representable, then N is a func-tion monotone by part. Proof: If N is n -representable, then there exist fuzzy negations N ≤ . . . ≤ N n suchthat N = ^ N . . . N n . Based on the antitonicity of N i ′ s and by property N2 , if π n − i +1 ( x ) ≥ π n − i +1 ( y ) for some i = 1 , . . . , n and x , y ∈ L n ([0 , N i ( π n − i +1 ( x )) ≤ N i ( π n − i +1 ( y )). Therefore, by Eq.(12), π i ( N ( x )) ≤ π i ( N ( y )). Hence, N is a monotone bypart fuzzy negation on L n ([0 , (cid:3) The partial order on fuzzy negations can be extended for n -dimensional fuzzy negations.For that, let N and N be n -dimensional fuzzy negations, then the following holds: N (cid:22) N iff for each x ∈ L n ([0 , , N ( x ) ≤ N ( x ) . (19) Lemma 12
Let N , . . . , N n be fuzzy negations. If N ≤ . . . ≤ N n , then f N (cid:22) ^ N . . . N n (cid:22) f N n . roof: By Eq.(12), we have that f N ( x ) = ( N ( π n ( x )) , . . . , N ( π ( x ))) and ^ N . . . N n ( x ) =( N ( π n ( x )) , . . . , N n ( π ( x ))). Since N ≤ . . . ≤ N n , then N ( π j ( x )) ≤ N i ( π j ( x )) with i, j =1 , . . . , n . So, by Eq.(19), f N (cid:22) ^ N . . . N n . Analogously we proof that ^ N . . . N n (cid:22) f N n . (cid:3) Proposition 13
Let N be an n -dimensional fuzzy negation. If N is ⊆ -monotone, then g N ⊥ (cid:22) N (cid:22) g N ⊤ . Proof:
Straightforward by Theorem 10 and Lemma 12. (cid:3)
Proposition 14
Let N be an n -dimensional fuzzy negation. Then, it holds that N ⊥ (cid:22) N (cid:22)N ⊤ whereas N ⊥ ( x ) = (cid:26) / /, if x = / / ; / /, if x = / / ; and N ⊤ ( x ) = (cid:26) / / if , x = / / ; / / if , x = / /. Proof: If x = / / , then N ⊥ ( x ) = / / and so N ⊥ ( x ) ≤ N ( x ). If x = / / , then N ( / / ) = / / and so N ⊥ ( x ) ≤ N ( x ). Analogously we proof when x = / / and x = / / . Therefore, N ⊥ (cid:22) N (cid:22) N ⊤ . (cid:3) Remark 2
Note that N ⊥ = e N ⊥ and N ⊤ = e N ⊤ . n -Dimensional strong fuzzy negations If an n -dimensional fuzzy negation N satisfies N3 N ( N ( x )) = x , ∀ x ∈ L n ([0 , n -dimensional strong fuzzy negation . Additionally, an n -dimensional fuzzynegation N is strict if it is continuous and strictly decreasing, i.e., N ( x ) < N ( y ) when y < x . Proposition 15
Let C k ( x ) = k √ − x k , for k ∈ { , , . . . , n } . Then C ( x ) = f C k ( x ) = ( C k ( π n ( x )) , . . . , C k ( π ( x ))) (20) is an n -representable strong fuzzy negation on ( L n ([0 , . Proof:
Let x ∈ [0 , N1: f C k ( / / ) = ( C k ( π n ( / / )) , . . . , C k ( π ( / / ))) = ( C k (0) , . . . , C k (0)) = / / f C k ( / / ) = ( C k ( π n ( / / )) , . . . , C k ( π ( / / ))) = ( C k (1) , . . . , C k (1)) = / / . N2:
Based on monotonicity of projection-functions, if x ≥ y then( C k ( π n ( x )) , . . . , C k ( π ( x ))) ≤ ( C k ( π n ( y )) , . . . , C k ( π ( y ))) , therefore, f C k ( x ) ≤ f C k ( y ) meaning that C ( x ) ≤ C ( y ). N3:
For all x ∈ L n ([0 , C ( C ( x )) = C ( f C k ( x )) = C ( C k ( π n ( x )) , . . . , C k ( π ( x )))= ( C k ( C k ( π ( x ))) , . . . , C k ( C k ( π n ( x ))))= x . Therefore, C = f C k is an n -representable strong fuzzy negation on L n ([0 , (cid:3) emma 16 Let N be an n -dimensional fuzzy negation. If N is strong then it is bijective. Proof:
Trivially if N is strong then it is injective and surjective. (cid:3) Proposition 17
Let N be an n -dimensional fuzzy negation. If N is strong then it is strict. Proof:
By Lemma 16, N is a strictly decreasing function. Therefore, if N is not continuous,then, by the continuity of R n , there exists y ∈ L n ([0 , x ∈ L n ([0 , (cid:3) Lemma 18
Let x , y ∈ L n ([0 , . Then, x ∨ y ∈ D n ( x ∧ y ∈ D n ) iff either x ∨ y = x or x ∨ y = y ( x ∧ y = x or x ∧ y = y ). Proof:
Straightforward from Eqs.(8) and (9). (cid:3)
Proposition 19
Let N be an n -dimensional strong fuzzy negation and x , y ∈ L n ([0 , .Then, the following holds:(i) N ( x ) = / / iff x = / / ;(ii) N ( x ) = / / iff x = / / ;(iii) N ( x ∨ y ) = N ( x ) ∧ N ( y ) ;(iv) N ( x ∧ y ) = N ( x ) ∨ N ( y ) . Proof:
Items ( i ) and ( ii ) are straightforward by Lemma 16 and N1 . ( iii ) Let N be an n -dimensional strong fuzzy negation and x , y ∈ L n ([0 , N ( x ∨ y ) ≤N ( x ) and N ( x ∨ y ) ≤ N ( y ). So, N ( x ∨ y ) ≤ N ( x ) ∧ N ( y ). Suppose that N ( x ∨ y ) < N ( x ) ∧ N ( y ). Then there exists z ∈ L n ([0 , N ( x ∨ y ) < z < N ( x ) ∧ N ( y )and therefore, z < N ( x ) and z < N ( y ). So, by Proposition 17, N ( z ) > N ( N ( x )) and N ( z ) > N ( N ( y )). Since N is strong we have that N ( z ) > x and N ( z ) > y . Hence, N ( z ) ≥ x ∨ y . Thus, by N2 and N3 , z ≤ N ( x ∨ y ) which is a contradiction and therefore N ( x ∨ y ) = N ( x ) ∧ N ( y ). ( iv ) Analogous to the above prove of item ( iii ). (cid:3) Lemma 20
Let N be an n -dimensional strong fuzzy negation. If for a x D n we have that N ( x ) ∈ D n then for some j = 1 , . . . , n − : x = (0 ( j ) , ( n − j ) ) , (21) where (0 ( j ) , ( n − j ) ) denotes (0 , . . . , | {z } j − times , , . . . , | {z } ( n − j ) − times ) . Proof:
Suppose that for some x D n , there exists z ∈ [0 ,
1] such that N ( x ) = /z/ . ByProposition 19, z ∈ (0 ,
1) and n ^ i =1 N ( e x i ) = /z/ once x = n _ i =1 e x i , where e x i = (0 ( i − , x ( n − i +1) i ).So by Lemma 18, /z/ = N ( f x k ) for some k = 1 , . . . , n . But, once N is bijective, then x = f x k .Analogously, since x = n ^ i =1 b x i , where b x i = ( x ( i ) i , ( n − i ) ), then by Proposition 19 we havethat n _ i =1 N ( b x i ) = /z/ . So by Lemma 18, /z/ = N ( e x j ) for some j = 1 , . . . , n . But, once N isbijective, then x = b x j . Hence, f x k = b x j and consequently k = j + 1. Therefore, the Equation(21) holds. (cid:3) heorem 21 Let N be an n -dimensional strong fuzzy negation. Then for each x D n , N ( x ) D n . Proof:
Let J = { j ∈ { , . . . , n − } : N ( x j ) ∈ D n } where x j = (0 ( j ) , ( n − j ) ). Observethat if j ≤ i then x i ≤ x j . If N ( x ) D n for some x D n , then by Lemma 20, J is afinite and not empty set. Let j = min J and z ∈ (0 ,
1) such that N ( x j ) = /z/ . For each y ∈ (0 , z ) we have that /y/ < /z/ and so, because N is strong, x j = N ( /z/ ) < N ( /y/ ).Therefore, since (0 ( j ) , ( n − j ) ) < N ( /y/ ), we have that N ( /y/ ) = ( a , . . . , a j , ( n − j ) ) for some a , . . . , a j ∈ [0 , i ∈ J we have that x i ≤ x j , then byLemma 20, N ( /y/ ) ∈ D n . Therefore, N ( /y/ ) = / / which is a contradiction with Proposition19. (cid:3) Corollary 22 If N is an n -dimensional strong fuzzy negation then N satisfies DP . Proof:
Suppose that there exists /x/ ∈ D n such that y = N ( /x/ )
6∈ D n . Since N is strong, N ( y ) = /x/ , i.e., N map a non-degenerate in a degenerate element which is a contradictionby Theorem 21. (cid:3) Lemma 23 [8, Theorem 3.2] Let N : L n ([0 , → L n ([0 , . N is an n -dimensional strongfuzzy negation satisfying the property DP iff there exists a strong fuzzy negation N such that N = e N . Theorem 24 N is an n -dimensional strong fuzzy negation iff there exists a strong fuzzynegation N such that N = e N . Proof:
Straightforward from Corollary 22 and Lemma 23. (cid:3) n -Dimensional equilibrium points Analogous to fuzzy negations, we will define an n -dimensional equilibrium point as the fol-lowing: Definition 25
An element e ∈ L n ([0 , is an n -dimensional equilibrium point for an n -dimensional fuzzy negation N if N ( e ) = e . Remark 3
Let N be a strict n -dimensional fuzzy negation. If x < e then N ( x ) > e and if e < x then N ( x ) < e . Proposition 26
Let N be a fuzzy negation with the equilibrium point e . Then, /e/ is an n -dimensional equilibrium point of e N . Proof:
Straightforward. (cid:3)
Corollary 27
Let N be an n -dimensional strong fuzzy negation. Then, there exists an ele-ment /e/ ∈ D n such that /e/ is an n -dimensional equilibrium point of N . Proof:
Straightforward from Corollary 22, Theorem 24, and Proposition 26. (cid:3)
Corollary 28
Let N be the strong fuzzy negation and e ∈ (0 , . Then, /e/ is an n -dimensional equilibrium point of e N iff e is an equilibrium point of N . Proof:
Straightforward from Theorem 24, and Proposition 26. (cid:3) emark 4 For k ∈ { , , . . . , n } , consider the strong fuzzy negation given by Eq.(2) in Ex-ample 1, C k ( x ) = n − k +1 √ − x n − k +1 and its corresponding equilibrium point n − k +1 √ . . Bytaking x = ( n − k +1 √ . , n − k +1 √ . , . . . , n − k +1 √ . ∈ L n ([0 , , one can observe that C ( x ) = f C k ( x ) = ( C k ( π n ( x )) , . . . , C k ( π ( x )))= ( n − k +1 √ . , n − k +1 √ . , . . . , n − k +1 √ . x , meaning that such operator preserves distinct equilibrium point of component-functions C k ofan n -representable fuzzy negation C = f C k . Remark 5
For k ∈ { , , . . . , n } , consider the fuzzy negation given by Eq.(3) in Proposi-tion 1, C k ( x ) = 1 − x k and its corresponding equilibrium point e k . By N2, e ≤ . . . ≤ e n ,then x = ( e , . . . , e n ) ∈ L n ([0 , . So, it is immediate observing that C ( x ) = ^ C . . . C n ( x ) = ( C ( π n ( x )) , . . . , C n ( π ( x )))= ( e , . . . , e n )= x , meaning that such operator preserves the equilibrium points of component-functions C , . . . , C of the n -representable fuzzy negation C = ^ C . . . C n . n -Dimensional automorphisms In this section we briefly recall some well-known results of automorphism on L ([0 , n -dimensional approach, mainly connected to representablefuzzy negation on L n ([0 , N -preserving n -dimensional fuzzy automorphismis also discussed.In [8], an n -dimensional automorphism is defined as follows: Definition 29
A function ϕ : L n ([0 , → L n ([0 , is an n -dimensional automorphism if ϕ is bijective and the following condition is satisfied x ≤ y iff ϕ ( x ) ≤ ϕ ( y ) . Theorem 30 [8, Theorem 3.4] Let ϕ : L n ([0 , → L n ([0 , . A function ϕ ∈ Aut ( L n ([0 , iff there exists ψ ∈ Aut ([0 , such that ϕ ( π ( x ) , . . . , π n ( x )) = ( ψ ( π ( x )) , . . . , ψ ( π n ( x ))) .In this case, we will denote ϕ by e ψ . Thus, the following holds e ψ ( π ( x ) , . . . , π n ( x )) = ( ψ ( π ( x )) , . . . , ψ ( π n ( x ))) . (22) Corollary 31 If ϕ ∈ Aut ( L n ([0 , then it is continuous, strictly increasing, ϕ ( / / ) = / / and ϕ ( / / ) = / / . Proposition 32 [8, Proposition 3.4] Let ψ ∈ Aut ([0 , . Then, the following holds: g ψ − = e ψ − . (23)12 roposition 33 Let ϕ ∈ Aut ( L n ([0 , . N is n -dimensional (strict, strong) fuzzy negationiff N ϕ is an n -dimensional (strict, strong) fuzzy negation such that, for all x ∈ L n ([0 , ,the following holds: N ϕ ( x ) = ϕ − ( N ( ϕ ( x ))) . Proof: ( ⇒ ) N1 : Let N be an n -dimensional fuzzy negation. Then, the following holds: N ϕ ( / / ) = ϕ − ( N ( ϕ ( / / ))) = ϕ − ( N ( / / ))= ϕ − ( / / ) = / /. Analogously we proof that N ϕ ( / / ) = / / . N2 : If x ≤ y then ϕ ( x ) ≤ ϕ ( y ) and the following holds: N ( ϕ ( x )) ≥ N ( ϕ ( y )) ⇒ ϕ − ( N ( ϕ ( x ))) ≥ ϕ − ( N ( ϕ ( y ))) ⇒ N ϕ ( x ) ≥ N ϕ ( y ) . Therefore, N ϕ is an n -dimensional fuzzy negation. In addition, if N is strictly decreasingthen, as the n -dimensional automorphism, trivially, N ϕ is strictly decreasing. If N is continu-ous then, by Corollary 31, ϕ and ϕ − are continuous. Since the composition of the continuousfunction is continuous, then N ϕ is continuous. Thus, if N is strict, then N ϕ is also strict.Moreover, if N is an n -dimensional strong fuzzy negation then the following holds: N ϕ ( N ϕ ( x )) = N ϕ ( ϕ − ( N ( ϕ ( x ))))= ϕ − ( N ( ϕ ( ϕ − ( N ( ϕ ( x ))))))= ϕ − ( N ( N ( ϕ ( x ))))= ϕ − ( ϕ ( x ))= x . ( ⇐ ) Let N ϕ be an n -dimensional (strict, strong) fuzzy negation. By the above proof,( N ϕ ) ϕ − also is an n -dimensional (strict, strong) fuzzy negation. Since N = ( N ϕ ) ϕ − , then N is an n -dimensional (strict, strong) fuzzy negation. (cid:3) Proposition 34
Let N . . . N n be fuzzy negations and ψ be an automorphism. Then, thefollowing holds: ^ N . . . N n e ψ = ^ N ψ . . . N ψn . Proof:
For all x ∈ L n ([0 , ^ N ψ . . . N ψn ( x ) = ( N ψ ( π n ( x )) , . . . , N ψn ( π ( x ))) by Eq.(5)= ( ψ − ( N ( ψ ( π n ( x )))) , . . . , ψ − ( N n ( ψ ( π ( x ))))) by Eq.(22)= g ψ − ( N ( ψ ( π n ( x ))) , . . . , N n ( ψ ( π ( x )))) by Eq.(12)= g ψ − ( ^ N , . . . , N n ( ψ ( π ( x )) , . . . , ψ ( π n ( x )))) by Eq.(22)= g ψ − ( ^ N , . . . , N n ( e ψ ( π ( x ) , . . . , π n ( x )))) by Eq.(23)= e ψ − ( ^ N , . . . , N n ( e ψ ( π ( x ) , . . . , π n ( x )))) by Prop. 32= ^ N . . . N n e ψ ( x ) . Therefore, Proposition 34 is verified. (cid:3) emma 35 [8, Corollary 3.1] A function N : L n ([0 , → L n ([0 , is an n -dimensionalstrong fuzzy negation satisfying the property DP iff there exists an n -dimensional automor-phism ϕ such that N = N ϕS , where N S ( x ) = (1 − π n ( x ) , − π n − ( x ) , . . . , − π ( x )) . The following theorem is a generalization of the Trillas theorem [26] and a generalizationfor the interval case given by Bedregal in [5].
Theorem 36
A function N : L n ([0 , → L n ([0 , is an n -dimensional strong fuzzy nega-tion iff there exists an n -dimensional automorphism ϕ such that N = N ϕS . Proof:
Straightforward from Corollary 22 and Lemma 35. (cid:3)
Proposition 37
Let N be an n -dimensional strict (strong) fuzzy negation and the n -dimensionalautomorphism ϕ ( x ) = x , i.e., ϕ ( x ) = (( π ( x )) , . . . , ( π n ( x )) ) . Then, N < N ϕ and ( N ϕ ) − < N . Proof:
Clearly, ϕ − ( x ) = √ x , i.e., ϕ − ( x ) = ( p π ( x ) , . . . , p π n ( x )). Since x < x foreach x ∈ L n ([0 , − { / /, / / } , then because N is strict we have that N ( x ) < N ( x ).So, ϕ − ( N ( x )) < ϕ − ( N ( ϕ ( x ))) = N ϕ ( x ). But, since x < √ x for each x ∈ L n ([0 , −{ / /, / / } , then N ( x ) < p N ( x ). Therefore, N ( x ) < N ϕ ( x ). Analogously we proof that( N ϕ ) − < N . (cid:3) Corollary 38
There exists neither a lesser nor greater n -dimensional strict (strong) fuzzynegation. Proof:
Straightforward from Proposition 37. (cid:3) N -Preserving n -dimensional automorphisms Let N be an n -dimensional fuzzy negation. An n -dimensional automorphism ϕ is N -preserving n -dimensional automorphism if, for each x ∈ L n ([0 , ϕ ( N ( x )) = N ( ϕ ( x )) . (24)The following theorem shows us that N -Preserving n -dimensional automorphisms arestrongly related with the notion of N -preserving automorphisms. Theorem 39
Let ϕ be an n -dimensional automorphism, N be a representable n -dimensionalfuzzy negation, ψ be the automorphism such that ϕ = e ψ and N , . . . , N n be fuzzy negationssuch that N = ^ N . . . N n . Then, ϕ is a N -preserving n -dimensional automorphism iff ψ isan N i -preserving automorphism, for each i = 1 , . . . , n . Proof: ( ⇒ ) Let x ∈ [0 , ψ ( N i ( x )) = ψ ( π i ( N ( /x/ ))) by Eq.(18)= π i ( e ψ ( N ( /x/ ))) by Eq.(22)= π i ( N ( e ψ ( /x/ ))) by Eq.(24)= π i ( N ( /ψ ( x ) / )) by Eq.(22)= N i ( ψ ( x )) by Eq.(18)14 ⇐ ) Let x ∈ L n ([0 , e ψ ( N ( x )) = e ψ ( ^ N . . . N n ( x ))= e ψ ( N ( π n ( x )) , . . . , N n ( π ( x ))) by Eq.(12)= ( ψ ( N ( π n ( x ))) , . . . , ψ ( N n ( π ( x )))) by Eq.(22)= ( N ( ψ ( π n ( x ))) , . . . , N n ( ψ ( π ( x )))) by Eq.(24)= ^ N . . . N n ( ψ ( π ( x )) , . . . , ψ ( π n ( x ))) by Eq.(12)= ^ N . . . N n ( e ψ ( x )) by Eq.(22)= N ( e ψ ( x ))Therefore, Proposition 39 holds. (cid:3) The following theorem is an n -dimensional version of Proposition 2 which extends [22,Proposition 4.2] and [5, Proposition 7.5] for interval case. It provides an expression for all N -preserving n -dimensional automorphisms in L n ([0 , Theorem 40
Let N be an n -dimensional strong fuzzy negation with /e/ as the degenerateequilibrium point and ϕ be an n -dimensional automorphism on L n ([0 , e ]) = { x ∈ L n ([0 , π n ( x ) ≤ e } . Then, ϕ N : L n ([0 , → L n ([0 , defined by ϕ N ( x )= ϕ ( x ) if x ≤ /e/ N ( ϕ ( N ( x ))) if x > /e/ ( π ( ϕ ( x )) , . . . ,π i ( ϕ ( x )) ,π i +1 ( N ( ϕ ( N ( x )))) , . . . ,π n ( N ( ϕ ( N ( x ))))) if π i ( x ) ≤ e < π i +1 ( x ) (25) is an N -preserving n -dimensional automorphism. Proof:
By Theorem 30, there exists an automorphism ψ such that ϕ = e ψ . Analogously, byTheorem 24, there exists a strong fuzzy negation N such that N = e N . Thus, it holds thatIf x = /e/ and since /e/ = N ( /e/ ), then N ( /e/ ) = N ( x ). So, ϕ N ( N ( x )) = ϕ ( N ( x )) since N ( x ) = /e/ = N ( ϕ ( x ) by Eq. (24)= N ( ϕ N ( x )) since x = /e/ If x < /e/ , then since N is strict, /e/ = N ( /e/ ) < N ( x ) and so, ϕ N ( N ( x )) = N ( ϕ ( N ( N ( x )))) since N ( x ) > /e/ = N ( ϕ ( x )) since N is strong= N ( ϕ N ( x )) since x < /e/ If x > /e/ then since N is strict, N ( x ) < /e/ implying new results as follows ϕ N ( N ( x )) = ϕ ( N ( x )) = N ( N ( ϕ ( N ( x )))) since N is strong . = N ( ϕ N ( x )) since x > /e/ If π i ( x ) < e < π i +1 ( x ) then N ( π i +1 ( x )) < N ( e ) < N ( π i ( x )) and by Corollary 28, N ( e ) = e . All definitions and results described in the beginning of this section (until Proposition 5.1) can be adaptedfor L n ([0 , e ]).
15n addition,( π ( ϕ ( x )) , . . . ,π i ( ϕ ( x )) ,π i +1 ( N ( ϕ ( N ( x )))) , . . . ,π n ( N ( ϕ ( N ( x ))))) (26)= ( ψ ( π ( x )) , . . . , ψ ( π i ( x )) , N ( π n − i ( ϕ ( N ( x )))) , . . . ,N ( π ( ϕ ( N ( x )))))= ( ψ ( π ( x )) , . . . , ψ ( π i ( x )) , N ( ψ ( π n − i ( N ( x )))) , . . . ,N ( ψ ( π ( N ( x )))))= ( ψ ( π ( x )) , . . . , ψ ( π i ( x )) , N ( ψ ( N ( π i +1 ( x )))) , . . . , N ( ψ ( N ( π n ( x ))))) . So, the following holds ϕ N ( N ( x ))= ( ψ ( π ( N ( x ))) , . . . , ψ ( π n − i ( N ( x ))) , N ( ψ ( N ( π n − i +1 ( N ( x ))))) , . . . , N ( ψ ( N ( π n ( N ( x ))))))by Eq.(26)= ( ψ ( N ( π n ( x ))) , . . . , ψ ( N ( π i +1 ( x ))) , N ( ψ ( N ( N ( π i ( x ))))) , . . . , N ( ψ ( N ( N ( π ( x ))))))= ( N ( N ( ψ ( N ( π n ( x ))))) , . . . , N ( N ( ψ ( N ( π i +1 ( x ))))) , N ( ψ ( π i ( x ))) , . . . , N ( ψ ( π ( x )))since N is strong= e N ( ψ ( π ( x )) , . . . , ψ ( π i ( x )) , N ( ψ ( N ( π i +1 ( x )))) , . . . , N ( ψ ( N ( π n ( x ))))) by Eq.(12)= N ( ψ ( π ( x )) , . . . , ψ ( π i ( x )) , N ( ψ ( N ( π i +1 ( x )))) , . . . , N ( ψ ( N ( π n ( x )))))= N ( π ( ϕ ( x )) , . . . , π i ( ϕ ( x )) , π i +1 ( N ( ϕ ( N ( x )))) , . . . , π n ( N ( ϕ ( N ( x ))))) by Eq.(26),based on results of Theorems 30 and 24= N ( ϕ N ( x )) . If π j ( x ) < π j +1 ( x ) = . . . = π i ( x ) = e < π i +1 ( x ) then the following holds N ( π i +1 ( x )) < e = N ( π i ( x )) = . . . = N ( π j +1 ( x )) < N ( π j ( x ))with j + 1 ≤ i and j ≥
0. Hence, the equations as follows are verified: ϕ N ( N ( x ))= ( ψ ( π ( N ( x ))) , . . . , ψ ( π n − i ( N ( x ))) , e, . . . , e, | {z } i − ( j +1) times N ( ψ ( N ( π n − j +1 ( N ( x ))))) , . . . , N ( ψ ( N ( π n ( N ( x ))))))by Eq.(26)= ( ψ ( N ( π n ( x ))) , . . . , ψ ( N ( π i +1 ( x ))) , e, . . . , e, | {z } i − ( j +1) times N ( ψ ( N ( N ( π j ( x ))))) , . . . , N ( ψ ( N ( N ( π ( x ))))))since N is strong= ( N ( N ( ψ ( N ( π n ( x ))))) , . . . , N ( N ( ψ ( N ( π i +1 ( x ))))) , e, . . . , e, | {z } i − ( j +1) times N ( ψ ( π j ( x ))) , . . . , N ( ψ ( π ( x ))))since N is strong= e N ( ψ ( π ( x ))) , . . . , ψ ( π j ( x )) , ψ ( e ) , . . . , ψ ( e ) , | {z } i − ( j +1) times N ( ψ ( N ( π i +1 ( x )))) , . . . , N ( ψ ( N ( π n ( x )))) by Eq.(12)= N ( ψ ( π ( x )) , . . . , ψ ( π i ( x )) , N ( ψ ( N ( π i +1 ( x )))) , . . . , N ( ψ ( N ( π n ( x )))))= N ( π ( ϕ ( x )) , . . . , π i ( ϕ ( x )) , π i +1 ( N ( ϕ ( N ( x )))) , . . . , π n ( N ( ϕ ( N ( x ))))) by Eq.(26),based on results of Theorem 40= N ( ϕ N ( x )) . Therefore, ϕ N is N -preserving n -dimensional automorphism. Now we will proof thatall N -preserving n -dimensional automorphisms have the form of Equation (25). Suppose16hat there exists an N -preserving n -dimensional automorphism ϕ ′ : L n ([0 , → L n ([0 , ψ ′ : [0 , → [0 ,
1] defined by ψ ′ ( x ) = π ( ϕ ′ ( /x/ )) is an N -preservingautomorphism. But, by Proposition 2, there exists an automorphism ψ ′′ : [0 , e ] → [0 , e ] suchthat ψ ′ = ψ ′′ N . Let ϕ ′′ = e ψ ′′ . Hence, if x ≤ /e/ , then π i ( x ) ≤ e and so ϕ ′ ( x ) = ( ψ ′ ( π ( x )) , . . . , ψ ′ ( π n ( x ))) based on results of Theorem 39= ( ψ ′′ N ( π ( x )) , . . . , ψ ′′ N ( π n ( x ))) based on results of Proposition 2= ( ψ ′′ ( π ( x )) , . . . , ψ ′′ ( π n ( x ))) by Eq.(5)= e ψ ′′ ( π ( x ) , . . . , π n ( x )) by Eq.(22)= ϕ ′′ ( x )= ϕ ′′N ( x ) by Eq.(25)If /e/ < x , then ϕ ′ ( x ) = ( ψ ′ ( π ( x )) , . . . , ψ ′ ( π n ( x ))) based on results of Theorem 39= ( ψ ′′ N ( π ( x )) , . . . , ψ ′′ N ( π n ( x ))) based on results of Proposition 2= ( N ( ψ ′′ ( N ( π ( x )))) , . . . , N ( ψ ′′ ( N ( π n ( x ))))) by Eq.(4)= e N ( ψ ′′ ( N ( π n ( x ))) , . . . , ψ ′′ ( N ( π ( x )))) by Eq.(12)= N ( ψ ′′ ( N ( π n ( x ))) , . . . , ψ ′′ ( N ( π ( x )))) based on Theorem 24= N ( e ψ ′′ ( N ( π n ( x )) , . . . , N ( π ( x )))) by Eq.(22)= N ( ϕ ′′ ( N ( π n ( x )) , . . . , N ( π ( x )))) based on results of Theorem 30= N ( ϕ ′′ ( e N ( π ( x )) , . . . , π n ( x ))) by Eq.(12)= N ( ϕ ′′ ( N ( x )))) based on results of Theorem 24= ϕ ′′N ( x ) by Eq.(25)If π i ( x ) ≤ /e/ < π i +1 ( x ) then ϕ ′ ( x ) = ( ψ ′ ( π ( x )) , . . . , ψ ′ ( π n ( x ))) based on Theorem 39= ( ψ ′′ N ( π ( x )) , . . . , ψ ′′ N ( π n ( x ))) based on results of Proposition 2= ( ψ ′′ ( π ( x )) , . . . , ψ ′′ ( π i ( x )) , N ( ψ ′′ ( N ( π i +1 ( x )))) , . . . , N ( ψ ′′ ( N ( π n ( x )))))= ( π ( ϕ ′′ ( x )) , . . . , π i ( ϕ ′′ ( x )) , π i +1 ( N ( ϕ ′′ ( N ( x )))) , . . . , π n ( N ( ϕ ′′ ( N ( x ))))) by Eq.(26)= ϕ ′′N ( x ) . Therefore, ϕ ′ = ϕ ′′N , i.e., all N -preserving n -dimensional automorphisms have the form ofEquation (25). (cid:3) The following result is analogous to Proposition 3.
Proposition 41
Let N be an n -dimensional strong fuzzy negation. Then, ( ϕ N ) − is an N -preserving n -dimensional automorphism. Proof:
By Theorem 40, ϕ N is an N -preserving n -dimensional automorphism. Let x ∈ L n ([0 , ϕ N ) − ( N ( x )) = ( ϕ N ) − ( N ( ϕ N (( ϕ N ) − ( x ))))= ( ϕ N ) − ( ϕ N ( N (( ϕ N ) − ( x )))) by Eq.(24)= N (( ϕ N ) − ( x ))Therefore, by Eq.(24), ( ϕ N ) − is also an N -preserving n -dimensional automorphism. (cid:3) Conclusion
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