key: cord-0846695-8x619wua
authors: Zanotelli, Rosana; Reiser, Renata; Bedregal, Benjamin
title: n-Dimensional (S,N)-implications
date: 2020-08-12
journal: Int J Approx Reason
DOI: 10.1016/j.ijar.2020.07.002
sha: daf324b6a0a9aef387c23b216afbe794d530a3cb
doc_id: 846695
cord_uid: 8x619wua

The n-dimensional fuzzy logic (n-DFL) has been contributed to overcome the insufficiency of traditional fuzzy logic in modelling imperfect and imprecise information, coming from different opinions of many experts by considering the possibility to model not only ordered but also repeated membership degrees. Thus, n-DFL provides a consolidated logical strategy for applied technologies since the ordered evaluations provided by decision makers impact not only by selecting the best solutions for a decision making problem, but also by enabling their comparisons. In such context, this paper studies the n-dimensional fuzzy implications (n-DI) following distinct approaches: (i) analytical studies, presenting the most desirable properties as neutrality, ordering, (contra-)symmetry, exchange and identity principles, discussing their interrelations and exemplifications; (ii) algebraic aspects mainly related to left- and right-continuity of representable n-dimensional fuzzy t-conorms; and (iii) generating n-DI from existing fuzzy implications. As the most relevant contribution, the prospective studies in the class of n-dimensional interval (S,N)-implications include results obtained from t-representable n-dimensional conorms and involutive n-dimensional fuzzy negations. And, these theoretical results are applied to model approximate reasoning of inference schemes, dealing with based-rule in n-dimensional interval fuzzy systems. A synthetic case-study illustrates the solution for a decision-making problem in medical diagnoses.

Zadeh introduced in 1975 the type-n fuzzy sets [56] (Tn-FSs) whose relevance emerges from the insufficiency of the traditional fuzzy logic (FL) in modeling inherent imperfect information related to distinct opinions of specialists in order to define antecedent and consequent of membership functions in inference systems [20] . Currently, many extensions of fuzzy sets are known, e.g. L-fuzzy sets as proposed by Goguen [29] , and the Hesitant Fuzzy Sets introduced by Torra [49, 50] .

In [47] , the notion of an n-dimensional fuzzy set (n-DFS) on L n -fuzzy set theory was introduced by Shang as a special class of Tn-FSs, generalizing the theories underlying many other multivalued fuzzy logics: the Interval-valued Fuzzy Sets [30, 46] , the Intuitionistic Fuzzy Sets [2, 3] and the Interval-valued Intuitionistic Fuzzy Sets [5] . In L n -fuzzy set theory [47] , the n-dimensional fuzzy sets membership values are n-tuples of real numbers on U = [0, 1], ordered in increasing order and called n-dimensional intervals.

Lately, in [15] , Bedregal et al. notice that in most applications the Typical Hesitant Fuzzy Elements (THFE) are used, i.e., considering finite and non-empty subsets of unitary interval (U = [0, 1]) as hesitant fuzzy degrees. In addition, even when the repetition of element in n-tuples on the hesitant membership

In [47] , the definitions of cut set on an n-dimensional fuzzy set and its corresponding n-dimensional vector level cut set are presented according to Zadeh fuzzy set approach. It also studies the decomposition and representation theorems of the n-dimensional fuzzy sets.

The construction of bounded lattice negations from bounded lattice t-norms is considered in [13] , together with a discussion under which these connective conditions are preserved by automorphisms and corresponding conjugate negations and t-norms.

In [12] , the authors consider the study of aggregation operators for these new concepts of n-dimensional fuzzy sets, starting from the usual aggregation operator theory and also including a new class of aggregation operators containing an L n (U )-extension of the OWA operator. The results presented in such context allow to extend fuzzy sets to interval-valued Atanassov's intuitionistic fuzzy sets and also preserve their main properties.

The results in [38] provide the class of representable n-dimensional strict fuzzy negations, i.e., an ndimensional strict fuzzy negation which is determined by strict fuzzy negations.

The authors in [39] and [16] consider the definitions and results obtained for n-dimensional fuzzy negations, applying these studies mainly on natural n-dimensional fuzzy negations for n-dimensional t-norms and n-dimensional t-conorms. And, in [40] Moore Continuous n-dimensional interval fuzzy negations are also discussed.

In [37] the triples formed by a t-norm, t-conorm and standard complement is called De Morgan triples if it fulfills De Morgan laws. Some new important results about t-norm and t-conorm theory are discussed and many of them are not readily found in the literature.

More recently, we can highlight an n-dimensional interval extension of uninorms in [41] , a preliminary study in the class of n-dimensional R-implications obtained from representable n-dimensional t-norms are discussed in [58] and the inference schemes making use of n-dimensional fuzzy logic in [59] .

Following the results above cited, this paper studies the possibility of dealing with main properties of representable n-dimensional S-implications on L n (U ), exploring their main properties.

The remaining of the paper is set as follows. Section2 introduces some definitions needed throughout this paper, reporting the main characteristics of fuzzy negations, t-conorms and fuzzy implications.

The concepts structuring the distributive complete lattice L n (U ) of n-dimensional fuzzy set are reported in Section 3, focusing on the supremum and infimum, both defined w.r.t. the partial natural order, also covering the projection operators and degenerate elements such as the top and botton elements. In addition, an n-dimensional automorphism on L(U ) and their well-known results are both reported.

In Section 4, fuzzy negations on L n (U ) are briefly discussed based on extensions of the main results from [12] , including the class of representale and conjugate n-dimensional fuzzy negations.

Section 5 is devoted to the new propositions discussing main properties n-dimensional fuzzy t-conorms, dual and conjugate constructions,projections and examples.

The core of the paper sits in the next three section. Firstly, in Section 6, the development of the concepts and reasonable properties of n-dimensional fuzzy implications on L n (U ) such as the Moore-continuity, as well as the evidence on properties assuring representability of n-DFI are presented. This section also considers new specific results in the analysis of conjugation operators. In sequence, Section 7 concerns the study of n-dimensional interval fuzzy (S, N )-implications, main characterization of such operators, duality and action of n-dimensional automorphisms. And, Section 7, exploring n-dimensional fuzzy (S, N )-implication in AR, presenting inference schemes, compositional rule-base and exemplification.

The Conclusion highlights main results and briefly comments on further work.

In this section, we will briefly review some basic concepts of FL, concerning the study of n-dimensional intervals, which can be found in [11] and [16] .

A function N : U → U is a fuzzy negation (shortly FN) if N1 N (0) = 1 and N (1) = 0;

N2 If x ≥ y then N (x) ≤ N (y), ∀x, y ∈ U .

And, a continuous FN is strict [31] , when N3a x > y then N (x) < N(y), ∀x, y ∈ U .

Involutive FNs are called strong FN (shortly SFN):

Definition 2.1. Let N be a FN and f : U n → U be a real function. The N −dual function of f is given by the expression:

where

A function S : U 2 → U is a triangular-conorm (t-conorm) if and only if it satisfies, for all x, y, z ∈ U , the following properties. S1 : S(x, 0) = x (neutral element); S2 : S(x, y) = S(y, x) (commutativity); S3 : S(x, S(y, z)) = S(S(x, y), z) (associativity);

The notion of a triangular t-norm T : U 2 → U can be analogously defined by properties from T2 to T4, with the property S1 replaced by T1: T (x, 1) = x, for all x, y, z ∈ U . 

The following comparisons can be requested:

(i) By [32] , the following holds:

for any t-conorm S and S and fuzzy negation N , we have that Let I(L(U )) be the family of fuzzy implication on L(U ).

Several reasonable properties may be required for fuzzy implications. The properties considered in this paper are listed below and have been extensively studied, see more details in [8, 23, 48] : 

An (S, N )-implication I S,N : U 2 → U is defined by the expression:

whenever S is a t-conorm and N is a fuzzy negation. This function is a fuzzy implication which generalizes the following classical logical equivalence: p → q ≡ ¬p ∨ q. When N is a strong fuzzy negation, then I S,N is a strong implication referred as S-implication. The name S-implication was firstly introduced in the fuzzy logic framework by [51] . 

In [47] , You-guang Shang et al. introduce a new extension of fuzzy sets, namely n-dimensional fuzzy sets in order to generalize in a natural way other two extensions: Interval-valued fuzzy sets [56, 46, 43] and 3-dimensional fuzzy sets [33] . In sequence, Benjamin Bedregal et al. proposed in [12] the following alternative definition for n-dimensional fuzzy sets:

Let X be a nonempty set, U = [0, 1], n ∈ N − {0} and N n = {1, 2, . . . , n}. An n-dimensional fuzzy set A over X is given by

when, for i = 1, . . . , n, the i-th membership degree of A denoted as μ Ai : X → U verifies the condition

In [11] , for n ≥ 1, n-dimensional upper simplex is given as

and its elements are called n-dimensional intervals. For each i = 1, . . . , n, the function π i :

so, a degenerate element (x, . . . , x) ∈ L n (U ) will be denoted by /x/.

Remark 3.1. The natural order also called the product order on L n (U ) is defined for each x, y ∈ L n (U ), as following:

x ≤ y if and only if π i (x) ≤ π i (y), ∀i ∈ N n .

In addition, (L n (U ), ≤) is a distributive complete lattice [11] . Additionally, for each i = 1, . . . , n and for all x, y ∈ L n (u) the following partial order is also considered

Moreover, one can easily observe that is more restrictive than ≤, meaning that x y ⇒ x ≤ y.

According to Bedregal et al. in [12] , L n (U ) = (L n (U ), ∨, ∧, /0/, /1/) is a distributive complete lattice with /0/ and /1/ being their bottom and top element, respectively, and ∨ and ∧ the supremum and infimum w.r.t. the product order. By [11] , for all x, y ∈ L n (U ), the supremum and infimum on L n (U ) is given as:

x ∨ y = (max(π 1 (x), π 1 (y)), . . . , max(π n (x), π n (y))) (7)

x ∧ y = (min(π 1 (x), π 1 (y)), . . . , min(π n (x), π n (y))).

According to [16] and [44] , an n-dimensional automorphism on L(U ) and their well-known results are both reported below: Definition 3.1. A function ϕ : L n (U ) → L n (U ) is an n-dimensional automorphism, (n-DA) if ϕ is bijective and the following condition is satisfied

The family of all automorphism on U and L n (U ) are denoted by Aut(U ) and Aut(L n (U )), respectively. Proposition 3.1. [12, Theorem 3.4 ] Given a function ϕ : L n (U ) → L n (U ), ϕ ∈ Aut(L n (U )) if and only if there exists ψ ∈ Aut(U ) such that ϕ(x) = (ψ(π 1 (x)), . . . , ψ(π n (x))), ∀x ∈ L n (U )

and, in this case, denote ϕ by ψ.

Corollary 3.1. Each n-DA is continuous and strictly increasing.

According with [12, Proposition 3.4] , given a ψ ∈ Aut(U ), we have that ψ −1 = ψ −1 and therefore ψ −1 ∈ Aut(L n (U )), i.e. the inverse of n-dimensional automorphism always exists and it is also an n-dimensional automorphism.

Moreover, when ϕ ∈ Aut(L n (U )) and F, F ϕ : (L n (U )) k → L n (U ), the function F ϕ is called the conjugate of F if for each x 1 , . . . , x k ∈ L n (U ) is verified that

The notion of fuzzy negation on U was extended to L n (U ) in [12] , as follows: Based on [38] , a n-DN N is strict if it is a continuous function 2 verifying the strict inequality:

Additionally, N is a strong n-DN if N verifies the involutive property:

In [14, Prop. 3.8] was proved that each strong n-DN is also strict. 

Notice that N S and N R are strong n-DN whereas N SR and N S 2 are strict n-DN. Moreover, N ⊥ is a non-continuous n-DN. In addition, one can observe that not all n-DN has a right or left inverse, e.g. N ⊥ . In addition, a (left) right inverse of a n-DN N , if there exists one, it can not be an n-DN. Indeed, consider the n-DN N 1 : L n (U ) → L n (U ) given as follows:

The function N 1 is the right inverse of the function N 2 (x) = N S (/0.8/ · x + /0.1/), which is not an n-DN because N 2 (/0/) = /0.9/.

So, the results from Remark4.1 motivate us to the following definition of a (left) right invertible operator: 

is a representable n-DN and (N 1 , . . . , N n ) their representants. Proof. (⇒) Suppose that N −r be the n-DN which is the right inverse of N . By Proposition 4.2, (N −r ) (n−i+1) , is a fuzzy negation for each i ∈ N n . In addition, for each x ∈ [0, 1], (11) , the following holds: N ( N −r n . . . N −r 1 (x)) = N (N −r n (π n (x)), . . . , N −r 1 (π 1 (x))) = (N (1) (N −r 1 (π 1 (x))), . . . , N (n) (N −r n (π n (x)))) = x, for each x ∈ L n (U ). Proof. Since N is a strict and representable n-DN, then by Remark 4.2, N (x) = (N (1) , . . . , N (n) )(x) and for each i ∈ N n , N (i) is a strict fuzzy negation and therefore has an inverse N −1

. So, by Proposition 4.1 and Remark 4.2, N −1 (1) . . . N −1 (n) is an strict representable n-DN. Then, we obtain that

The family of all n-DN will be denoted by N (L n (U )). Let N be fuzzy negations and N . . . N will be denoted just as N . 

In [12] , the notion of aggregation function was extended for n-dimensional intervals, as follows:

Definition 5.1. [12] Let m and n be positive natural numbers such that m ≥ 2. A function P : (L n (U )) m → L n (U ) is a n-dimensional m-ary aggregation function, if P (/0/, . . . , /0/) = /0/, P (/1/, . . . , /1/) = /1/ and for each

Based on the relevance of the t-norm and t-conorm classes as bivariate aggregation operators, their extension on L n (U ) were presented in [39] . Thus, their main concepts and results are reported as follows:

for all x, y, z ∈ L n (U ) , the following properties:

y) (monotonicity related to the product order in Eq. (5)).

Let S be a n-DS and N a n-DN. A pair (S, N ) satisfies the law of excluded middle (LEM) if

Analogously, an n-dimensional t-norm (n-DT) T : L n (U ) 2 → L n (U ) has /1/ as the neutral element, is commutative, associative and a monotonic function with respect to the product order.

According to [12] , the conditions under which an n-DS can be obtained from a finite subset of t-conorm S i : U 2 → U , for i ∈ N n−1 , are reported below.

and

are, respectively, an n-DT and n-DS called as representable operators.

Proof. Trivially, S N is commutative and has /1/ as neutral element. If y ≤ z then N (z) ≤ N (y) and therefore,

So, S N is associative and therefore is n-DT. In addition, since N is a strong n-DN by Theorem 4.1, there exists a strong fuzzy negation N such that

, N (y))) = (N (S 1 (N (π 1 (x)), N(π 1 (y)))), . . . , N(S n (N (π n (x)), N(π n (y))))

So, by Remark 2.1 and Proposition 5.1, S N is a representable n-DT.

Let S be a t-conorm and T be a t-norm. We will denote S . . . S and T . . . T just as S and T , respectively. Proof. Suppose that S = S 1 . . . S n and S = S 1 . . . S n . Then, from Eq. (14) , for each x, y ∈ U , S i (x, y) = π i (S(/x/, /y/)) = S i (x, y).

Then, for i ∈ N n , the function S (i) : U 2 → U given by

Proof. Since S is a representable n-DS then there exist t-conorms S 1 , . . . , S n such that S = S 1 . . . S n . The proposition follows, once clearly S (i) = S i for each i ∈ N n . In fact, for each x, y ∈ U , S (i) (x, y) = π i (S(/x/, /y/)) = S i (x, y). Therefore, Proposition 5.4 is verified. Proposition 5.5. [12, Theorem 3.6 ] Let S be a n-DS and ϕ be an n-DA. Then S ϕ is also a n-DS.

Then, Proposition 5.6 is verified.

Since, each n-dimensional t-norm T and t-conorm S are associative operators, then for each natural number m ≥ 2, they can be naturally extended for an m-ary n-dimensional aggregation function, as follows:

respectively.

This section studies n-dimensional fuzzy implications on the lattice (L n (U ), ≤) introduced in [57] extending this work investigating construction methods of n-dimensional fuzzy implications from fuzzy implications preserving their main properties. Additionally, if n=2, the n-dimensional fuzzy implications are the usual interval-valued fuzzy implications as investigated in [1, 10, 17] and therefore, their corresponding properties are investigated in the more general n-dimensional interval space. 2. An n-dimensional fuzzy implicator I is a n-dimensional fuzzy implication (n-DI), if it also satisfies the properties:

We also consider the following extra properties for n-DIs: I12: x ≤ y ⇔ I(x, y) = /1/ (ordering principle);

Moreover, additional conditions are required in I13, meaning that new properties related to natural negations can be discussed as follows: Concluding, Proposition 6.2 is verified.

. Therefore, Proposition 6.3 is verified.

Proposition 6.4. If an n-DI I satisfies I5 and I12 then for each x ∈ L n (U ) we have that

Proof. Let x ∈ L n (U ), the following holds: So, I(x, N I (N I (x))) = /1/ and then, by I12, x ≤ N I (N I (x)). In addition, since N I is decreasing, it implies that N I (N I (N I (x))) ≤ N I (x) for each x ∈ L n (U ). Therefore, Proposition 6.4 is verified.

6.1. Representable n-DI on L n (U ) Proposition 6.5. [57, Prop. 6] Let I 1 , . . . , I n : U 2 → U be functions such that I 1 ≤ . . . ≤ I n . Then, for all x, y ∈ L n (U ), the function I 1 . . . I n : L n (U ) 2 → L n (U ) given by

. . I n (x, y)=(I 1 (π n (x), π 1 (y)),. . . ,I n (π 1 (x), π n (y))),

is I(x, y) .

The next proposition shows that a conjungate operation w.r.t. an n-DI also e is an n-DI. Proposition 6.6. Let I be a n-DI and ϕ ∈ Aut(L n (U )). Then I ϕ also is a n-DI.

Proof. Trivial, once ϕ(/0/) = /0/ = ϕ −1 (/0/), ϕ(/1/) = /1/ = ϕ −1 (/1/) and both, ϕ and ϕ −1 , are increasing functions. 

is a fuzzy implication (implicator).

Proof. We have that I (i) (0, x) = π i (I(/0/, /x/)) = π(/1/) = 1, I (i) (x, 1) = π i (I(/x/, /1/)) = π(/1/) = 1 and I (i) (1, 0) = π i (I(/1/, /0/)) = π(/0/) = 0. So, I (i) satisfies the boundary conditions of fuzzy implications, i.e. it is a fuzzy implicator when I is a n-dimensional fuzzy implicator. Now, if x ≤ z then /x/ ≤ /z/ and therefore by I1, it is holds that I (i) (x, y) = π i (I(/x/, /y/)) ≥ π i (I(/z/, /y/)) = I (i) (z, y). Analogously, it is possible to prove that I (i) (x, y) ≤ I (i) (x, z) whenever y ≤ z. And, Lemma 6.2 holds. Proposition 6.7. Let I be a n-DI (n-dimensional fuzzy implicator), ψ ∈ Aut(U ) and ϕ = ψ ∈ Aut(L n (U )). Then the following statements are equivalent: (2 ⇒ 3) Since, by Proposition 6.6, I ϕ is an n-DI then, by Lemma 6.2, (I ϕ ) (i) for each i ∈ N n is a fuzzy implication (implicator) and ϕ −1 = ψ −1 = ψ −1 then (I ϕ ) (i) (x, y) = π i (ϕ −1 (I(ϕ(/x/), ϕ(/y/)))) = π i ( ψ −1 (I(/ψ(x)/, /ψ(y)/))) = ψ −1 (π i (I(/ψ(x)/, /ψ(y)/))) = ψ −1 (I (i) (ψ(x), ψ(y)))) = (I (i) ) ψ (x, y).

On the other hand, for each x, y ∈ L n (U ), it is holds that π 1 ( ψ(y) ))), . . . , I (n) (π 1 ( ψ(x), π n ( ψ(y))))) = ψ −1 (I (1) (ψ(π n (x), ψ(π 1 (y)))), . . . , I (n) (ψ(π 1 (x), ψ(π n (y))))) = ((I (1) ) ψ (π n (x), π 1 (y)), . . . , (I (n) ) ψ (π 1 (x), π n (y)) , y) .

Therefore, I ϕ is representable.

(3 ⇒ 1) Since ϕ −1 ∈ Aut(L n (U )), then I = (I ϕ ) ϕ −1 and therefore, since I ϕ is representable, then there exist fuzzy implications (implicators) I 1 ≤ . . . ≤ I n such that I ϕ = I 1 . . . I n and by 3.1 there exists an automorphism ψ such that ϕ = ψ. So, for each x, y ∈ L n (U ) we have that

, ψ −1 (y))) by Remark 3.2 = (I ψ 1 (π n (x), π 1 (y)), . . . , I ψ n (π 1 (x), π n (y))) and since each I ψ i is a fuzzy implication (implicator) and I ψ 1 ≤ . . . ≤ I ψ n then I is representable. 

The condition under which an n-dimensional interval fuzzy implication verifies the continuity on I(L n (U )) based on the continuity of family I(U n ) of fuzzy implications on U n is considered in the following. Definition 6.3. Let φ : U n → L n (U ) be the (U n , L n (U ))-permutation expressed by the increase ordering,

is continuous in the usual sense.

Observe that, since L n (U ) ⊂ U n , then F φ is well defined. (⇐) If I (i) is continuous for each i ∈ N n , then I (1) × . . . × I (n) also is continuous. Therefore, since

and δ as well as ψ are continuous,

Hence, by Definition 6.3, I (1) . . . I (n) is continuous.

In the following, main properties of fuzzy implications on L(U ) are preserved by the representable ndimensional fuzzy implications on L n (U ). (1, 0) , I (2) (1, 1) , . . . , I (n) (1, 1)) = I(x, x) = x = /1/. Therefore, Proposition 6.10 is verified. I9(a) : I(x, N (y)) = (I (1) (π n (x), N (1) (π n (y))), . . . , I (n) (π 1 (x), N (n) (π 1 (y)))) by (11) and (17) = (I (1) (π n (y), N (1) (π n (x))), . . . , I (n) (π 1 (y), N (n) (π 1 (x)))) by I9(a) = I(y, N (x)) by (17) and (11).

Since, the pair (N (n−i+1) , I (i) ) verifies I9(b), for each i = 1, . . . , n, then we have that:

, π 1 (x)), . . . , I (n) (N (1) (π n (y)), π n (x))) by (11) and (17) = (I (1) (N (n) (π 1 (x)), π 1 (y)), . . . , I (n) (N (1) (π n (x)), π n (y))) by I9(b)

= I(N (x), y) by (11) and (17).

In addition, since the pair (N, I (i) ) verifies I9, for each i = 1, . . . , n, it holds that:

I9 : I( N (y), N (x)) = (I (1) (N (π 1 (y)), N(π n (x))), . . . , I (n) (N (π n (y)), N(π 1 (x)))) by (11) and (17) = (I (1) (π n (x), π 1 (y)), . . . , I (n) (π 1 (x), π n (y))) by I9 = I(x, y) by (17) .

(⇒) Conversely, since (N , I) verifies I9(a), (I9(b), I9), we have the following results:

: I (i) (y, N (i) (x)) = π i (I (1) (y, N (1) (x)), . . . , I (n) (y, N (n) (x))) = π i (I(/y/, N (/x/))) by (17) and (11) = π i (I(/x/, N (/y/))) by I9 = π i (I (1) (x, N (1) (y)), . . . , I (n) (x, N (n) (y)))I (i) (y, N (i) (x)) by (11) and (17) .

: N (1) (x) , . . . , N (n) (x)), /y/))) by (17) = π i (I(N (/x/), /y/)) by (11) = π i (I(N (/y/), /x/)) by I9(b) (11) and (17) .

: I (i) (N (y), N(x)) = π i (I (1) (N (y), N(x) ), . . . , I (n) (N (y), N(x))) = π i (I( N (/y/), N (/x/)))by (17) and (11) = π i (I(/x/, /y/)) = I (i) (x, y) by I9 and (17).

Therefore, Proposition 6.11 is verified. 

It is an example of representable n-DI which neither satisfies I13(a) nor satisfies I13(c). However, each I LK i satisfies I13(a) and therefore I13(c), since N I LK i (x) = i √ 1 − x i is a strong fuzzy negation. Finally, one can also easily verify that the converse of the other item holds, meaning that I13(b) ⇒ I13(b).

Properties in the class of (S, N )-Implication on L n (U ) are analysed in the following propositions.

Proposition 7.1. Let S be a n-DS and N be a n-DN. The function I S,N : L n (U ) 2 → L n (U ) given by

is an n-dimensional fuzzy implication called as n-dimensional (S, N )-implication.

Proof. Let S be a n-DITS and N be a n-DIFN. The following holds: Proof. For all x, y, z, the following holds:

(i) For each y ∈ L n (U ) we have that I(/1/, y) = S(/0/, y) = y and therefore I S,N verify I3. Since S verifies the S2 and S3 properties, the following holds for each x, y ∈ L n (U ): S(N (x), S(N (y), z))) = S(S(N (x), N (y)), z))) = S(S(N (y), N (x)), z))) = S(N (y), S(N (x), z))). Therefore, I(x, I(y, z)) = I(y, I(x, z)). So I verifies I5. 

The Proposition 6.10 and Proposition 6.12 claim that each representable n-dimensional (S, N )-implication satisfies neither the first axiom of Hilbert system nor the identity principle. Nevertheless, there are ndimensional (S, N )-implication satisfying I7 and I11, for instance, the n-dimensional version of the Weberimplication: (N (x), x) , N (y))) = S (I(x, x) , N (y)) = S(/1/, N (y)) = /1/. Therefore, I satisfies I7.

Since not all (S, N )-implication, even S-implications, satisfy the identity principle, we analyse this property for this family in the following propositions. Proposition 7.6. For a n-DS S and a n-DN N the following statements are equivalent: From the results achieved in Proposition 7.5, the Propositions 7.6 and 7.7 also hold when I11 is substituted by I7.

As noted earlier, not all natural generalizations of the classical implication to multi-valued logic satisfy ordering property I12. In the following section we discuss results on (S, N )-implications with respect to their ordering property. (ii) N I is a strong negation and I satisfies I11.

Proof. (i) ⇒ (ii) Since I12 implies I11 and, by Proposition 7.2, I satisfies I3 and I5, then, by Proposition 6.4(2), we have that N I (N I (N I (x) )) ≤ N I (x) for each x ∈ L n (U ). But N I is right invertible and so has a right inverse, denoted by N −r . Then for each x ∈ L n (U ), x = N I (N −r (x)) ≥ N I (N I (N I (N −r (x) ))) = N I (N I (x) ). Therefore, by Proposition 6.4(1), we conclude that N I (N I (x)) = x.

(ii) ⇒ (i) Since N I is strong, trivially is right invertible. Let x, y ∈ L n (U ), I(x, y) = /1/. So, since N I is strong, it holds that y ≥ inf{z ∈ L n (U ) : I(x, z) = /1/} = inf{z ∈ L n (U ) : I (N (N (x) ), z) = /1/}. N 1 (π n (x) ), . . . , N n (π 1 (x))), (π 1 (y), . . . , π n (y))) = (S 1 (N 1 (π n (x)), π 1 (y)), . . . , S n (N n (π 1 (x)), π n (y))) = (I S1,N1 (π n (x), π 1 (y)), . . . , I Sn,Nn (π 1 (x), π n (y))) = I S1,N1 , . . . , I Sn,Nn (x, y).

Therefore, I S,N is also a representable function on L n (U ) and so, Proposition 7.8 holds. Proposition 7.9. Let S be a n-DS and N be a n-DN. If I S,N is representable then N is representable. In addition, if N is right invertible then S is representable.

Proof. Since I = I S,N is a representable n-dimensional (S, N )-implication then, by Proposition 6.7, each I (i) with i ∈ N n is an n-DI and I = I (1) . . . I (n) . Then for each x ∈ L n (U ), we have that N (x) = S(N (x), /0/) = I(x, /0/) = (I (1) (π n (x), 0), . . . I (n) (π 1 (x), 0)) = (N I (1) (π n (x)), . . . , N I (n) (π 1 (x))) = N I (1) . . . N I (n) (x), that is, N is representable. In addition, let N −r the right inverse of N . By Proposition 4.3, N (i) is right invertible and, Remark 4.3, holds that N −r is representable. Then, for x, y ∈ L n (U ), N (N −r (x)) = x, and we obtained the results below: π n (x) ), . . . , (N −r ) (n) (π 1 (x))), (π 1 (y), . . . , π n (y)) = (I (1) ((N −r ) (n) (π 1 (x)), π 1 (y)), . . . , I (n) ((N −r ) (1) (π n (x)), π n (y))) = (S 1 (π 1 (x), π 1 (y)), . . . , S n (π n (x), π n (y))) with S i (x, y) = I (i) ((N −r ) (n−i+1) (x), y) for each i ∈ N n . Moreover, for each x, y, z ∈ [0, 1] and i ∈ N n , we have that: S i (x, 0) = π i (S(/x/, /0/)) = π i (/x/) = x, S i (x, y) = π i (S(/x/, /y/)) = π i (S(/y/, /x/)) = S i (y, x), if y ≤ z then /y/ ≤ /z/ and so S i (x, y) = π i (S(/x/, /y/)) ≤ π i (S(/x/, /z/)) = S i (x, z). And, 21 finally, since I (i) is an (S.N )-implication and I satisfies I5. Consequently, by Proposition 6.9, I (i) satisfies I5. So, we obtained the following results:

Therefore, S i is associative and therefore, it is a t-conorm. And, Proposition 7.9 holds. (ii) Based on results in [7] and [6] we have that I KD ≤ I RC ≤ I LK . So, see two examples of representable n-DI obtained from Eq. (17) :

Therefore, the following results are obtained: (iii) Analogously, by results from [7] and [6] , I KD ≤ I RC ≤ I LK ≤ I F D and the following holds: N ϕ (x, y) . (21) In addition, given ϕ ∈ Aut(L n (U )), we have that I S,N is representable if and only if (I S,N ) ϕ is representable.

Proof. Let ϕ ∈ Aut(L n (U )) and let S, N be an n-DS and an n-DN, respectively. So, by Propositions 5.5 and 4.5, the functions S ϕ , N ϕ are also a n-DS and an n-DN, thus N (ϕ(x) ), ϕ(y))) by Eqs. (10) and (19) = ϕ −1 (S(ϕ(ϕ −1 (N (ϕ(x) ))), ϕ(y))) = S ϕ (N ϕ (x), y) by Eq.(10) = I S ϕ ,N ϕ (x, y) by Eq. (19) .

Therefore, (I S,N ) ϕ is an n-dimensional (S, N )-implication. In addition, by Proposition 6.7, I S,N is representable if and only if (I S,N ) ϕ is also representable. Therefore, Proposition 7.10 is verified.

Owing to the effective and reasonable description to the uncertainty information, the expression ability related to the concepts in the n-dimensional simplex L n (U ) is stronger than Zadeh's fuzzy sets. So, in this section, first results in the extension of the basic concepts of AR are considered, by using n-dimensional intervals. In particular, the class of n-dimensional fuzzy (S, N )-implication can be employed to relate fuzzy propositional formulae in n-dimensional fuzzy logic inference schemes. For example, if A, B are any ndimensional fuzzy logic propositional formulae, then A → B is called an n-dimensional fuzzy conditional statement or more commonly, as an n-dimensional fuzzy IF-THEN rule and it is again interpreted as "A implies B". This construction can be carried out considering both aspects:

(i) n-dimensional intervals and fuzzy statements An expression of the form "x is A" is termed as a fuzzy statement, where A is a n-dimensional fuzzy set on the n-dimensional simplex L n (U ), with reference to the context. Thus, we can say that the above statement can be interpreted as follows:

-Let "x is A" and also that x assumes the precise value, let us say, μ A (x)=u ∈ L n (U ), the domain of A. Then the truth value of the above fuzzy statement is obtained as t(x is A) = A(u). Thus, the greater the membership degree of x in the concept A is, the higher the truth value of the fuzzy statement.

While in the above case a fuzzy statement was looked upon as a fuzzy proposition to be evaluated based on some precise information, it can also be used to express something precise when the only information regarding the variable x is imprecise.

(ii) n-dimensional intervals compounding n-dimensional IF-THEN rules

We can also interpret an n-dimensional fuzzy statement as a linguistic statement on the suitable domain L n (U ). Then A represents a concept and hence can be thought of as a linguistic value. Then a symbol x can assume or be assigned to a linguistic value. Then a linguistic statement "x is A" is interpreted as the linguistic variable x taking the linguistic value A.

This section describes a structure in the fuzzy rules of deduction for inference schemes in AR on the n-dimensional simplex domain, which is analogous to the fuzzy logic approach.

In the GMP methodology, a fuzzy logic rule of deduction considers an inequality explicit by a conjunction, defined as an n-dimensional t-norm T together with an n-dimensional fuzzy (S, N )-implication.

The inference schemes is performed based on the combination-projection principle, providing the Compositional Rules of Inference (CRI) [55] , which has the structure fuzzy rules based on the GMP inference patterns as follows:

(i) the fuzzy rule has the form "IF x is A THEN y is B", and the fact "x is A ";

(ii) a conclusion to be drawn has the form "y is B " when A, A ∈ F and B, B ∈ F.

In fuzzy approach, neither A is necessarily identical to A nor B is also necessarily identical to B.

This section describes the application of compositional rules of inference (CRI) systems on L n (U ). For that, let F S χ be the set of all n-dimensional fuzzy sets w.r.t. a universe χ.

The Cartesian Product among n-DFS is given in the next definition. Let χ 1 , . . . , χ m be non-empty and finite universe-sets and, for each i ∈ N m , A i ∈ F S χi . Then, the Cartesian Product

In particular, let A 1 , . . . , A m ∈ F S(χ) w.r.t. the same universe χ = {x j : j ∈ N #χ }. So, when i ∈ N m , for each A i ∈ F S(χ) the related membership function μ Ai : χ → L n (U ) is given as μ Ai (x j ) ≡ A i (x j ) = x ij , ∀j ∈ N #χ . The Cartesian Product of these n-DFS is given in the next definition. 

Thus, the relation Π j∈Nm (A j ) can be expressed as a matrix X on (L n (U )) m×l given as

where l = #χ and whose elements x ij = A i (x j ) ∈ L n (U ), for all i ∈ N m and j ∈ N l . Definition 8.3. Let P = {P 1 , . . . , P l } be a family of n-dimensional m-ary aggregation functions, χ be a non-empty and finite universe-set, l = #χ and A = {A 1 , . . . , A m } ⊆ F S χ . An operator P : (L n (U )) m×l → (L n (U )) l is defined as follows

The expression given by Eq. (22) provides a method to generate new members on F S χ , based on the action of a family of aggregation operators. Now, taking the associative operators , : (L 4 (U )) 3×2 → (L 4 (U )) 2 we have that 

Analogously, an IF-THEN rule is represented by a binary n-dimensional fuzzy relation R I (A 1 , A 2 ) : (L n (U )) 2 → L n (U ) given as:

when I is usually a n-dimensional fuzzy (S, N )-implication and A 1 , A 2 are n-dimensional fuzzy sets on their respective universe domains χ 1 , χ 2 . Therefore, given a fact "x 1 is A 1 ", the inferred output "x 2 is A 2 " is obtained as sup-T composition of A 1 (x 1 ) and R I (A 1 , A 2 )(x 1 , x 2 ), as follows:

Let A 1 , A 2 , A 3 be n-DFS on their respective universe domains χ 1 , χ 2 , χ 3 . So, considering the two following cases.

1. Firstly, considering a SISO system given by Eq. (24) attaining normality at an x 1 ∈ χ 1 , then the related output constructing when the input A 1 is the singleton n-dimensional fuzzy set A 1 (x) = /1/ for each x ∈ χ 1 , is obtained as follows:

2. And, in the another case, considering the rule-base in a Multi-Input Single-Output (MISO) system, the relation R is given by

where operator , called the n-dimensional antecedent combiner, is usually given as an m-ary ndimensional t-norm as in Eq. (15) . Thus, we have

A i 's with respect to an m-ary n-dimensional t-norm . So, given a multiple-input (A 1 , A 2 ) and taking the sup-T M composition, the inferred output A m+1 is given by the following expression:

Then, by applying results of Eq. (26), for all x m+1 ∈ χ m=1 , we obtain the following expression for an output in the IF-THEN base-rule in a MISO system:

So, whe m = 2, is in fact an n-dimensional t-norm, just denoted by . In the following, an example exploring the structure presented in Eq. (27) for m = 2 and considering the n-DFS A 1 and A 2 as singleton inputs, is presented.

Consider a virtual application in developing method to medical diagnosis for a patient-analysis with the given five symptoms: fever (a 1 ), sore throat (a 2 ), (head)ache (a 3 ), (dry)cough (a 4 ), anosmia (a 5 ), which are described in terms of L 3 (U )-fuzzy set theory by χ A = {a 1 , . . . , a 5 } in order to contemplate the opinions of three experts from distinct researches areas (infectology, epidemology, and pneumology).

In addition, consider the medical knowledge base components: Influenzavirus Subtype A-H1N1 (b 1 ), COVID-19 (b 2 ) and Atopic Bacterial Pneumonia (b 3 ), which can enable a proper diagnosis from the set χ B = {b 1 , b 2 , b 3 }. The resulting data provide the worst, moderate and best estimates to each one of diagnoses, modelled by χ C = {c 1 , c 2 , c 3 } in L 3 (U ).

The proposed computational evaluation process is conceived to add degrees of freedom and to directly model uncertainty levels of experts knowledge, also including uncertain words from natural language and possible repetition of parameters related to the collected data.

So, let χ A = {a 1 , . . . , a 5 }, χ B = {b 1 , b 2 , b 3 } and χ C = {c 1 , c 2 , c 3 } be universe-sets related to the membership function A : χ 1 → (L 3 (U )) 5 , B : χ 2 → (L 3 (U )) 3 and A : χ 3 → (L 3 (U )) 3 , defining the corresponding 3-DFS in the following: 

The steps to consolidate Eq. (28) are described in the following.

(I) Firstly, the Cartesian Product A × B considering T LK , T P , T M is defined as follows:

w ij = T LK , T P , T M (x i , y j ) = (T LK (π 1 (x i ), π 1 (y j )), T P (π 2 (x i ), π 2 (y j )), T M ((π 3 (x i ), π 3 (y j ))) , ∀i ∈ N 5 , j ∈ N 3 .

where x i = A(a i ) and y j = B(b j ) for each i ∈ N 5 and j ∈ N 3 . Thus, for example, the first component, taking i = j = 1 is given as Analogously, the other components can be obtained. They are described as a matrix structure below: 

where z k = C(c k ) for each k ∈ N 3 . For k = i = 1 and j ∈ N 3 :

(v 1j ) 1 = I KD , I RC , I LK (w 1j , c 1 ) = (I KD (π 3 (w 1j ), π 1 (c 1 )), I RC (π 2 (w 1j ), π 2 (c 1 )), I LK ((π 3 (w 1j ), π 3 (c 1 ))) 

X The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☐ The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

It is worth mentioning that we considered the case and provided a characterization in Theorem 7.1, for n-dimensional interval (S, N )-implications on L n (U ) when the n-DN is right reversible. Moreover, since n-DFS generalize fuzzy sets and interval-valued fuzzy set and such class of (interval-valued) (S, N )-implications were not studied, then Theorem 7.1 can also contribute in the study of (interval-valued) (S, N )-implications

Further work also considers studying other special classes of fuzzy implications as D-, QL-and Rimplications and others as power-implications

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(0.3500, 0.5920, 0.9000), (0.5500, 0.7960, 1.0000), (0.3500, 0.6940, 1.0000) (0.4500, 0.5400, 1.0000), (0.5500, 0.8300, 1.0000), (0.4500, 0.7450, 1.0000) (0.5500, 0.7280, 1.0000), (0.5500, 0.8640, 1.0000), (0.5500, 0.79601.0000) (0.5500, 0.7280, 1.0000), (0.5500, 0.8640, 1.0000), (0.5500, 0.79601.0000) (0.3000, 0.5580, 0.8500), (0.5500, 0.7790, 1.0000), (0.35000.6685, 1.0000)(0.3500, 0.6160, 0, 9500) (0, 5500, 0.8080, 1.0000) (1, 0000, 0.7120, 1.0000) (0.4500, 0.6800, 1.0000) (0.5500, 0.8400, 1.0000) (1, 0000, 0.7600, 1.0000) (0.5500, 0.7440, 1.0000) (0.5500, 0.8720, 1.0000) (1, 0000, 0.8080, 1, 0000) (0.5500, 0.7440, 1.0000) (0.5500, 0.8720, 1.0000) (1, 0000, 0.8080, 1, 0000) (0.3000, 0.5840, 0.9000) (0.7323, 0.7920, 0.8425) (1, 0000, 0.6880, 1, 0000) operator is defined by as follows:and graphically represented by the matrix below:resulting on the matrices below:(V) Concluding, in the − T M composition, we apply the operator : (L 3 (U )) 5 → (L 3 (U )) considering the five lines of each matrix [t 1 ], [t 2 ] and [t 3 ]. It results on the following n-DFS: ((0.550, 0.864, 1.000), (0.550, 0.872, 1.000), (0.200, 0.880, 1.000 ))The constructor of IF-THEN base-rules in MISO n-DFL can be obtained considering other three operators, as reported in Table 8