The existing generating methods of fuzzy implications are closed in the set of all fuzzy implications, but not when applied to some families of these operators. In this paper, some binary operations are defined on some well-established families of fuzzy implications. Namely, the families of (S,N)-implications with continuous t-conorms, R-implications obtained from continuous t-norms, Yager's f- and g-generated implications, h- and generalized (h,e)-implications and k-implications are considered and it is proved that with these operations, they are lattices. Moreover, some other lattice substructures are defined in some of the families when some restrictions on the underlying generators of the implications are imposed. Furthermore, it is emphasized that these generating methods preserve important properties like the exchange principle and the law of importation.

The research on fuzzy implication functions has exponentially grown in the last decades becoming one of the main topics studied in the Fuzzy Logic community. These efforts have led to significant advances on the theoretical (mainly) and applied aspects, but the overwhelming number of papers hinder the track of the new results and the problems that remain unsolved. The goal of this paper is to determine the valuable practices in the research on classes of fuzzy implication functions and the desirable characteristics that a paper on this topic should have to be useful for a reader. Clearly appointing these features can accelerate the pace of the future research and be constructive for new researchers.

In this work, we discuss the law of α-migrativity as applied to fuzzy implication functions in a meaningful way. A generalisation of this law leads us to Pexider-type functional equations connected with the law of importation, viz., the generalised law of importation I(C(x,α),y)=I(x,J(α,y)) (GLI) and the generalised cross-law of importation I(C(x,α),y)=J(x,I(α,y)) (CLI), where C is a generalised conjunction. In this article we investigate only (GLI). We begin by showing that the satisfaction of law of importation by the pairs (C, I) and/or (C, J) does not necessarily lead to the satisfaction of (GLI). Hence, we study the conditions under which these three laws are related.

High dimensional data analysis gives rise to many challenges. One such that has come to gain a lot of attention recently is the concentration of distances (CoD) phenomenon, which is the inability of distance functions to distinguish points well in high dimensions. CoD affects almost every machine learning and data analysis algorithm in high dimensions. In this work, we present a novel efficient and effective empirical index that not only illustrates whether a distance function tends to concentrate for a given data set, but also enables us to measure the rate of concentration and allows us to compare different distance functions vis-a-vis their rate of concentration. As opposed to existing empirical indices, the proposed empirical measure uses only the internal characteristics of a given data set and hence is applicable on real data sets, which was hitherto not possible.

A fuzzy relational inference (FRI) mechanism is appraised based on the different desirable properties it possesses. Among these properties, monotonicity of an FRI has not received much attention. In this work, we investigate the monotonicity of a single-input single-output FRI with an implicative form of the rule base. In all the previous works that deal with monotonicity of an FRI with the implicative form of rule base, the employed fuzzy implications come from a residuated lattice. It can be noticed that this rich underlying structure plays a major role in proving the results. Further, we also modify the given monotone rule base. This work differs from the previous works in that: 1) the fuzzy implications employed in it do not come from any known residuated structure on [0, 1]; and 2) the original rule base is employed without any alteration. We determine conditions under which monotonicity of an FRI, where the rule base is modeled by a strict fuzzy implication, can be ensured without transforming the original rule base. Thus, the results in this work further augment the case for considering fuzzy implications, other than those from the residuated setting, to be used in applications.

The Bandler-Kohout subproduct (BKS) inference mechanism is one of the two established fuzzy relational inference (FRI) mechanisms; the other one being Zadeh's compositional rule of inference (CRI). Both these FRIs are known to possess many desirable properties. It can be seen that many of these desirable properties are due to the rich underlying structure, viz., the residuated algebra, from which the employed operations come. In this study, we discuss the BKS relational inference system, with the fuzzy implication interpreted as Yager's classes of implications, which do not form a residuated structure on [0,1] . We show that many of the desirable properties, viz., interpolativity, continuity, robustness, which are known for the BKS with residuated implications, are also available under this framework, thus expanding the choice of operations available to practitioners. Note that, to the best of the authors' knowledge, this is the first attempt at studying the suitability of an FRI where the operations come from a nonresiduated structure.

The curse of dimensionality, which refers to both the combinatorial explosion in dimensions and the concentration of distances or norms in high dimensions, affects most of the clustering techniques. Recent studies on the concentration of norms suggest the use of a correlation measure instead of distances to more effectively judge (dis)similarity in high dimensions. In this work, based on these observations, we propose a robust rank correlation based clustering method. Specifically, we employ the recently proposed fuzzy gamma rank correlation measure. We show that this intuitively simple algorithm has the following advantages: (i) It requires very few parameters to be set, (ii) the number of clusters need not be apriori known, (iii) while there is an indirect dependence on the underlying distance measure, its makes use of both global and local information, (iv) it can be robust to noise depending on the correlation measure employed and, (v) as it is shown, performs well with high dimensional data. We illustrate the algorithm on some datasets where the traditional Fuzzy C-Means algorithm is known to fail.

In this work, we show that fuzzy inference systems based on Fuzzy Relational Inference (FRI) with implicative interpretation of the rule base are universal approximators under suitable choice of operations for the rest of the components of the fuzzy system. The presented proofs make no assumption on the form or representations of the considered fuzzy implications and hence show that a much larger class of fuzzy implications other than what is typically considered in the literature can be employed meaningfully in FRIs based on implicative models.

In this work we propose and study a particular type of lattice and semigroup homomorphisms on the monoid (II, ⊗) of the set of all fuzzy implications proposed in [1]. We show that the subclass of neutral implications which generate homomorphisms of the defined form and the set of such homomorphisms themselves form abelian groups, suggesting that the investigated homomorphisms form the group of inner semigroup homomorphisms. Finally, investigating the images of the studied homomorphisms, we present some natural partitions on I and orderings on these equivalence classes. Our investigations have led us to obtain a group structure on a subset of I. Note that, to the best of the authors' knowledge, this is the first work to present such a rich algebraic structure on the set of all fuzzy implications I.

Fuzzy relational inference (FRI) systems form an important part of approximate reasoning schemes using fuzzy sets. The compositional rule of inference (CRI), which was introduced by Zadeh, has attracted the most attention so far. In this paper, we show that the FRI scheme that is based on the Bandler–Kohout (BK) subproduct, along with a suitable realization of the fuzzy rules, possesses all the important properties that are cited in favor of using CRI, viz., equivalent and reasonable conditions for their solvability, their interpolative properties, and the preservation of the indistinguishability that may be inherent in the input fuzzy sets. Moreover, we show that under certain conditions, the equivalence offirst-infer-then-aggregate (FITA)andfirst-aggregate-then-infer (FATI)inference strategies can be shown for the BK subproduct, much like in the case of CRI. Finally, by addressing the computational complexity that may exist in the BK subproduct, we suggest a hierarchical inferencing scheme. Thus, this paper shows that the BK-subproduct-based FRI is as effective and efficient as the CRI itself.

The compositional rule of inference (CRI) is widely used in approximate reasoning schemes using fuzzy sets. In this work we discuss the suitability of the Bandler-Kohout subproduct for an alternative inference mechanism from the computational point of view.

Recently, many works have appeared in this very journal dealing with the distributivity of fuzzy implications over t-norms and t-conorms. These equations have a very important role to play in efficient inferencing in approximate reasoning, especially fuzzy control systems. Of all the four equations considered, the equation$I(x,S_1(y,z))=S_2(I(x,y),I(x,z))$, when$S_1,S_2$are both t-conorms and$I$is an$R$-implication obtained from a strict t-norm, was not solved. In this paper, we characterize functions$I$that satisfy the previous functional equation when$S_1,S_2$are either both strict or nilpotent t-conorms. Using the obtained characterizations, we show that the previous equation does not hold when$S_1,S_2$are either both strict or nilpotent t-conorms, and$I$is a continuous fuzzy implication. Moreover, the previous equation does not hold when$I$is an$R$-implication obtained from a strict t-norm, and$S_1,S_2$are both strict t-conorms, while it holds for an$R$-implication$I$obtained from a strict t-norm$T$if and only if the t-conorms$S_1 = S_2$are$\Phi$-conjugate to the Łukasiewicz t-conorm for some increasing bijection$\varphi$of the unit interval, which is also a multiplicative generator of$T$.

The law of importation, given by the equivalence (x Lambda y) rarr z equiv (xrarr (y rarr z)), is a tautology in classical logic. In A-implications defined by Turksen et aL, the above equivalence is taken as an axiom. In this paper, we investigate the general form of the law of importation J(T(x, y), z) = J(x, J(y, z)), where T is a t-norm and J is a fuzzy implication, for the three main classes of fuzzy implications, i.e., R-, S- and QL-implications and also for the recently proposed Yager's classes of fuzzy implications, i.e., f- and g-implications. We give necessary and sufficient conditions under which the law of importation holds for R-, S-, f- and g-implications. In the case of QL-implications, we investigate some specific families of QL-implications. Also, we investigate the general form of the law of importation in the more general setting of uninorms and t-operators for the above classes of fuzzy implications. Following this, we propose a novel modified scheme of compositional rule of inference (CRI) inferencing called the hierarchical CRI, which has some advantages over the classical CRI. Following this, we give some sufficient conditions on the operators employed under which the inference obtained from the classical CRI and the hierarchical CRI become identical, highlighting the significant role played by the law of importation.

In this paper, we explore the distributivity of implication operators [especially Residuated (R)- and Strong (S)-implications] over Takagi (T)- and Sugeno (S)-norms. The motivation behind this work is the on going discussion on the law [(p/spl and/q)/spl rarr/r]/spl equiv/[(p/spl rarr/r)/spl or/(q/spl rarr/r)] in fuzzy logic as given in the title of the paper by Trillas and Alsina. The above law is only one of the four basic distributive laws. The general form of the previous distributive law is J(T(p,q),r)/spl equiv/S(J(p,r),J(q,r)). Similarly, the other three basic distributive laws can be generalized to give equations concerning distribution of fuzzy implications J on T- and S- norms. In this paper, we study the validity of these equations under various conditions on the implication operator J. We also propose some sufficiency conditions on a binary operator under which the general distributive equations are reduced to the basic distributive equations and are satisfied. Also in this work, we have solved one of the open problems posed by M. Baczynski (2002).