Information Sciences 253 (2013) 126–146 Contents lists available at ScienceDirect Information Sciences journal homepage: www.elsevier.com/locate/ins Aggregating fuzzy implications Renata H.S. Reiser a,⇑, Benjamin Bedregal b, Michał Baczyński c a b c Center of Technological Development, Federal University of Pelotas, Pelotas, Brazil Department of Informatics and Applied Mathematics, Federal University of Rio Grande do Norte, Natal, Brazil Institute of Mathematics, University of Silesia, ul. Bankowa 14, 40-007 Katowice, Poland a r t i c l e i n f o Article history: Received 10 September 2012 Received in revised form 2 August 2013 Accepted 13 August 2013 Available online 23 August 2013 Keywords: Fuzzy connectives Fuzzy implication Aggregation function (S, N)-implication R-implication a b s t r a c t The aim of this work is to study the I A fuzzy implication obtained by composition of an aggregation function A and a family I of fuzzy implications. Thus, it discusses under which conditions such functions preserve the main properties of fuzzy implications. In addition, by aggregating conjugate fuzzy implications it is shown that an I A fuzzy implication can be preserved by action of an order automorphism. Finally, we introduce the family of I A fuzzy implications obtained by taking the extended classes of (S, N)-implications which are given by t-subconorms and of R-implications underlying left continuous t-subnorms. Their dual construction is also considered. Ó 2013 Elsevier Inc. All rights reserved. 1. Introduction Fuzzy implications play an important role in fuzzy reasoning allowing intelligent systems to be modeled by plausible conclusions in a similar way to human reasoning which are applied in many fields of approximate reasoning as fuzzy relational equations [40], fuzzy morphology [10] and image processing [6]. Additionally, fuzzy implications together with aggregation functions have been used in the inference schemes in approximate reasoning (see [1,26,13]). In the compositional rule of inference of Zadeh, a fuzzy if-then rule is represented by a fuzzy relation usually defined as a fuzzy implication [54]. Thus, given a fact, which can be expressed as a fuzzy singleton set or multiple fuzzy sets, an inferred output can be obtained. In such inference cases, the input(s) can be seen as a fuzzy set on a single domain or a Cartesian product of the domains, respectively. When the fuzzy rule base is compounded by multiple rules, it means that we need to aggregate over these rules and infer the output fuzzy set. This can be performed making use of an aggregation operator, in distinct ways. One can first either aggregate all the rules (or relations) into a single ‘‘aggregated’’ relation or compose the input with each relation to get a set of outputs. In the former we have a nonadditive fuzzy model (Mamdani model) and in the latter we infer the output individually from each fuzzy relation and then the overall output fuzzy set is obtained by aggregating those individually inferred outputs (Takagi–Sugeno–Kang additive fuzzy model). Considering fuzzy negations and aggregation functions, fundamental classes of fuzzy implications are implicit and explicitly represented, e.g., the R-implication class and the classes of S-implications [2], QL-implications [19], D-implications and many others [39,45,47]. The most used aggregation functions are t-norms and t-conorms [33], with some recent results presented in [15,27]. However, since properties as associativity or commutativity are unnecessarily restrictive conditions, the use of uninorms, pseudo-(co)norms, weak t-(co)norms, medians, arithmetic and weight means are extensively investigated in the literature, see also [18,38,30,41,7]. In this work we consider t-sub(co)norms [29,28]. ⇑ Corresponding author. Tel.: +55 5381110603. E-mail addresses: reiser@inf.ufpel.edu.br (R.H.S. Reiser), bedregal@dimap.ufrn.br (B. Bedregal), michal.baczynski@us.edu.pl (M. Baczyński). 0020-0255/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ins.2013.08.026 R.H.S. Reiser et al. / Information Sciences 253 (2013) 126–146 127 More recently, the work in [53] describes a number of commonly used smoothing techniques, moving average and exponential smoothing providing an extension of these methods to the case where the observations can have different credibility or importances. Such exponential smoothing method provides to the observations different importance weights in the smoothing process. In [22] a study integrating Dombi aggregative operators, uninorms, strict t-norms and t-conorms is considered together with a new representation theorem of strong negations that explicitly contains the neutral value. Some class of weighted aggregative operators (representable uninorms) which build a self-DeMorgan class with infinitely many negations are also investigated. The work in [8] uses the penalty-based representation of aggregation functions in order to investigate the conditions for weighting vectors associated with some important weighted families, and also extends the results already established for quasi-arithmetic means. It can be applied to practical applications dealing with aggregate data of varying dimension. 1.1. Aggregation operators and fuzzy implications – related work Let us relate some works issued from the literature on the use of aggregations to obtain new fuzzy implications, which have been explored in many fuzzy logical theoretic approaches. The class of binary aggregation operators satisfying the 2increasing property is investigated in [23] resulting in a methodology for constructing copulas. In [12], the authors show that a migrative aggregation function verifying associativity and bisymmetric properties provides partial information from different sources which need to be amalgamated into a global summary. In [14], the authors propose a general framework allowing to generate many measures of comparison as inclusion, similarity and distance, by using fuzzy implications and aggregation functions and studying their properties. A special class of residual fuzzy implications derived from non-associative aggregation functions is proposed in [15], allowing for new families of continuous fuzzy implications which satisfy the modus ponens for a given nilpotent t-norm. Additionally, the minimal conditions which must be satisfied for a binary aggregation A to generate a residual implication are presented in [41]. Another class of generated fuzzy implications is studied in [9], emphasizing the construction of a monotone generator triplet related to fuzzy preference structure. More recently, special classes of binary conic aggregation functions such as conic quasi-copulas and conic copulas are considered in [31]. Furthermore, in [42] the behavior of aggregation functions is analysed when the inputs are contradictory and bivariate aggregation functions are classified with special attention to t-norms, copulas, symmetric sums and uninorms. In paper [48], Riera and Torrens study residual implications defined on the set of discrete fuzzy numbers whose support is a set of consecutive natural numbers. A specific construction of these implications is given and some examples are presented showing in particular that such a construction generalizes the case of interval-valued residual implications. The present work in [43] reviews the most relevant aspects of this relation, which most of the times also involves negation functions. The author investigates how new families of aggregation functions and negation functions can be obtained from implications, the main equations and equations involving these three classes of functions are recalled. Such diversity of aggregation functions mainly used to generate other fuzzy implications motivates the study of the I A operator, obtained by composition of an n-ary aggregation function A and a family I of fuzzy implications. Aiming to study the classes of representable implications, some general concepts of t-sub(co)norms were also considered in the definition of the F A -operator, aggregating families of fuzzy connectives. 1.2. Main contributions The F A -operator brings about an interesting discussion of aggregating families of fuzzy connectives from aggregation operators. Our first contribution is the study under which conditions an I A -implicator can be obtained by an aggregation A and a family of implications I ¼ fIi : U 2 ! Ugi2f1;2;...;ng , preserving main properties of fuzzy implications. In particular, with respect to a list of (eighteen) selected algebraic properties, it is shown that such I A -operator preserves the exchange principle when fuzzy implications in I verify both properties, the generalized exchange principles and the distributive property with respect to a (non-necessarily binary) aggregation A. We also investigate the relationship between the properties which provides a characterization of (S,N)- and R-implication classes in I and other relevant properties such as the generalized associativity and k-homogeneity related to an aggregation A. This investigation guarantees the main conditions for an ðS A ÞT A -operator be closed under the ((S, N)-) R-implication class. The N-dual construction of an ðA; F Þ-operator F A is obtained by composition of an aggregation function A and the corresponding family F N of the N-dual fuzzy connectives, also named F N -cooperator. Additionally, by aggregating a family of conjugate fuzzy connectives F q the conjugate ðF q ÞAq -operator is defined and it is also the conjugate operation ðF A Þq of an -F A -operator, meaning that an F A -operator is preserved by action of an order automorphism q on U. Moreover, another research topic is related to an aggregation A together with the families S ¼ fSI : U 2 ! Ugi2f1;2;...;ng of t-subnorms verifying distributivity and generalized associativity and N ¼ fN I : U ! Ugi2f1;2;...;ng of fuzzy negations, we introduce two families of fuzzy implications, denoted by I A;S;N and I A;S;N . The former is obtained as a class of (S, N)-implications 128 R.H.S. Reiser et al. / Information Sciences 253 (2013) 126–146 underlying a fuzzy negation N and a family S of continuous t-subconorms on S. The latter is obtained when a class of (S, N)implications underlying a fuzzy t-subconorm S and a family N , which is distributive with respect to A. And finally, R-implications underlying left-continuous t-subnorms and their dual and conjugate constructions are defined, discussing the preserved properties. 1.3. Outline of the paper This paper is organized as follows. After Preliminaries, in Section 3, we report main characteristics of aggregation functions based on reasonable properties (from A1 to A6) including definition of t-sub(co)norms and their conjugate functions. Section 4 focusses on fuzzy implications, their dual construction, main properties and the fundamental classes of (S, N)implications and R-implications. Section 5 introduces the F A operator, named as an ðA; F Þ-operator and obtained by an aggregation function A and a family of functions F . Thus, we present minimal conditions under which we obtained F A as an ðA; I Þ-implication by analyzing properties (from I1 to I18) which are verified by implications in the related family I . In particular, a discussion about the conjugated functions obtained by the action of automorphisms on ðA; I Þ-implications is presented. Our study also deals with the aggregation fuzzy connectives T A ðS A Þ related to family of t-sub(co)norms. Section 6 shows that such connectives are preserved by considering the distributivity and generalized associativity. Finally, by taking into consideration idempotent aggregation functions and corresponding families of implications, both the aggregation fuzzy (S, N)-implications, underlying fuzzy negations together with t-subconorms, and the aggregation fuzzy R-implications are considered in Sections 7 and 8, respectively. The conjugate and dual constructions are also discussed. The Conclusion reports main results and further work. 2. Preliminaries 2.1. Continuous functions In this article we will denote the unit interval by U = [0, 1]. Taking n 2 N, a monotone function F Un ? U is continuous if and only if it is continuous in each component (see [33, p. 15]). Continuity of a function F:Un ? U can be given in one component only. Thus, a function F:U2 ? U verifies the second place (left-) right-continuous property if, for all (non-decreasing) non-increasing sequence ðxn Þn2N 2 U n , it holds that lim Fðy; xn Þ ¼ Fðy; lim xn Þ: n!1 n!1 2.2. Order automorphisms Definition 2.1 [11, Definition 0]. A mapping q:U ? U is an order automorphism if it is continuous, strictly increasing and verifies the boundary conditions q(0) = 0 and q(1) = 1, i.e., if it is an increasing bijection on U. Order automorphisms are closed under composition, i.e., denoting by AðUÞ the set of all order automorphisms on U, if q; q0 2 AðUÞ then q  q0 ðxÞ ¼ qðq0 ðxÞÞ 2 AðUÞ. In addition, the inverse q1 of an order automorphism q is also an order automorphism. Based on [11,39,40,1], the action of an order automorphism q on a function f:Un ? U, denoted by fq and named the qconjugate of f is defined as: f q ð~ xÞ ¼ q1 ðf ðqð~ xÞÞÞ; ~ x ¼ ðx1 ; . . . ; xn Þ; 2 U n ; n ð1Þ n where qð~ xÞ ¼ ðqðx1 Þ; . . . ; qðxn ÞÞ 2 U . Two mappings f1, f2:U ? U are conjugated to each other, if there exists an order automorphism q such that f2 ð~ xÞ ¼ f1q ð~ xÞ ¼ q1 ðf1 ðqð~ xÞÞÞ; ~ x 2 Un : 1 Notice that, if f2 ¼ f1q then f1 ¼ f2q . Proposition 2.2 [5, Proposition 2.1]. Let q 2 Aut(U), q(xy) = q(x)q(y), for all x, y 2 U, if and only if there exists a real number r > 0 such that q(x) = xr, for all x 2 U. 2.3. Fuzzy negations A function N:U ? U is a fuzzy negation (shortly FN) if. R.H.S. Reiser et al. / Information Sciences 253 (2013) 126–146 129 N1 N(0) = 1 and N(1) = 0; N2 If x P y then N(x) 6 N(y), x, y 2 U. Fuzzy negations satisfying the involutive property are called strong FN: N3 N(N(x)) = x, x 2 U. And, a continuous FN is strict [33], when. N4 if x > y then N(x) < N(y), x, y 2 U. Strong FNs are also strict negations [11]. The standard strong FN is defined as NS(x) = 1  x. Based on Eq. (1), the action of an order automorphism q on a negation N is given by Nq ðxÞ ¼ q1 ðNðqðxÞÞÞ: ð2Þ In addition, (strict, strong) fuzzy negation class is closed under the action of an order automorphism, i.e., if N is a (strict, strong) fuzzy negation then Nq is also a (strict, strong) fuzzy negation. For details, see [4, Proposition 1]. It was also proved in [51] that each strong fuzzy negation is a monotone transformation of the standard negation. Proposition 2.3 [51]. A function N:U ? U is a strong FN if and only if there exists an order automorphism q such that NðxÞ ¼ NqS ðxÞ ¼ q1 ð1  qðxÞÞ; x 2 U: A less known class of fuzzy negations is the following: a fuzzy negation N is non-vanishing if N(x) = 0 implies that x = 1, see [1] for details. Definition 2.4. Let N be a FN and f:Un ? U be a real function. The N-dual function of f is given by the expression: fN ð~ xÞ ¼ Nðf ðNð~ xÞÞÞ; ~ x ¼ ðx1 ; . . . ; xn Þ 2 U n ; ð3Þ n where Nð~ xÞ ¼ ðNðx1 Þ; . . . ; Nðxn ÞÞ 2 U . Notice that, when N is involutive, (fN)N = f, that is the N-dual function of fN coincides with f. In addition, if f = fN then it is clear that f is a self-dual function. 3. Aggregation functions: properties, conjugate and dual constructions This section reviews some basic properties (from A1 to A7) of an n-ary aggregation function including propositions presenting the conditions under which such properties can be preserved by corresponding conjugate and dual constructors. We also presented the subclasses of aggregation functions named t-sub(co)norms, 3.1. Definition and properties of aggregation functions Based on [52,18,20,13,7], the general meaning of an n-ary aggregation function in FL is to assign a single real number on U to any n-tuple of real numbers belonging to Un, such that it is a non-decreasing and idempotent (i.e., it is the identity when an n-tuple is unary) function satisfying boundary conditions. Among several definitions we will use the following one. An n-ary aggregation is a function A:Un ? U demanding, for all ~ y ¼ ðy1 ; y2 ; . . . ; yn Þ 2 U n , the following conditions: x ¼ ðx1 ; x2 ; . . . ; xn Þ; ~ A1: Að~ 0Þ ¼ Að0; 0; . . . ; 0Þ ¼ 0 and Að~ 1Þ ¼ Að1; 1; . . . ; 1Þ ¼ 1; x 6~ y then Að~ xÞ 6 Að~ yÞ; A2: If ~ ƒ! A3: Að xr Þ ¼ Aðxr1 ; xr2 ; . . . ; xrn Þ ¼ Aðx1 ; x2 ; . . . ; xn Þ ¼ Að~ xÞ. Hereafter, since we use only n-ary aggregation functions, they will be called simply aggregation functions and related extra properties are reported below: A4a: A(x, x, . . . , x) = x, for all x 2 U (idempotency property); A4b: A(x, x, . . . , x) 6 x, for all x 2 U; A5a: If Að~ xÞ ¼ 1, then there exists 1 6 i 6 n, such that xi = 1 (necessity of one); A5b: If Að~ xÞ ¼ 0, then there exists 1 6 i 6 n, such that xi = 0 (necessity of zero); A6: If ~ x <~ y, then Að~ xÞ < Að~ yÞ. A7: Let k2]0, 1[, for all a 2 [0, 1[ such that akx1, akx2, . . . , akxn 2 U, A(ax1, ax2, . . . , a xn) = akA(x1, x2, . . . , xn) (k-homogeneity) Remark 3.1. Let _, ^:U2 ? U be the binary idempotent aggregation functions defined as _(x, y) = max (x, y) and ^(x, y) = min(x, y). So, when A is an idempotent aggregation function, then ^ðx; yÞ 6 Aðx; yÞ 6 _ðx; yÞ; for all x; y 2 U: 130 R.H.S. Reiser et al. / Information Sciences 253 (2013) 126–146 3.2. Conjugate aggregation functions The properties considered previously, usually demanded from an aggregation function A, can also be naturally extended to its conjugate function Aq. Proposition 3.2. Let A:Un ? U be a function and q be an order automorphism. For i = {1, 2, 3, 4a, 4b, 5a, 5b, 6, 7}, the conjugate function Aq satisfies Ai if and only if A satisfies Ai. xÞ ¼ ðqðx1 Þ; . . . ; qðxn ÞÞ. x 2 U n , qð~ Proof. (Ü) Let us recall that for all ~ A1: Once for any order automorphism q, q(0) = 0 and q(1) = 1, we have that Aq ð~ 0Þ ¼ q1 ðAðqð~ 0ÞÞÞ ¼ q1 ðAð~ 0ÞÞ ¼ q1 ð0Þ ¼ 0 q ~ 1 1 1 ~ ~ and, analogously, A ð1Þ ¼ q ðAðqð1ÞÞÞ ¼ q ðAð1ÞÞ ¼ q ð1Þ ¼ 1. A2: If ~ x 6~ y, then qð~ xÞ 6 qð~ yÞ, so Aðqð~ xÞÞ 6 Aðqð~ yÞÞ. Therefore Aq ð~ xÞ ¼ q1 ðAðqð~ xÞÞÞ 6 q1 ðAðqð~ yÞÞÞ ¼ Aq ð~ yÞ. ƒ! q ƒ! q 1 1 A3: If A verifies A3, then A ð xr Þ ¼ q Aðqð xr ÞÞ ¼ q ðAðqð~ xÞÞÞ ¼ A ð~ xÞ. A4a: Since A is an idempotent function, we get Aq ðx; . . . ; xÞ ¼ q1 ðAðqðxÞ; . . . ; qðxÞÞÞ ¼ q1 ðqðxÞÞ ¼ x: A4b: When A verifies A4b, we have Aq(x, . . . , x) = q1(A(q(x), . . . , q(x))) 6 q1(q(x)) = x. A5a: When Aq ð~ xÞ ¼ 1, then q1 ðAðqð~ xÞÞÞ ¼ 1. By the definition of an order automorphism, Aðqð~ xÞÞ ¼ 1. So, because A verifies A5, there exists 1 6 i 6 n, such that q(xi) = 1 which means that xi = 1. A5b: Analogous to A5a. A6: Since q is a monotonic function, if ~ x <~ y then Aq ð~ xÞ ¼ q1 ðAðqð~ xÞÞÞ < q1 ðAðqð~ yÞÞÞ ¼ Aq ð~ yÞ. 1 ()) Straightforward, based on that q1 is an order automorphism and ðAq Þq ¼ A, for each order automorphism q. h Proposition 3.3. If A is an aggregation verifying the k-homogeneity property (A7), for all q 2 Aut(U), such that q(x) = xr, when r > 0, then Aq is an aggregation also verifying the k-homogeneity property. Proof. Consider A as an aggregation verifying the k-homogeneity property for all q(x) = xr 2 Aut(U) when r > 0, which means, 1 q1 ðxÞ ¼ x r . For all x, y, x 2 U, it holds that Aq ðxy1 ; xy2 ; . . . ; xyn Þ ¼ q1 ðAðqðxy1 Þ; qðxy2 Þ; . . . ; qðxyn ÞÞÞ ¼ q1 ðAðqðxÞqðy1 Þ; qðxÞqðy2 Þ; . . . ; qðxÞqðyn ÞÞ ¼ q1 ðððqðxÞÞk Aðqðy1 Þ; qðy2 Þ; . . . ; qðyn ÞÞÞ ¼ q1 ðqðxÞÞk Þq1 ðAðqðy1 Þ; qðy2 Þ; . . . ; qðyn ÞÞÞ ¼ xk Aq ðy1 ; y2 ; . . . ; yn ÞÞ Therefore, Aq is k-homogeneous aggregation. h be the arithmetic mean given by Mðx1 ; . . . ; xn Þ ¼ 1n ðx1 þ . . . þ xn Þ. Then Example 3.4. Let M:U2 ? U Mðax1 ; . . . ; axn Þ ¼ 1n ðax1 þ . . . þ axn Þ ¼ a 1n ðx1 þ . . . þ xn Þ ¼ aMðx1 ; . . . ; xn Þ. Therefore, M is 1-homogeneous aggregation. Corollary 3.5. Let A:Un ? U be an (idempotent) aggregation function and q be an order automorphism. The conjugate function of A, Aq:Un ? U, is also an (idempotent) aggregation function. Proof. Straightforward from Proposition 3.2. h 3.3. Dual aggregation functions In the following, a discussion under which conditions the properties of an aggregation function A can preserved by its Ndual function AN is developed in this subsection. Proposition 3.6. Let A:Un ? U be a function and N be a FN. The N-dual function of A, AN:Un ? U, satisfies Ai, for some i 2 {1, 2, 3, 6}, if and only if A satisfies Ai. Additionally, when N is involutive, AN, satisfies A4a and A4b, if and only if A satisfies A4a and A4b. R.H.S. Reiser et al. / Information Sciences 253 (2013) 126–146 131 Proof. (Ü) Let us recall that for all ~ x 2 U n ; Nð~ xÞ ¼ ðNðx1 Þ; . . . ; Nðxn ÞÞ. A1: Any fuzzy negation verifies N1, so AN ð~ 0Þ ¼ NðAðNð~ 0ÞÞÞ ¼ NðAð~ 1ÞÞ ¼ Nð1Þ ¼ 0 and, analogously, AN ð~ 1Þ ¼ NðAðNð~ 1ÞÞÞ ¼ NðAð~ 0ÞÞ ¼ Nð0Þ ¼ 1. A2: When ~ x 6~ y, then AN ð~ xÞ ¼ NðAðNð~ xÞÞÞ 6 NðAðNð~ yÞÞÞ ¼ AN ð~ yÞ. ƒ! ƒ! xÞ. xÞÞÞ ¼ AN ð~ A3: When A verifies A3, then AN ð xr Þ ¼ NðAðNð xr ÞÞÞ ¼ NðAðNð~ A4a: If A verifies A4a and N is a strong FN, AN(x, . . . , x) = N(A(N(x), . . . , N(x))) = N(N(x)) = x. A4b: If A verifies A4b and N is a strong FN, AN(x, . . . , x) = N(A(N(x)), . . . , N(x)) 6 N(N(x)) = x. A6: If ~ x <~ y then AN ð~ xÞ ¼ NðAðNðx1 Þ; Nðx2 Þ; . . . ; Nðxn ÞÞÞ 6 NðAðNðy1 Þ; Nðy2 Þ; . . . ; Nðyn ÞÞÞ ¼ AN ð~ xÞ. ()) Straightforward, based on that N is an involutive FN, which means (AN)N = A. h In case of Property A7, as stated in the following proposition, an special class of automorphisms is required to the dual constructor: Proposition 3.7. Let A: Un ? U be a function and N be a FN. The N-dual function of A, AN:Un ? U, satisfies A5a (A5b), if and only if A satisfies A5b (A5a). Proof. (Ü) Let us fix arbitrarily ~ x 2 Un . A5a: If AN ð~ xÞ ¼ 1, then NðAðNð~ xÞÞÞ ¼ 1 which means AðNð~ xÞÞ ¼ 0 and since A verifies A5b, it holds that there exists i, N(xi) = 0 or xi = 0. Therefore AN verifies A5a. A5b: If AN ð~ xÞ ¼ 0, then NðAðNð~ xÞÞÞ ¼ 0 which means AðNð~ xÞÞ ¼ 1 and since A verifies A5a, it holds that there exists i, N(xi) = 1 or xi = 0. Therefore, AN verifies A5b. ()) Analogously. h Corollary 3.8. Let A:Un ? U be a function and N be a FN. The N-dual function of A, AN:Un ? U, is also an (idempotent) aggregation function. Moreover, if N is a strong FN, then (A, AN) is a pair of N-mutual dual functions. Proof. Straightforward from Propositions 3.6 and 3.7. h 3.4. Triangular sub(co)norms A triangular sub(co)norm (t-sub(co)norm, for short) is a binary aggregation function (S)T:U2 ? U such that (S(x, y) P _ (x, y)) T(x, y) 6 ^ (x, y), satisfying the associativity property which are, respectively, given by the expressions: T1 : Tðx; Tðy; zÞÞ ¼ TðTðx; yÞ; zÞ; S1 : Sðx; Sðy; zÞÞ ¼ SðSðx; yÞ; zÞ: A t-sub(co)norm satisfying the boundary conditions T2 : Tðx; 1Þ ¼ x; S2 : Sðx; 0Þ ¼ x; is called a t-(co)norm. A t-sub(co)norm that is not a t-(co)norm is frequently called as a proper t-sub(co)norm. A t-sub(co)norm (S) T is (right) left-continuous if, for all y 2 U and, for each (decreasing) increasing chain ðxi Þi2N on U, the following equation is held lim Sðxi ; yÞ ¼ Sðlim xi ; yÞ; i!1 i!1 lim Tðxi ; yÞ ¼ Tðlim xi ; yÞ: i!1 i!1 Proposition 3.9. Let (S) T be a (right-) left-continuous t-sub(co)norm and q be an order automorphism on U. Then (Sq) Tq is also a (right-) left-continuous t-sub(co)norm given by Sq ðx; yÞ ¼ qðSðqðxÞ; qðyÞÞÞ; ð4Þ T q ðx; yÞ ¼ qðTðqðxÞ; qðyÞÞÞ: ð5Þ Proof. Straightforward from Proposition 3.2. h 132 R.H.S. 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In addition, when N is a FN, a function SqN ðx; yÞ T qN ðx; yÞ is also a t-sub(co)norm preserving its N-dual and conjugate function. Moreover, let N be a strong FN. The N-dual of a t-sub(co)norm (S) T is a (t-subnorm) t-subconorm (SN)TN:U2 ? U given by ðSN Þq ¼ SNq ; ð6Þ q ðT N Þ ¼ T Nq : ð7Þ Example 3.10. For a 2 {1, 2, . . . , n}, consider the family of t-subnorms T P=a ðx; yÞ ¼ a1 xy and their NS-dual functions, the family of t-subconorms SP=a ðx; yÞ ¼ 1a ða  ð1  xÞð1  yÞÞ ¼ a1 ða  1 þ x þ y  xyÞ. In particular, the functions TP, TP/2, TP/3:U2 ? U are t-subnorms defined as T P ðx; yÞ ¼ xy; T P=2 ðx; yÞ ¼ 12 ðxyÞ and T P=3 ðx; yÞ ¼ 13 ðxyÞ, respectively. Their corresponding dual constructions are the t-subconorms SP, SP/2, SP/3:U2 ? U expressed as SP ðx; yÞ ¼ x þ y  xy; SP=2 ðx; yÞ ¼ 12 ð1 þ x þ y  xyÞ and SP=3 ðx; yÞ ¼ 13 ð2 þ x þ y  xyÞ. 4. Fuzzy(co)implications This section reviews some basic properties (from I1 to I18) of implications which are preserved by their conjugate construction. In a dual approach, we consider related notions and properties (from J1 to J18) of coimplication functions, conceived as N-dual structures of implication functions extending the classical coimplications (see [24,26,25,11,40,2]). Definition 4.1 [50, Section 3.1]. The binary operation (J)I:U2 ? U is a (co)implication operator if it satisfies the boundary conditions: I1: I(1, 1) = I(0, 1) = I(0, 0) = 1 and I(1, 0) = 0; J1: J(1, 1) = J(1, 0) = J(0, 0) = 0 and J(0, 1) = 1. Based on Definition 4.1, several other extra properties of (co)implication operators are considered in the literature, in addition to (J1) I1. The most used ones are listed in Table 1. In this paper, the definition of a fuzzy (co)implication is the following one, which is equivalent to the definition proposed by Kitainik [32] (see also Fodor and Roubens [26] and Baczyński and Jayaram [2]). Definition 4.2 [16, Definition 6]. A fuzzy (co)implication I(J) is a (co)implication operator (J)I:U2 ? U which fulfills, for all x, y, z 2 U, the conditions stated from (J1) I1 to (J4) I4. The next result states how a fuzzy implication gives rise to a coimplication and vice versa, see also [17,37,35,49]. Proposition 4.3. Let N be a FN and (J) I be a (co)implication. Then (JN) IN defined by Eq. (3) is a (implication) coimplication. Table 1 The most important properties of (co)implications. Properties of coimplications Properties of implications J2: x 6 z ? J(x, y) P J(z, y); J3: y 6 z ? J(x, y) 6 J(x, z); J4: J(1, y) = 0; J5: J(x, 0) = 0; J6: J(0, y) = y; J7: J(x, J(y, z)) = J(y, J(x, z)); J8: x P y ? J(x, y) = 0; J9: NJ(x) = J(x, 1) is FN; J9b: NJ(x) = J(x, 1) is a continuous FN; J9c: NJ(x) = J(x, 1) is strong FN; J10: J(x, y) 6 y; J11: J(x, x) = 0; J12: J(x, y) = J(N(y), N(x)); J13: x < 1 ? J(x, 1) > 0; J14: y > 0 ? J(0, y) > 0; J15: J is continuous; J16: J(x, y) = J(x, J(x, y)); J17: J(x, J(y, x)) = 0; J18: J(N(x), x) = x; I2: x 6 z ? I(x, y) P I(z, y); I3: y 6 z ? I(x, y) 6 I(x, z); I4: I(0, y) = 1; I5: I(x, 1) = 1; I6: I(1, y) = y; I7: I(x, I(y, z)) = I(y, I(x, z)); I8: x 6 y ? I(x, y) = 1; I9: NI(x) = I(x, 0) is FN; I9b: NI(x) = I(x, 0) is a continuous FN; I9c: NI(x) = I(x, 0) is strong FN; I10: I(x, y) P y; I11: I(x, x) = 1; I12: I(x, y) = I(N(y), N(x)); I13: x > 0 ? I(x, 0) < 1; I14: y < 1 ? I(1, y) < 1; I15: I is continuous; I16: I(x, y) = I(x, I(x, y)); I17: I(x, I(y, x)) = 1; I18: I(x, N(x)) = N(x); R.H.S. Reiser et al. / Information Sciences 253 (2013) 126–146 133 Proof. Clearly, since (J) I satisfies (J1) I1 and N satisfies N1, (JN) IN satisfies (I1) J1 and so (JN) IN is the (implication) coimplication associated to (J) I by the fuzzy negation N. h Proposition 4.4 [46, Proposition 3.3]. Let N be a strong negation and I a fuzzy implication and IN its corresponding N-dual function. Then I satisfies the property Ii if and only if IN satisfies the property Ji, for all i, with 2 6 i 6 19. Corollary 4.5. Let N be a strong FN and (J) I be a fuzzy (co)implication. Then (JN)IN:U2 ? U defined by Eq. (3) is a fuzzy (implication) coimplication. Proposition 4.6 [1]. The conjugate function of a fuzzy (co)implication (J) I, denoted by (Jq)Iq:U2 ? U is also a fuzzy (co)implication. 4.1. Fuzzy (co)implication classes This section considers the results obtained previously by studying the main properties preserved by the classes of (co)implications named (S, N)-(co)implications and R-(co)implications, the former explicitly represented by classes of t-subconorms together with fuzzy negations, and the latter, implicitly represented by t-subnorms. The action of automorphisms and dual constructors are also analysed. 4.1.1. (S, N)-implication and (T, N)-coimplication classes Definition 4.7. A function (J)I:U2 ? U is called an ((T, N)-coimplication) (S, N)-implication if there exists a (t-subnorm T) tsubconorm S and a fuzzy negation N such that Jðx; yÞ ¼ TðNðxÞ; yÞ; ð8Þ Iðx; yÞ ¼ SðNðxÞ; yÞ; ð9Þ for all x, y 2 U. If N is a strong FN, then (J)I is called a strong (co)implication or (T-coimplication) S-implication. Moreover, if (J)I is an ((T, N)-coimplication) (S, N)-implication generated from (T)S and N, then it will be denoted by (JT,N) IS,N. Proposition 4.8. A conjugate of a ((T, N)-coimplication) (S, N)-implication (JT,N) IS,N is also a ((T, N)-coimplication) (S, N)-implication given as IqS;N ¼ ISq ;Nq ; J qT;N ¼ J T q ;Nq : Proof. This proof can be obtained analogously of [1, Theorem 2.4.5]. h Proposition 4.9. Let N be a strong FN, T and S be a t-subnorm and its corresponding dual t-subconorm, respectively, and (JT,N) IS,N:U2 ? U be a fuzzy ((T, N)) (S, N)-(co)implication. Then it follows that JT,N and IS,N are mutual N-dual functions given as ðIS;N ÞN ¼ J T;N ; ðJ T;N ÞN ¼ IS;N : Proposition 4.10. For a function I:U2 ? U, the following statements are equivalent: 1. I is an (S, N)-implication underlying continuous fuzzy negation N and continuous t-subconorm S at point 0. 2. I satisfies I2, I7 and I9b and it is continuous at point x = 1 at the first component. Proof. This proof can be obtained analogously to [1, Theorem 2.4.10].()) By definition, I(1, x) = S(0, x) and then, by the continuity of S at 0, it follows that I is continuous at point x = 1 at the first component. Additionally, it also follows that: I2 If x1 6 x2, we have that I(x1, y) = S(N(x1), y) 6 S(N(x2), y) = I(x2, y). I7 I(x, I(y, z)) = S(N(x), S(N(y), z)) = S(S(N(y), N(x)), z) = S(N(x), S(N(y), z)) = I(y, I(x, z)). 134 R.H.S. Reiser et al. / Information Sciences 253 (2013) 126–146 I9(b) Firstly, NI(0) = I(0, 0) = 1 and NI(1) = I(1, 1) = 0. Suppose that x 6 y. By I2, NI(x) = I(x, 0) P I(y, 0) = NI(y). Since NI(x) = I(x, 0) = S(N(x), 0), based on continuity of such function, NI is a continuous fuzzy negations. (Ü) By [1, Theorem 2.4.10], we have that I is an (S,N)-implication with underlying fuzzy t-conorm and a continuous fuzzy negation. Then it is sufficient to prove that S is continuous at point 0. By the continuity of I and since I(1, x) = S(0, x) = S(x, 0), it follows that S is continuous at point x = 0. Therefore, Proposition 4.10 holds. h Remark 4.11. By comparing the conditions of Proposition 4.10 with the conditions stated in [1] by Theorem 2.4.10, one can observe that the continuity of the fuzzy negation NI, in this last case, follows from the equality NI = N. However, in this work, in order to prove Proposition 4.10, it was also necessary to require the right-continuity at zero of the t-subconorm S. Proposition 4.12. For a function I:U2 ? U, the following statements are equivalent: 1. I is an (S, N)-implication with underlying strict fuzzy negation (or just strict (S, N)-implication). 2. I satisfies I2, I7 and I9c. Proof. This proof can be obtained analogously to [1, Theorem 2.4.11]. h Proposition 4.13. A function I:U2 ? U is a strong S-implication if and only if it satisfies the properties: I2, I3, I6, I7 and I12. Proof. This proof can be obtained analogously to [26, Theorem 1.13]. h 4.1.2. R-(co)implication classes An R-implication arises from the notion of residuum in Intuitionistic Logic [2] or, equivalently, from the notion of residue in the theory of lattice-ordered semigroups [25]. This is well-defined only if the t-norm is left-continuous, or equivalent, if it satisfies the residuation condition, see Eq. (RP), in [1, Proposition 2.5.2]. Thus, the name residuum of R is because that Rimplication satisfies the residuation condition when the underlying t-norm R is left continuous: Rðx; zÞ 6 y if and only if IR ðx; yÞ P z: Generalizing the previous works (see [24,26,33,11,50,1]) related to the residuum of t-norm implications, a class of R-implications obtained by left-continuous t-subnorms are introduced. This extended approach has been investigated in many research works, as can be seen in [29]. We also study the corresponding dual constuction, the class R-coimplications obtained by right-continuous t-subconorms. According with [30], if (S) T is a t-(co)norm, then its residual is an R-(co)implication. However, if (S) T is a t-sub(co)norm, then their residual (IS) IT is not an R-(co)implication, in general. In the following, the next proposition provides conditions in order to guarantee such constructions. Proposition 4.14. Let (S) T be a t-sub(co)norm such that (S(x, 0) = 1 if and only if x = 1) T(x, 1) = 0 if and only if x = 0. Then, for all x, y 2 [0, 1], the function (IS) IT defined by IS ðx; yÞ ¼ inffz 2 U : Sðx; zÞ P yg; ð10Þ IT ðx; yÞ ¼ supfz 2 U : Tðx; zÞ 6 yg; ð11Þ is a fuzzy (co)implication, called R-(co)implication obtained by a sub(co)norm (S) T. Proof. For all x, y 2 U, IT(0, 0) = sup{z 2 U:T(0, z) 6 0} = supU = 1, since T(x, y) 6 ^ (x, y); IT(0, 1) = sup{z 2 U:T(0, z) 6 1} = supU = 1; IT(1, 1) = sup{z 2 U:T(1, z) 6 1} = supU = 1; IT(1, 0) = sup{z 2 U:T(1, z) 6 0} = sup{0} = 0, since T(x, 1) = 0 if, and only if, x = 0. And so, it holds that IT verifies the boundary conditions. Additionally, for all z 2 U, when x1 6 x2, T(x1, z) 6 T(x2, y). Then, for all x 2 U, sup{z 2 [0, 1]:T(x1, z) 6 y} P sup{z 2 [0, 1]:T(x2, z) 6 y} and by Eq. (11), it follows that IT(x1, y) 6 IT(x2, y). Therefore, IT verifies the first place antitonicity property. Finally, if y1 6 y2, T(x, z1) 6 y1 6 y2. Then it follows that sup{z 2 [0, 1]:T(x, z) 6 y1} 6 sup{z 2 [0, 1]: T(x, z) 6 y2} resulting in the inequality IT(x, y1) 6 IT(x, y2) and implying that IT verifies the second place monotonicity property. Analogously, it can be done to prove Eq. (10). h R.H.S. Reiser et al. / Information Sciences 253 (2013) 126–146 135 The main results for the development of this work, and related to R-(co)implication class obtained by left-continuous underlying t-sub(co)norms and its properties are presented below. Proposition 4.15. Let I:U2 ? U be a fuzzy implication. A function (IS)IT:U2 ? U, defined as an R-(co)implication with a leftcontinuous underlying t-sub(co)norm (S) T, satisfies the properties (J3, J7, J8) I3, I7, I8 and it is right-continuous in the second place. Proof. Properties I3 and I7 follow from Proposition 4.14 and [29, Theorem 2.4], respectively. Since T 6 ^ (x, y), when x 6 y it follows that T(x, 1) 6 x 6 y. Therefore, IT(x, y) = 1 which means that Property I8 is verified. Analogously, it can be done to prove that an R-coimplication IS verifies the properties J3, J7, J8 and it is right-continuous in the second place. h For a characterization of an R-implication underlying left-continuous t-norm, see [1, Theorem 2,5,17]. It is not the case of t-subnorm because the converse of Property I8 is not satisfied. Observe the implication IT P=2 : U 2 ! U, given by IT P=2 ðx; yÞ ¼ 1 if x 6 2y and IT p=2 ðx; yÞ ¼ 2y , otherwise. Thus, this converse x construction fails. According with Król [34], IT(x, y) = 1 if, and only if, x 6 y is verified just when T is a left-continuous t-norm. Proposition 4.16. Let q be an order automorphism and (IS) IT is an R-(co)implication underlying a t-sub(co)norm T. A conjugate of an R-(co)implication (IS) IT is also an R-(co)implication given as ðIT Þq ¼ IT q ; ð12Þ ðIS Þq ¼ ISq : ð13Þ Proof. Let q 2 Aut(U). For all w 2 U, there exists z 2 U such that w = q(z). Thus, for all x,y 2 U, the following holds: IqT ðx; yÞ ¼ q1 IT ðqðxÞ; qðyÞÞ by Eq: ð1Þ ¼ supfw 2 U : TðqðxÞ; wÞ 6 qðyÞg by Eq: ð11Þ ¼ ¼ supfqðzÞ 2 U : TðqðxÞ; qðzÞÞ 6 qðyÞg supfqðzÞ 2 U : qðT q ðx; zÞÞ 6 qðyÞg by Eq: ð6bÞ ¼ supfz 2 U : T q ðx; zÞ 6 qðyÞg by Eq: ð11Þ ¼ IT q ðx; yÞ: Analogously, it can be done to prove Eq. (13). h 5. Aggregating fuzzy implications Definition 5.1. Let A:Un ? U be an aggregation function and F ¼ fF i : U k ! Ug, for i 2 {1, 2, . . . , n} be a family of k-ary functions. An ðA; F Þ-operator on U, denoted by F A : U k ! U, is obtained as the composition given by: F A ðx1 ; . . . ; xk Þ ¼ AðF 1 ðx1 ; . . . ; xk Þ; F 2 ðx1 ; . . . ; xk Þ; . . . ; F n ðx1 ; . . . ; xk ÞÞ: ð14Þ Consider A:Un ? U as an aggregation function and F ¼ fF i : U 2 ! Ug, with i 2 {1, 2, . . . , n} as a family of binary functions in the following results in this section. Theorem 5.2. For all j 2 {1, . . . , 5, 9, 11, 12}, the ðA; F Þ-operator F A verifies the property Ij if each F i 2 F verifies the property Ij, for all i 2 {1, 2, . . . , n}. Proof. For all x, y, z 2 U it holds that: I1: By A1, since Fi verifies the property I1, we have that: F A ð0; 0Þ ¼ AðF 1 ð0; 0Þ; F 2 ð0; 0Þ; . . . ; F n ð0; 0ÞÞ ¼ Að1; 1; . . . ; 1Þ ¼ 1; F A ð1; 1Þ ¼ AðF 1 ð1; 1Þ; F 2 ð1; 1Þ; . . . ; F n ð1; 1ÞÞ ¼ Að1; 1; . . . ; 1Þ ¼ 1; F A ð0; 1Þ ¼ AðF 1 ð0; 1Þ; F 2 ð0; 1Þ; . . . ; F n ð0; 1ÞÞ ¼ Að1; 1; . . . ; 1Þ ¼ 1; F A ð1; 0Þ ¼ AðF 1 ð1; 0Þ; F 2 ð1; 0Þ; . . . ; F n ð1; 1ÞÞ ¼ Að0; 0; . . . ; 0Þ ¼ 0: Therefore, F A also verifies I1. I2: Let us assume that x 6 z. Since A and Fi, for all i 2 {1, 2, . . . , n}, verify the property A2 and I2, respectively, it holds that F A ðx; yÞ ¼ AðF 1 ðx; yÞ; F 2 ðx; yÞ; . . . ; F n ðx; yÞÞ P AðF 1 ðz; yÞ; F 2 ðz; yÞ; . . . ; F n ðz; yÞÞ ¼ F A ðz; yÞ: 136 R.H.S. Reiser et al. / Information Sciences 253 (2013) 126–146 I3: Let us assume that y 6 z. Since A and Fi, for all i 2 {1, 2, . . . , n}, verify the property A2 and I3, respectively, it holds that F A ðx; yÞ ¼ AðF 1 ðx; yÞ; F 2 ðx; yÞ; . . . ; F n ðx; yÞÞ 6 AðF 1 ðx; zÞ; F 2 ðx; zÞ; . . . ; F n ðx; zÞÞ ¼ F A ðx; zÞ: I4: By hypothesis, Fi satisfies I4, for all i 2 {1, 2, . . . , n}, thus by A1 we get F A ð0; yÞ ¼ AðF 1 ð0; yÞ; F 2 ð0; yÞ; . . . ; F n ð0; yÞÞ ¼ Að1; 1; . . . ; 1Þ ¼ 1: I5: By hypothesis, Fi verifies I5, for all i 2 {1, 2, . . . , n}, thus by A1 we get F A ðx; 1Þ ¼ AðF 1 ðx; 1Þ; F 2 ðx; 1Þ; . . . ; F n ðx; 1ÞÞ ¼ Að1; 1; . . . ; 1Þ ¼ 1: I9: By A1, it holds that F A ð0; 0Þ ¼ AðF 1 ð0; 0Þ; F 2 ð0; 0Þ; . . . ; F n ð0; 0ÞÞ ¼ Að1; 1; . . . ; 1Þ ¼ 1; and F A ð1; 1Þ ¼ AðF 1 ð1; 1Þ; F 2 ð1; 1Þ; . . . ; F n ð1; 1ÞÞ ¼ Að0; 0; . . . ; 0Þ ¼ 0: Let us assume that x 6 y. Therefore, by A2 and N2 we also have AðN1 ðxÞ; N2 ðxÞ; . . . ; Nn ðxÞÞ P AðN 1 ðyÞ; N2 ðyÞ; . . . ; Nn ðyÞÞ: By I9 we get F A ðx; 0Þ ¼ AðF 1 ðx; 0Þ; F 2 ðx; 0Þ; . . . ; F n ðx; 0ÞÞ P AðF 1 ðy; 0Þ; F 2 ðy; 0Þ; . . . ; F n ðy; 0ÞÞ ¼ F A ðy; 0Þ: Concluding, we proved that F A ð; 0Þ is a fuzzy negation. I11: If Fi(x, x) = 1 for all x 2 U and all i 2 {1, 2, . . . , n}, then by I1 we get F A ðx; yÞ ¼ AðF 1 ðx; xÞ; F 2 ðx; xÞ; . . . ; F n ðx; xÞÞ ¼ Að1; 1; . . . ; 1Þ ¼ 1 Therefore, Theorem 5.2 is proved. h Proposition 5.3. Let N be a fuzzy negation. The ðA; F Þ-operator F A verifies the property I12 with respect to N each F i 2 F , for all i 2 {1, 2, . . . , n}, verify the property I12 with respect to N. Proof. For all y 2 U, if Fi(x,y) = Fi(N(y), N(x)) for all i 2 {1, 2, . . . , n} and some fuzzy negation N, then it holds that F A ðx; yÞ ¼ AðF 1 ðx; yÞ; F 2 ðx; yÞ; . . . ; F n ðx; yÞÞ by Eq: ð14Þ ¼ AðF 1 ðNðyÞ; NðxÞÞ; F 2 ðNðyÞ; NðxÞÞ; . . . ; F n ðNðyÞ; NðxÞÞÞ by I12: Therefore, F A ðx; yÞ ¼ F A ðNðyÞ; NðxÞÞ. h Proposition 5.4. For j 2 {6, 10, 18} the ðA; F Þ-operator F A verifies the property Ij if A is an idempotent function and F i 2 F verifies the property Ij, for all i 2 {1, 2, . . . , n}. Proof. For all i 2 {1, 2, . . . , n} and x, y, z 2 U, it holds that: I6: By hypothesis, Fi verifies I6, for all i 2 {1, 2, . . . , n}, thus F A ð1; yÞ ¼ AðF 1 ð1; yÞ; F 2 ð1; yÞ; . . . ; F n ð1; yÞÞ ¼ Aðy; y; . . . ; yÞ: Since A is idempotent aggregation function, we conclude that F A ð1; yÞ ¼ y. I10: By A2 and A4a, if Fi verifies I10, for all i 2 {1, 2, . . . , n}, then F A ðx; yÞ ¼ AðF 1 ðx; yÞ; F 2 ðx; yÞ; . . . ; F n ðx; yÞÞ P Aðy; y; . . . ; yÞ ¼ y: I18: Since Fi verifies I18, for all i 2 {1, 2, . . . , n}, then F A ðx; NðxÞÞ ¼ AðF 1 ðx; NðxÞÞ; F 2 ðx; NðxÞÞ; . . . ; F n ðx; NðxÞÞÞ ¼ AðNðxÞ; NðxÞ; . . . ; NðxÞÞ ¼ NðxÞ:  Proposition 5.5. F A verifies I7 when F i 2 F verifies, for all i 2 {1, 2, . . . , n}, the next two conditions: 1. the distributive property, which means, for all x, y1, y2, . . . , yn 2 U: AðF i ðx; y1 Þ; F i ðx; y2 Þ; . . . ; F i ðx; yn ÞÞ ¼ F i ðx; Aðy1 ; y2 ; . . . ; yn ÞÞ; ð15Þ R.H.S. Reiser et al. / Information Sciences 253 (2013) 126–146 137 2. the generalized exchange principle,1meaning that, for all 0 6 i, j 6 n: F i ðx; F j ðy; zÞÞ ¼ F i ðy; F j ðx; zÞÞ: ð16Þ Proof. For all x, y, z 2 U we have F A ðx; F A ðy; zÞÞ ¼ AðF 1 ðx; AðF 1 ðy; zÞ . . . ; F n ðy; zÞÞÞ; . . . ; F n ðx; AðF 1 ðy; zÞ . . . ; F n ðy; zÞÞÞÞ by Eq: ð14Þ ¼ AðAðF 1 ðx; F 1 ðy; zÞÞ; . . . ; F 1 ðx; F n ðy; zÞÞÞ; . . . ; AðF n ðx; F 1 ðy; zÞÞ; . . . ; F n ðx; F n ðy; zÞÞÞÞ by Eq: ð15Þ ¼ AðAðF 1 ðy; F 1 ðx; zÞÞ; . . . ; F 1 ðy; F n ðx; zÞÞÞ; . . . ; AðF n ðy; F 1 ðx; zÞÞ; . . . ; F n ðy; F n ðx; zÞÞÞÞ by Eq: ð16Þ ¼ AðF 1 ðy; AðF 1 ðx; zÞ; . . . ; F n ðx; zÞÞÞ; . . . ; F n ðy; AðF 1 ðx; zÞ; . . . ; F n ðx; zÞÞÞÞ by Eq: ð15Þ ¼ F A ðy; F A ðx; zÞÞ: by Eq: ð14Þ: Therefore, F A verifies I7. h Proposition 5.6. F A verifies I8 if A satisfies A5a and F i 2 F verifies I8, for all i 2 {1, 2, . . . , n}. verifies I13, when x6y we get Proof. Let us fix arbitrarily x, y 2 U. By A1, since Fi F A ðx; yÞ ¼ AðF 1 ðx; yÞ; F 2 ðx; yÞ; . . . ; F n ðx; yÞÞ ¼ Að1; 1; . . . ; 1Þ ¼ 1. By the converse, if F A ðx; yÞ ¼ 1, then by A5a there exists i 2 {1, . . . , n} such that Fi(x, y) = 1, which implies that x 6 y. Therefore, F A verifies I8. h Proposition 5.7. For j 2 {13, 14}, the operator F A verifies Ij if A verifies A6 and F i 2 F verifies the property Ij, for all i 2 {1, 2, . . . , n}. Proof. Consider an aggregation A verifying A6 and F i 2 F verifying the properties I13 and I14. Then, for all x, y 2 U, we have the following: I13: By A6 and since Fi verifies I13 when x > 0, it holds that F A ðx; 0Þ ¼ AðF 1 ðx; 0Þ; F 2 ðx; 0Þ; . . . ; F n ðx; 0ÞÞ < Að1; 1; . . . ; 1Þ ¼ 1: I14: By A6 and since Fi verifies I14, when x < 1, it holds that F A ð1; yÞ ¼ AðF 1 ð1; yÞ; F 2 ð1; yÞ; . . . ; F n ð1; yÞÞ < Að1; 1; . . . ; 1Þ ¼ 1: Therefore, the operator F A verifies I13 and I14. h Proposition 5.8. F A verifies I15 if A is a continuous aggregation function and Fi verifies I15 for all i 2 {1, 2, . . . , n}. Proof. It follows from the composition of continuous functions. h Proposition 5.9. F A satisfies the property I16 whenever A is an idempotent aggregation function satisfying Eq. (15) and for all x, y 2 U and both F i ; F j 2 F verify the next generalized property: F i ðx; F j ðx; yÞÞ ¼ F i ðx; yÞ: Proof. Let us fix arbitrarily x, y 2 U. Since each Fi verifies Eq. (17), for all i, j 2 {1, 2, . . . , n}, it holds that 1 This property can also be considered as a generalization of the extended migrative property, see [27, Definition 2]. ð17Þ 138 R.H.S. Reiser et al. / Information Sciences 253 (2013) 126–146 F A ðx; F A ðx; yÞÞ ¼ AðF 1 ðx; AðF 1 ðx; yÞ; . . . ; F n ðx; yÞÞÞ; . . . ; F n ðx; AðF 1 ðx; yÞ; . . . ; F n ðx; yÞÞÞÞ; by Eq: ð14Þ ¼ AðAðF 1 ðx; F 1 ðx; yÞÞ; . . . ; F 1 ðx; F n ðx; yÞÞÞ; . . . ; AðF n ðx; F 1 ðx; yÞÞ; . . . ; F n ðx; F n ðx; yÞÞÞÞ; by Eq: ð15Þ ¼ AðAðF 1 ðx; yÞ; . . . ; F 1 ðx; yÞÞ; . . . ; AðF n ðx; yÞ; . . . ; F n ðx; yÞÞÞ; by Eq: ð17Þ ¼ AðF 1 ðx; yÞ; . . . ; F n ðx; yÞÞ by A4 ¼ F A ðx; yÞ; by Eq: ð14Þ: Therefore, F A satisfies the property I16. h Proposition 5.10. F A satisfies the property I17 whenever A verifies A5 and F i 2 F is a distributive function in the sense of Eq. (15) and also verifying I17, for 1 6 i 6 n. Proof. Let us fix arbitrarily x, y 2 U. We get F A ðx; F A ðy; xÞÞ ¼ AðF 1 ðx; AðF 1 ðy; xÞ; . . . ; F n ðy; xÞÞÞ; . . . ; F n ðx; AðF 1 ðy; xÞ; . . . ; F n ðy; xÞÞÞÞ; by Eq: ð14Þ ¼ AðAðF 1 ðx; F 1 ðy; xÞÞ; . . . ; F 1 ðx; F n ðy; xÞÞÞ; . . . ; AðF n ðx; F 1 ðy; xÞÞ; . . . ; F n ðx; F n ðy; xÞÞÞÞ; by Eq: ð15Þ ¼ AðAð1; . . . ; F 1 ðx; F n ðy; xÞÞÞ; . . . ; AðF n ðx; F 1 ðy; xÞÞ; . . . ; 1ÞÞ; by I17 ¼ Að1; . . . ; 1Þ ¼ 1 by A1 and A5 Therefore F A satisfies the property I17. h Corollary 5.11. Let I ¼ fIi : U 2 ! Ug, with i 2 {1, 2, . . . , n} be a family of fuzzy implications. Then the operator I A : U 2 ! U is also a fuzzy implication named fuzzy ðA; I Þ-implication. Proof. Follows from Theorem 5.2. h 5.1. Automorphisms acting on aggregating fuzzy implications In the following propositions, in order to analyze the action of an order automorphisms on aggregating fuzzy implications, consider q as an order automorphism and A:Un ? U as an aggregation function. n o q Proposition 5.12. Let F q ¼ F i : U k ! U F i 2 F be a family of conjugate functions. Then ðF A Þq : U k ! U is an ðA; F Þ-operator given as: ðF A Þq ¼ ðF q ÞAq : Proof. By Proposition 3.2 and Corollary 3.5, for all x1, . . . , xk 2 U, we have
ðF q ÞAq ðx1 ; . . . ; xk Þ ¼ Aq F q1 ðx1 ; . . . ; xk Þ; . . . ; F qn ðx1 ; . . . ; xk Þ by Eq: ð14Þ



¼ q1 A q F q1 ðx1 ; . . . ; xk Þ ; . . . ; q F qn ðx1 ; . . . ; xk Þ by Eq: ð1Þ ¼ q1 ðAððq  q1 ÞF 1 ðqðx1 Þ; . . . ; qðxk ÞÞ; . . . ; ðq  q1 ÞF n ðqðx1 Þ; . . . ; qðxk ÞÞÞÞ by Eq: ð1Þ ¼ q1 ðAðF 1 ðqðx1 Þ; . . . ; qðxk ÞÞ; . . . ; F n ðqðx1 Þ; . . . ; qðxk ÞÞÞÞ by Eq: ð14Þ ¼ ðF A Þq ðx1 ; . . . ; xk Þ: Therefore, ðF A Þq ¼ ðF q ÞAq . h n o Corollary 5.13. Let I q ¼ Iqi : U k ! U Ii 2 I be a family of conjugate fuzzy implications. Then ðI A Þq : U 2 ! U is an ðA; I Þ-implication given as: ðI A Þq ¼ ðI q ÞAq : Proof. Follows from Corollaries 3.5 and 5.11 and Propositions 4.6 and 5.12. h ð18Þ R.H.S. Reiser et al. / Information Sciences 253 (2013) 126–146 139 5.2. Duality principle acting on aggregation fuzzy implications Let F ¼ fF i : U k ! Ug, with i 2 {1, 2, . . . , n} be a family of functions and A:Un ? U be an aggregation function. In order to analyze the N-dual construction of an ðA; F Þ-operator F A , which is considered in the next proposition and corollary, we assume that N is a strong FN. Proposition 5.14. The ðF A ÞN : U k ! U is an ðA; F Þ-operator named as N-dual operator of F A and defined as: ðF A ÞN ¼ ðF N ÞAN ; where F N ¼ fðF i ÞN : U k ! Ug. Proof. Let ~ x ¼ ðx1 ; . . . ; xk Þ 2 U k . It holds that ðF A ÞN ð~ xÞ ¼ NðF A ðNð~ xÞÞÞ by Eq: ð3Þ ¼ NðAðF 1 ðNð~ xÞÞ; . . . ; F n ðNð~ xÞÞÞÞ by Eq: ð14Þ xÞ; . . . ; NðF nN ð~ xÞÞÞÞÞ by Eq: ð3Þ ¼ NðAðNðF 1N ð~ ~ ~ ¼ AN ððF 1 ÞN ðxÞ; . . . ; ðF n ÞN ðxÞÞ by Eq: ð14Þ ¼ ðF N ÞAN ð~ xÞ: Therefore, ðF A ÞN ¼ ðF N ÞAN . h Corollary 5.15. Let I ¼ fIi : U 2 ! Ug, with i 2 {1, 2, . . . , n} be a family of fuzzy implications. Then ðI A ÞN : U 2 ! U is a fuzzy ðA; I Þ-implication named as the N-dual fuzzy implication (or coimplication) of I A and characterized as: ðI A ÞN ¼ I N AN : Proof. Follows from Corollary 5.11 and Proposition 5.14. h 6. Aggregating fuzzy connectives Proposition 6.1. Let A:Un ? U be an aggregation function and ðSÞT ¼ fðSi ÞT i : U 2 ! Ug, with i 2 {1, 2, . . . , n} be a family of tsub(co)norms. Then (S A ) T A is a t-sub(co)norm whenever (Si) Ti verifies the distributivity and generalized associativity,2 which means that for all i, j such that 0 6 i, j 6 n and x, y, z 2 [0, 1], we have the following: Si ðx; Sj ðy; zÞÞ ¼ Si ðSj ðx; yÞ; zÞ; ð19Þ T i ðx; T j ðy; zÞÞ ¼ T i ðT j ðx; yÞ; zÞ: ð20Þ Proof. A1: We easily get S A ð0; 0Þ ¼ AðS1 ð0; 0Þ; . . . ; Sn ð0; 0ÞÞ ¼ Að~ 0Þ ¼ 0 and similarly S A ð1; 1Þ ¼ AðS1 ð1; 1Þ; . . . ; Sn ð1; 1ÞÞ ¼ Að~ 1Þ ¼ 1. A2: For all x, y 2 U we get S A ðx; yÞ ¼ AðS1 ðx; yÞ; . . . ; Sn ðx; yÞÞ ¼ AðS1 ðy; xÞ; . . . ; Sn ðy; xÞÞ ¼ S A ðy; xÞ. A3: Let A and Si, for 0 6 i 6 n, be functions verifying A2 and S4, respectively. When x 6 z, then it follows that S A ðx; yÞ ¼ AðS1 ðx; yÞ; . . . ; Sn ðx; yÞÞ 6 AðS1 ðz; yÞ; . . . ; Sn ðz; yÞÞ ¼ S A ðz; yÞ. S1: If x, y, z 2 ]0, 1], then S A ðx; S A ðy; zÞÞ ¼ AðS1 ðx; AðS1 ðy; zÞ; . . . ; Sn ðy; zÞÞÞ; . . . ; Sn ðx; AðS1 ðy; zÞ; . . . ; Sn ðy; zÞÞÞÞ by Eq: ð14Þ and A3 ¼ AðAðS1 ðx; S1 ðy; zÞÞ; . . . ; S1 ðx; Sn ðy; zÞÞÞ; . . . ; AðSn ðx; S1 ðy; zÞÞ; . . . ; Sn ðx; Sn ðy; zÞÞÞÞ by Eq: ð15Þ ¼ AðAðS1 ðS1 ðx; yÞ; zÞ; . . . ; S1 ðSn ðx; yÞ; zÞÞ; . . . ; AðSn ðS1 ðx; yÞ; zÞ; . . . ; Sn ðSn ðx; yÞ; zÞÞÞ by Eq: ð19Þ ¼ AðAðS1 ðz; S1 ðx; yÞÞ; . . . ; S1 ðz; Sn ðx; yÞÞÞ; . . . ; AðSn ðz; S1 ðx; yÞÞ; . . . ; Sn ðz; Sn ðx; yÞÞÞÞ by S1 and A3 2 Eqs. (19) and (20) are a particular cases of Eq. (GA) in [38]. 140 R.H.S. Reiser et al. / Information Sciences 253 (2013) 126–146 Sn ðz; AðS1 ðx; yÞ; . . . ; Sn ðx; yÞÞÞÞ by Eq: ð15Þ AðS1 ðz; AðS1 ðx; yÞ; . . . ; Sn ðx; yÞÞÞ; . . . ; ¼ ¼ S A ðS A ðx; yÞ; zÞ by S1; Eq:ð14Þ and A3 The analogous construction can be done in order to prove Eq. (20). h A t-sub(co)norm (Si) Ti is associative with respect to a t-sub(co)norm (Sj) Tj when the pair of t-sub(co)norms ((Si, Sj)) (Ti, Tj) satisfies (Eqs. (19) and (20). Let (S1, S2) T1, T2 be a t-(co)norms. It is immediate to verify that a t-(co)norm (S1) T1 is associative with respect to a t-(co)norm (S2) T2 iff (S1 = S2) T1 = T2. And by Proposition 6.1, if a pair of t-subnorms T1, T2 verifies the generalized associativity, it follows that at least one of them is not a t-norm on U. Example 6.2. Let A be a 1-homogeneous aggregation, e.g. the arithmetic mean aggregation M in Example 3.4 and t-subnorms T P=k1 and T P=k2 as given in Example 3.10. (i) For all x, y, z 2 U and k1, k2 2 {1, 2, . . . }, we obtain that       1 1 1 1 1 1 1 T P=k1 ðx; T P=k2 ðy; zÞÞ ¼ T P=k1 x; ðyzÞ ¼ x ðyzÞ ¼ xðyzÞ ¼ ðxyÞz ¼ ðxyÞz : k2 k1 k2 k1 k2 k2 k1 k2 k1 Therefore T P=k1 ðx; T P=k2 ðy; zÞÞ ¼ T P=k2 ðT P=k1 ðx; yÞ; zÞ, concluding that T ¼ fT P ; T P=2 g verifies Eq. (20) which means T P=k1 is associative with respect to a t-sub(co)norm T P=k2 and vice versa. (ii) For all x, y1, . . . , yn 2 U and k1 2 {1, 2, . . . }, it follows that: AðT P=k1 ðx; y1 Þ; T P=k1 ðx; y2 Þ; . . . ; T P=k1 ðx; yn ÞÞ ¼ A  1 1 1 xy ; xy ; . . . ; xyn k1 1 k1 2 k1  ¼ 1 xAðy1 ; y2 ; . . . ; yn Þ: k1 Therefore, AðT P=k1 ðx; y1 Þ; T P=k1 ðx; y2 Þ; . . . ; T P=k1 ðx; yn ÞÞ ¼ T P=k1 ðx; Aðy1 ; y2 ; . . . ; yn ÞÞ. So, for all k 2 {1,2, . . . }, TP/k verifies the distributivity property. In the case of the arithmetic mean M, whenever T ¼ fT P ; T P=2 g, we have the following t-subnorm: T M ðx; yÞ ¼ 3 xy; for all x; y 2 U: 4 ð21Þ Analogously, we can extend this example to the family of NS-dual functions of TP/a, the t-subnorms SP/a also introduced in Example 3.10. In particular, whenever S ¼ fSP ; SP=2 ; SP=3 g by the arithmetic mean M we have that S M ðx; yÞ ¼ MðSP;N ððx; yÞ; SP=2;N ðx; yÞ; SP=3;N ðx; yÞÞ resulting on the t-subconorm: S M ðx; yÞ ¼ 7 11 þ ðx þ y  xyÞ; for all x; y 2 U: 18 18 ð22Þ Proposition 6.3. If a pair of t-sub(co)norms ((S1, S2)) (T1, T
2
) satisfies
the generalized associativity then for all q 2 Aut(U) it holds that the corresponding pair of conjugate functions Sq1 ; Sq2 T q1 ; T q2 also satisfies the generalized associativity. Proof. For all x, y, x 2 U, if (T1, T2) satisfies the generalized associativity, it holds that
T q1 x; T q2 ðy; zÞ ¼ q1 ðT 1 ðqðxÞ; T 2 ðqðyÞ; qðzÞÞÞ by Eq: ð5Þ ¼ q1 ðT 1 ðT 2 ðqðxÞ; qðyÞÞ; qðzÞÞÞ by Eq: ð20Þ

¼ T q1 T q2 ðx; yÞ; z by Eq: ð5Þ:
Therefore, T q1 ; T q2 satisfies the generalized associativity. h Proposition 6.4. If A is an aggregation verifying the distributivity with respect to a t-sub(co)norms (S) T then, for all q 2 Aut(U) the corresponding conjugate aggregation Aq is an aggregation verifying the distributivity with respect to the corresponding conjugate t-sub(co)norm ((Sq)) Tq. Proof. For all x, y, x 2 U, let A be an aggregation verifying the distributivity with respect to a t-subnorm T. Then, it holds that: Aq ðT q ðx; y1 Þ; . . . ; T q ðx; yn ÞÞ ¼ q1 ðAðTðqðxÞ; qðy1 ÞÞ; . . . ; TðqðxÞ; qðyn ÞÞÞ by Eq: ð5Þ ¼ q1 ðTðqðxÞ; Aðqðy1 ÞÞ; . . . ; qðyn ÞÞÞ byProposition 6:1 ¼ q1 TðqðxÞ; qðAq ðy1 ; . . . ; yn ÞÞÞÞ by Eq: ð5Þ ¼ T q ðx; Aqðy1 ; . . . ; yn ÞÞ R.H.S. Reiser et al. / Information Sciences 253 (2013) 126–146 141 Therefore, Aq verifies the distributivity with respect to the Tq. h Corollary
6.5. Let q be an order automorphism, A:Un ? U be an aggregation function and ðSÞT be a family of t-sub(co)norms. Then S qA T qA : U 2 ! U is t-sub(co)norm given as ðS A Þq ¼ ðSq ÞAq ; ðT A Þq ¼ ðT q ÞAq ; whenever A and ðSÞT verifies A4 and (Eq. (19)) Eq. (20), respectively. Proof. Follows from Corollaries 3.5, 5.11 and Propositions 3.2, 5.12, 6.1, 6.3 and 6.4. h Based on results reported above, in Corollary 6.3 and Proposition 3.3, by considering Example 6.2, we can obtain an infin
ity family S qAq T qAq of t-sub(co)norms. Proposition 6.6. Let A:Un ? U be an aggregation function and N ¼ fN i : U ! Ug, with i 2 {1, 2, . . . , n} be a family of fuzzy negations. Then N A is a fuzzy negation. Moreover, when A and Ni are (strict) continuous, then N A is also a (strict) continuous fuzzy negation. Proof. N1: By A1 we get N A ð0Þ ¼ AðN 1 ð0Þ; . . . ; N n ð0ÞÞ ¼ Að1; . . . ; 1Þ ¼ 1 and similarly N A ð1Þ ¼ AðN 1 ð1Þ; . . . ; N n ð1ÞÞ ¼ Að0; . . . ; 0Þ ¼ 0. N2: Let A and Ni, for 0 6 i 6 n, be functions verifying A2 and N2, respectively. When x 6 z, it follows that N A ðx; yÞ ¼ AðN 1 ðxÞ; . . . ; N n ðxÞÞ P AðN 1 ðyÞ; . . . ; N n ðyÞÞ ¼ N A ðyÞ. In addition, when A and N are continuous functions, then, by composition, N A is also a continuous function. And, when A and N are both strict functions and we assume that x < z, then N A ðx; yÞ ¼ AðN 1 ðxÞ; . . . ; N n ðxÞÞ < AðN 1 ðyÞ; . . . ; N n ðyÞÞ ¼ N A ðyÞ. h Corollary 6.7. Let q be an order automorphism, A:Un ? U be an aggregation function and N ¼ fN i : U ! Ug, with i 2 {1, 2, . . . , n} q be a family of fuzzy negations. Then it holds that an ðA; N Þ-operator defined by N A : U ! U is a fuzzy negation if A verifies the idempotent property. Proof. Follows from Corollary 5.11 and Propositions 2.3 and 5.12. h 7. Aggregating fuzzy (S, N)-(co)implications obtained by fuzzy negations and t-sub(co)norms Proposition 7.1. Let A:Un ? U be an aggregation function and I ¼ fISi ;Ni : U 2 ! Ug, with i 2 {1, 2, . . . , n} be a family of (S, N)implications given by Definition 4.7. Then an I A -operator is an (S, N)-implication according with Definition 4.7 whenever one of the two conditions hold: 1. (Si, Sj) verifies Eq. (19), for all x, y 2 U and Ni = N, for 0 6 i,j 6 n, 2. (A, S) verifies Eq. (15), for all (x, yi) 2 U2 and Si = S, for all 0 6 i 6 n. Such I A -operator is denoted as I A;S;N or I A;S;N , depending on which of these two conditions are verified. Proof. Let I ¼ fISi ;Ni : U 2 ! Ug, with i 2 {1, 2, . . . , n} be a family of (S, N)-implications. 1. Firstly, if (Si, Sj) verifies Eq. (19), for all x, y 2 U and Ni = N, for 0 6 i, j 6 n, then it holds that: I A;S;N ðx; yÞ ¼ AðIS1 ;N ðx; yÞ; . . . ; ISn ;N ðx; yÞÞ Eq: ð14Þ ¼ AðS1 ðNðxÞ; yÞ; . . . ; Sn ðNðxÞ; yÞÞ Eq: ð9Þ ¼ S A ðNðxÞ; yÞ Eq: ð14Þ: Implying that I A;S;N ¼ I SA ;N . 2. Now, if (A, S) verifies Eq. (15), for all (x, yi) 2 U2 and Si = S, for all 0 6 i 6 n, we have that: 142 R.H.S. Reiser et al. / Information Sciences 253 (2013) 126–146 I A;S;N ðx; yÞ ¼ AðIS;N1 ðx; yÞ; . . . ; IS;Nn ðx; yÞÞ Eq: ð14Þ ¼ AðSðN1 ðxÞ; yÞ; . . . ; SðNn ðxÞ; yÞÞ Eq: ð14Þ ¼ AðSðy; N1 ðxÞÞ; . . . ; Sðy; Nn ðxÞÞ by S1 ¼ Sðy; AðN1 ðxÞÞ; . . . ; Nn ðxÞÞÞ Eq: ð15Þ ¼ SðN A ðxÞ; yÞ Eq: ð14Þ: Implying that I A;S;N ¼ I S;NA . Therefore I A -operator is an (S, N)-implication. h Proposition 7.2. Let A:Un ? U be an aggregation function and I ¼ fIi : U 2 ! Ug, with i 2 {1, 2, . . . , n} be a family of functions satisfying Eqs. (15) and (16). I A is an (S, N)-implication underlying a continuous fuzzy negation and a t-subconorm S at point 0, if Ii verifies I2,I7, and I9b. Proof. Follows from Propositions 4.10 and 5.5 together with Theorem 5.2. h Example 7.3. Let M be the media aggregation and SP/a be the family of t-subconorms given in Example 3.10. Taking N = NS, for all x, y 2 U we have that I M;SM ;N ðx; yÞ ¼ MðISP ;N ðx; yÞ; ISP=2 ;N ðx; yÞ; ISP=3 ;N ðx; yÞÞ   1 1 1 11 1  x þ xy þ ð2  x þ xyÞ þ ð3  x þ xyÞ ¼ 1  ðx  xyÞ ¼ 3 2 3 18 Moreover, by Eq. (22) it holds that S M ðNðxÞ; yÞ ¼ MðSP;N ððNðxÞ; yÞ; SP=2;N ðNðxÞ; yÞ; SP=3;N ðNðxÞ; yÞÞ ¼ 1  11 ðx  xyÞ: 18 Therefore, I M;S M ;N ðx; yÞ ¼ S M ðNðxÞ; yÞ. Moreover, since ðT M ; S M Þ is a pair of N-dual operators we can also conclude that I M;SM ;N ðNðxÞ; yÞ ¼ NðT M ðx; NðyÞÞ. Additionally, according to Proposition 7.1 the I A -operator given by I M;S;N ðx; yÞ ¼ ( 1  11 ðx  xyÞ; if x 2 ½0; 1½and y 20; 1; 18 0; otherwise: ð23Þ is an (S,N)-implication. Corollary 7.4. Let q be an order automorphism, A:Un ? U be an aggregation function and I ¼ fIi : U 2 ! Ug, with i 2 {1, 2, . . . , n} be a family of ((T, N)) (S, N)-implications according with Definition 4.7. Then I qA : U 2 ! U is an ((T, N)) (S, N)-implication. Proof. Follows from Propositions 4.8, 5.12 and 7.1. h Proposition 7.5. Let q be an order automorphism, A:Un ? U be an aggregation function and I ¼ fIi : U 2 ! Ug, with i 2 {1, 2, . . . , n} be a family of ((T, N)) (S, N)-implications given by Definition 4.7. Then the following equations are true: ðI A;S;N Þq ¼ I Aq ;Sq ;Nq ; ð24Þ ðI A;T;N Þq ¼ I Aq ;T q ;N q : ð25Þ Proof. Let I ¼ fIi : U 2 ! Ug, with i 2 {1, 2, . . . , n} be a family of (S, N)-implications according with Proposition 7.1. For all x, y 2 U we get ðI A;S;N Þq ðx; yÞ ¼ q1 ðI A;S;N ÞðqðxÞ; qðyÞÞ by Eq: ð18Þ ¼ q1 ðAðIS;N1 ðqðxÞ; qðyÞÞ; . . . ; IS;Nn ðqðxÞ; qðyÞÞÞÞ by Eq: ð14Þ ¼ q1 ðAðSðN1 ðqðxÞÞ; qðyÞÞ; . . . ; SðN n ðqðxÞÞ; qðyÞÞÞÞ Eq: ð2Þ ¼ q1 ðAðSðqðNq1 ðxÞÞ; qðyÞÞ; . . . ; SðqðNqn ðxÞÞ; qðyÞÞÞÞ Eq: ð4Þ



¼ q1 A q Sq Nq1 ðxÞ; y ; . . . ; qðSq ðNqn ðxÞ; yÞÞ Eq: ð1Þ


q q q q q ¼ A S N1 ðxÞ; y ; . . . ; S Nn ðxÞ; y Eq: ð18Þ ¼ I Aq ;Sq ;N q ðx; yÞ: 143 R.H.S. Reiser et al. / Information Sciences 253 (2013) 126–146 Therefore Eq. (24) holds. The other equation can be analogously proved. h Corollary 7.6. Let N be a strong FN, A:Un ? U be an aggregation function and I ¼ fIi : U 2 ! Ug, with i 2 {1, 2, . . . , n} be a family of (T) S-implications in Definition 4.7. Then it holds that ðI A ÞN : U 2 ! U is an (T-implication) S-coimplication by Definition 4.7. Proof. Follows from Propositions 4.9 and 7.1 together with Corollary 5.11. h Proposition 7.7. Let N be a strong FN, A:Un ? U be an aggregation function and I ¼ fIi : U 2 ! Ug, with i 2 {1, 2, . . . , n} be a family of ((T, N)) (S, N)-implications verifying the first condition (item 1.) in Proposition 7.1. Then it holds that ðI A;S;N ÞN ¼ I AN ;SN ;N ; ð26Þ ðI A;T ;N ÞN ¼ I AN ;T N ;N : ð27Þ Proof. For all x, y 2 U, it holds that: ðI A;S;N ÞN ðx; yÞ ¼ NðAðIS1 ;N ðNðxÞ; NðyÞÞ; . . . ; ISn ;N ðNðxÞ; NðyÞÞÞÞ Eq: ð14Þ ¼ NðAðS1 ðN2 ðxÞ; NðyÞÞ; . . . ; Sn ðN2 ðxÞ; NðyÞÞÞÞ by N3 ¼ NðAðNðS1N ðNðxÞ; yÞÞ; . . . ; NðSnN ðNðxÞ; yÞÞÞ by Eq:ð3Þ ¼ AN ðS1N ðNðxÞ; yÞÞ; . . . ; SnN ðNðxÞ; yÞÞ by Eq: ð3Þ ¼ I AN ;SN ;N ðx; yÞ Eq: ð14Þ: Therefore, Eq. (26) holds. And, Eq. (27) can be analogously proved. h 8. Aggregating fuzzy R-(co)implications obtained by t-sub(co)norms We have obtained a new method of generating R-(co)implications called aggregation fuzzy R-(co)implications by considering an aggregation function A:Un ? U and a family of R-(co)implications: J R ¼ fJ Si : U 2 ! Ug; underlying right-continuous t-subconorms fSi gi2f1;2;...;ng ; I R ¼ fIT i : U 2 ! Ug; underlying left-continuous t-subnorms fT i gi2f1;2;...;ng : The following results related to a family of R-implications I can be obtained analogously to a family of R-coimplications J . Proposition 8.1. Let A:Un ? U be an aggregation function and I R ¼ fIT i : U 2 ! Ug be a family of R-implications underlying leftcontinuous t-subnorms Ti, with i 2 {1, 2, . . . , n}. I A -operator is an R-implication, denoted by I A;T , if one of the two cases holds: 1. A is an idempotent aggregation and Ti = T for all 1 6 i 6 n; 2. A is a continuous aggregation such that A 6 min and T verifies the generalized associativity. Proof 1. Since A is an idempotent aggregation, it holds that I A;T ðx; yÞ ¼ AðIT ðx; yÞ; . . . ; IT ðx; yÞÞ ¼ IT ðx; yÞ. Therefore, I A;T is an Rimplication. 2. Consider that A 6 min and t = A(t1, . . . , tn). When Ti(x, ti) 6 y, Ti(x, t) 6 y. So, A(T1(x, t), . . . , Tn(x, t)) 6 y. Therefore, we have the following inclusion: ftjAðT 1 ðx; tÞ; . . . ; T n ðx; tÞÞ 6 yg # ftjAðT 1 ðx; t 1 Þ; . . . ; T n ðx; t n ÞÞ 6 yg: And, as a consequence, IT A ðx; yÞ ¼ supftjAðT 1 ðx; tÞ; . . . ; T n ðx; tÞÞ 6 yg T n ðx; t n ÞÞ 6 yg ¼ I A;T ðx; yÞ. On the other hand, since A is a continuous aggregation, we have that which means IT A ðx; yÞ 6 supftjAðT 1 ðx; t 1 Þ; . . . ; 144 R.H.S. Reiser et al. / Information Sciences 253 (2013) 126–146 I A;T ðx; yÞ ¼ AðIT 1 ðx; yÞ; . . . ; IT n ðx; yÞÞ Eq: ð14Þ ¼ Aðsupft 1 jT 1 ðx; t1 Þ 6 yg; . . . ; supft n jT n ðx; tn Þ 6 ygÞ by Eq: ð11Þ ¼ AðsupftjT i ðx; t i Þ 6 y and 1 6 i 6 ngÞ 6 6 supftjAðT 1 ðx; t1 Þ; . . . ; T n ðx; t n ÞÞ 6 yg supftjAðT 1 ðx; tÞ; . . . ; T n ðx; tÞÞ 6 yg Eq: ð14Þ ¼ IT A ðx; yÞ: Thus I A;T ¼ IT A . In addition, by Proposition 6.1, T A is a t-subnorm. Therefore, we can conclude that I A;T is an Rimplication. h Example 8.2. Let TP and TP/2 be t-subnorms as presented in Example 3.10, and M be the arithmetic mean in Example 3.4, which is distributed over TP and TP/2 and verifies 1-homogeneity property, as requested in Proposition 6.1. Therefore, taking T ¼ fT P ; T P=2 g, by the previous result, for all x, y 2 U, we have that I M;T ðx; yÞ ¼ IT M ðx; yÞ ¼ ( 4y ; 3x if y < 34 x; 1; otherwise: ð28Þ Remark 8.3. Consider now the correspondence between the elements in the image sets of a fuzzy implication I and an aggregation function A with annihilator 0, verifying the following condition Iðx; yÞ ¼ NS ðAðx; NS ðyÞÞ; for all x; y; 2 U: ð29Þ Taking the fuzzy implication I M;T presented in Example 8.2, for all x, y 2 U we have that Aðx; yÞ ¼ NS ðI M;T ðx; NðyÞÞÞ ¼ ( 1  4ð1yÞ ; if 4ð1  yÞ < 3x; 3x 0; otherwise: ð30Þ One can observe that A is not an aggregation since we have that (i) neither the associativity property:            1 15 15 11 8 1 11 1 15 A A ;1 ; – ¼A ; ¼ A 1; ¼ ¼ A ; A 1; ; 8 16 16 12 9 8 12 8 16 (ii) nor the commutativity property:     1 1 ¼ 0; A ; 1 ¼ 1 – A 1; 8 8 are verified. Concluding, I M;T cannot be explicitly represented by Eq. (29). Proposition 8.4. Let A:Un ? U be a right-continuous aggregation function verifying A5 and I ¼ fIi : U 2 ! Ug, with i 2 {1, 2, . . . , n} be a family of R-implications underlying left-continuous t-subnorms Ti. If I A is a R-implication then I A verifies I3, I5, and I8 and it is right-continuous function in the second place. Proof. By Proposition 4.15, each Ii satisfies I3,I5, and I8 and it is right-continuous in the second place. Thus, by Theorem 5.2 and Proposition 5.6, I A also satisfies I3, I5, and I8. In order to prove that I A is right-continuous function in the second place, consider a non-decreasing sequence ðxj Þj2N . Since A is right-continuous aggregation function and Ii is right-continuous in the second place, we have that lim I A ðy; xj Þ ¼ lim AðI1 ðy; xj Þ; . . . ; In ðy; xj ÞÞ j!1 j!1 ¼ Aðlim I1 ðy; xj Þ; . . . ; lim In ðy; xj ÞÞ j!1 j!1 ¼ AðI1 ðy; lim xj Þ; . . . ; In ðy; lim xj ÞÞ ¼ I A ðy; lim xj Þ; for all 1 6 i 6 n:  j!1 j!1 j!1 Corollary 8.5. Let q be an order automorphism, A:Un ? U be an aggregation function and I ¼ fIi : U 2 ! Ug, with i 2 {1, 2, . . . , n} be a family of R-implications underlying left-continuous t-subnorms Ti. Then it holds that I qA : U 2 ! U is an R-implication. R.H.S. Reiser et al. / Information Sciences 253 (2013) 126–146 145 Proof. Follows from Propositions 4.16, 5.12 and 8.1. h Proposition 8.6. Let q be an order automorphism, A:Un ? U be an aggregation function and I ¼ fIi : U 2 ! Ug, with i 2 {1, 2, . . . , n} be a family of R-implications underlying left-continuous t-subnorms Ti. Then it holds that ðI A;T Þq ¼ I Aq ;T q : ð31Þ Proof. For all x, y 2 U, we have that ðI A;T Þq ðx; yÞ ¼ q1 ðI A;T ðqðxÞ; qðyÞÞÞ by Eq: ð1Þ ¼ q1 ðAðIT 1 ðqðxÞ; qðyÞÞ; IT 2 ðqðxÞ; qðyÞÞ; . . . ; IT n ðqðxÞ; qðyÞÞÞÞ         q q q ¼ q1 A q IT 1 ðx; yÞ ; q IT 2 ðx; yÞ ; . . . ; q IT n ðx; yÞ ¼ Aq ððIT 1 Þq ðx; yÞ; ðIT 2 Þq ðx; yÞ; . . . ; ðIT n Þq ðx; yÞÞ   ¼ Aq IT q ðx; yÞ; IT q ðx; yÞ; . . . ; IT qn ðx; yÞ by Eq: ð12Þ 1 2 ¼ I Aq ;T q ðx; yÞ by Eq: ð12Þ: Therefore, Eq. (31) holds. h Example 8.7. Based on Corollary 6.5 and results presented in Example 8.2, it follows that ðI M;T Þq is an infinity family of aggregation fuzzy R-implications. Additionally, I Aq ;Sq ;Nq and I Aq ;Sq ;N q are also families of (S, N)-implications. 9. Conclusion and final remarks The main research question considered in this paper is the following: how to aggregate fuzzy connectives of the same family in such a way that the resulting fuzzy connective would also belong to this family? Our proposal was focussed on the aggregation of t-subnorms, t-subconorms, negations, implications and coimplications. We show that not all aggregations of a family of fuzzy t-sub(co)norms result in a t-sub(co)norm. It is well known that the aggregation of a family of fuzzy negations and implications does not always result on a negation and an implication, respectively. Therefore, we discuss about the conditions under which a family of fuzzy t-sub(co)norms is preserved by such construction and related classes of (S, N)-(co)implications and R-(co)implications. Moreover, the proposed methodology preserves main extra properties for these fuzzy connectives. Additionally, the effect caused by the action of order-automorphisms and N-duality construction when applied to these classes of (co)implications was discussed. Thus, this construction can be extended in order to generate new fuzzy connectives from a family of corresponding fuzzy connectives. 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