Gödel integral, along with Sugeno integral, are two of the well-known qualitative integrals. In this work, we discuss the utility of Gödel integrals both in providing an ordinal representation of a preference relation and as a classifier in the case of ordinal data. Typically, a preference relation defined over a scale is converted into equivalence classes and a monotone mapping of these classes into levels is done using a qualitative integral, which requires the existence of a common capacity. In this work, we give necessary and sufficient conditions for such a common capacity to exist for a Gödel integral, comparing its applicability with Sugeno integral. With the help of these conditions, we discuss the suitability of the Gödel integral as a classifier. Through a novel perspective of discussing the dependence of class-specific features in the data and their expression levels, we offer a commentary on the type of datasets that are amenable for classification through a threshold-type Gödel integral.

The study of functional equations involving fuzzy logic connectives, especially fuzzy implications, has found immense utility both in the advancement of theory and applications. In this work, we discuss the pseudo-homogeneity functional equation involving fuzzy implications. An interesting outcome of this work is a novel sufficient condition on the triple ( S,T_P,N ) such that the QL-operation obtained from it is also a QL-implication.

In the recent past, a special type of distance function, namely monometrics, has garnered a lot of attention because of its value and interest in both theory and applications. The key challenge lies in finding a monometric w.r.t. an arbitrary betweenness relation. To address this, the constructions of non-trivial monometrics on various types of betweenness relations have been explored. While it was shown that a non-trivial monometric may not always exist on arbitrary betweenness relations, the study of such betweenness relations has not been undertaken. In this paper, we take a step towards filling this gap by partially characterizing betweenness relations that will not give rise to non-trivial monometrics. By addressing this fundamental question, we hope to provide a clearer framework for understanding when monometrics can be defined.

The study of functional equations involving fuzzy logic operations has an immense impact both on the theoretical and applicational fronts. In this work, the pseudo-homogeneity functional equation involving fuzzy implications with the minimum t-norm is studied and some complete solutions are given under reasonable assumptions on the involved operations. Noting that the Godel implication is one such solution, we determine an equivalent formulation for the Godel integral based binary classifier, which offers an alternative perspective, especially on the interpretation of the threshold. employed.

While fuzzy logic connectives were seen as generalisations of classical logic connectives, their utility has extended beyond their intended use and context. One interesting avenue of exploration that began almost 4 decades ago is to obtain metrics from fuzzy logic connectives. Not only was this a fertile approach for obtaining metrics with myriad properties but such studies have also thrown up some interesting insights. In this work, we present a state-of-the-art survey of the different works detailing the multitude of operators used to obtain these distance functions, the host of properties they satisfy, the novel contexts in which they have been employed, and the insightful commentary that they have provided on the underlying structures.Recently, monometrics - distance functions compatible with the underlying order - have attracted scrutiny for their utility in the fields of rationalisation of ranking rules, penalty-based aggregation, and binary classification. In this work, adding to the survey, we examine if and when the existing distance functions yield a monometric. Further, by employing monotonic fuzzy logic connectives and fuzzy negations, we offer a construction of distance functions that always yield monometrics and helps us in providing a characterisation of symmetric monometrics on the unit interval. Our work showcases a close relationship between monometrics and fuzzy implications.

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 tracking 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 adopting these features can accelerate the pace of the future research and be constructive for new researchers.

Among the non-parametric classification methods, the nearest neighbour classifier (NNC) holds a pre-eminent position. Given a training or sample set 𝒮 the choice one needs to make is on the value of k and the distance function d to be employed. Towards improving the efficacy of an NNC, there are many works—both theoretical and empirical—that help in choosing a suitable value of k. However, works that deal with the appropriateness of a distance d for a given 𝒮 are largely empirical. In this work, we address the following two posers for a given 𝒮 : (1) How to identify a potentially appropriate distance d? (2) What qualities should an appropriate d possess? Our investigations show that every distance function d determines a landscape on the underlying data space and only if the class boundaries align with this landscape can this d be appropriate. In view of this, we construct a relational graph 𝒢_𝒮,d , in fact, a poset, on the given 𝒮 using d. With the help of 𝒢_𝒮,d , we choose a 𝒯⊂𝒮 to be used in a condensed-NN algorithm. Terming it the NEN algorithm, firstly, we show empirically that the training error of this NEN algorithm is reflective of the appropriateness of d. Towards providing a theoretical justification to our claims based on empiricism, we investigate the problem of classification in the setting of monometric spaces, wherein it emerges that the suitability of d is essentially related to the embeddability of 𝒢_𝒮,d in the monometric space ( 𝒳, ≼ _d,d ).

There have been a few works studying metrics obtained from fuzzy logic connectives such as t-norms and copulas which are either commutative, monotonically increasing, or associative. In this work, we define a distance function generated from a non-associative, non-commutative, and non-monotonic fuzzy logic connective, viz., a fuzzy implication. We consider fuzzy implication as a relation on [0, 1] and give a way to obtain metrics from S_LK - transitive relations that turn out to be monometrics w.r.t. the betweenness relation obtained from the underlying total order on [0, 1]. We also give some sufficient conditions under which certain families of fuzzy implications yield a metric. Our study, on the one hand highlights the usefulness of S-transitive fuzzy relations as much as T-transitive fuzzy relations, and on the other hand, illustrates emphatically the need for fuzzy logic operations on non-linear posets.

Fuzzy implications continue to remain an important class among the fuzzy logic connectives, having found utility in contexts outside of logical inference too. Recently, fuzzy implications have been shown to be a fertile source for obtaining distance functions with very beneficial properties. In this work, we show that fuzzy implications can also be a good wellspring of fuzzy compatibility relations with myriad properties. Quite interestingly, the converse also holds good, i.e., we also show that we can represent any fuzzy implication through a pair (d,E), where d is a distance function - called a pseudo-monometric - and E is a mono-similarity fuzzy relation. This representation helps us to propose an equivalent formulation of the problem of characterisation of QL-implications, that has the potential to offer computational savings in verifying if a QL-operation can be a QL-implication and also throws open further avenues of fruitful theoretical exploration.

In approximate query searching (AQS), the given query point ( q̅' ) can be seen as a noise ( η̅ ) corrupted version of one of the points ( q̅ ) in the existing database 𝒳 , i.e., q̅' = q̅ + η̅ . Thus deciding on an appropriate distance d that would return the correct match ( q̅ ) entails that the chosen distance should be aware of the type of distribution of the noise. In this work, we study the suitability of Minkowski-type distances in AQS when the q̅ is afflicted by both white and coloured noises to different extent. To this end, we employ a simple similarity search based scoring algorithm proposed in François et al. (ESANN 2005, 13th European Symposium on Artificial Neural Networks, Bruges, Belgium, April 27–29, 2005, Proceedings, pp 339–344, 2005). Our study reveals an interesting interplay of the following 3D’s in the quest for an appropriate distance: Dimensionality and Domain geometry of the data and the type of noise Distribution and has led us to explore this problem from a basic geometric perspective. Our main contribution herein is the proposal of a novel index called the Relative Contained Volume (RCV) that helps explain the performance of the considered distances.

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.

Inspired by the work of Clifford on obtaining order from semigroups, many works have proposed different ways of obtaining orders from associative fuzzy logic operations. However, unlike Clifford's relation, these were dependent on the subdomain of its arguments. Recently, it was shown that a property termed Quasi-Projectivity (QP) is necessary to obtain an order from Clifford's relation. Further, for the underlying domain [0,1] it was shown that while all t-norms, t-conorms and nullnorms satisfy (QP), giving rise to posets, not all classes of uninorms satisfy (QP). Several constructions of uninorms U exist on bounded lattices, which unlike [0,1] may neither be total nor complete. In this work, we investigate the satisfaction of (QP) for these constructions. This study merits attention since it offers an alternate perspective - that a uninorm U on a lattice L can be seen as a t-norm on the obtained U-poset.

In the literature, there have been a few works studying distance functions from fuzzy logic connectives (FLCs), such as t-norms, t-conorms, copulas, and quasi-copulas. Recently, Nanavati et al. defined a distance function d_I using a fuzzy implication I. They characterised fuzzy implications that would yield a metric and showed that d_I is a pseudo-monometric w.r.t. the usual order on [0, 1]. In this work, we generalise the definition of d_I with the aid of any t-conorm S. Denoting it by d_I,S , we investigate the conditions under which d_I,S yields metrics and pseudo-monometrics for the major t-conorms. We thus expand our armoury of practical distance functions using FLCs such as t-conorms and fuzzy implications.

Subgroup Discovery is a descriptive data mining technique for obtaining subgroups with unusual statistical characteristics with respect to a given target variable. In this paper, unlike existing approaches, we capture the data in the form of implicative-type fuzzy rules and propose an algorithm to determine sharp transitions in the consequent when there is a minimal change in the antecedent. Our study contained herein highlights the role and employability of fuzzy implication functions in such settings through illustrative examples with several real datasets.

In approximate reasoning (AR), inference mechanisms are functions that produce meaningful outcomes using imprecise input data [1]. To deal with the imprecision in the data, fuzzy sets are applied. Inference mechanisms that use fuzzy set theory for specific purposes are referred to as fuzzy inference mechanisms. The fuzzy inference mechanisms such as (i) Fuzzy Relational Inference (FRI), (ii) Similarity- Based Reasoning (SBR) [2–5], and (iii) Takagi–Sugeno (TS) fuzzy system [6] are very well known. Two of the well-known FRIs are the Compositional Rule of Inference (CRI) [7, 8], and the Bandler–Kohout Subproduct (BKS) [9] based on the works of Bandler and Kohout [10]. As part of this study, we primarily focus on BKS.

ABSTRACT Order-theoretic explorations of algebraic structures are known to lead to hitherto hidden insights. Two such relations that have stood out are those of Mitsch and Clifford – the former for the generality in its application and the latter for the insights it offers. In this work, our motivation is to study the converse: we want to explore the extent of the utility of Mitsch’s order and the applicability of Clifford’s order. Firstly, we show that if the Mitsch’s poset is either bounded or a chain, arguably a richer order theoretic structure, the semigroup reduces to one of a simple band. Secondly, noting that the special semigroups on which Clifford’s relation does give rise to an order has not been characterised so far, we solve this problem by proposing a property called Quasi-Projectivity that is essential in this context and also give necessary and sufficient conditions for the Clifford’s relation to give a total and compatible order, even if the semigroup is not commutative. Further, by showing some interesting connections between this relation and the orders obtained by Green’s relations, we further reaffirm the importance and naturalness of the order proposed by Clifford. Finally, by discussing the Clifford’s relations on ordered semigroups, we present some novel perspectives and also show that some of the assumptions in the often cited results of Clifford’s are not necessary. On the whole, our study argues favourably towards Clifford’s than that of the Mitsch’s relation, in so far as the structural information gained about the underlying semigroup.

In the recent past, monometrics w.r.t. a ternary relation defined on a set, called the betweenness relation, have garnered a lot of attention for their important role in decision making, penalty-based data aggregation, and binary classification. One of the major challenges herein is that of obtaining monometrics on a given betweenness set ( 𝒳,B ). In this work, we propose a couple of constructions of monometrics on lattice betweenness, the latter using fuzzy implications. Our work seems to suggest that fuzzy implications are rather a natural choice for constructing monometrics on lattice betweenness. It also justifies the exploration of fuzzy logic connectives on general posets, primarily bounded lattices.

Several works have proposed the construction of distance functions using fuzzy logic connectives to proffer further applications of the corresponding connectives. In these works, the authors define a distance function using t-norms, t-conorms, copulas, or quasi-copulas, all of which are either associative, commutative or monotonic fuzzy logic connectives. In this work, we define a distance function, denoted dI, from a non-associative, non-commutative, and mixed-monotonic fuzzy logic connective, viz., a fuzzy implication I, and study the above distance function along two aspects. Firstly, we investigate the necessary and sufficient conditions for dI to be a metric, wherein the role played by a transitivity type functional inequality involving the considered fuzzy implication and the Łukasiewicz t-conorm is highlighted. In the recent past, monometrics w.r.t. a ternary relation, called the betweenness relation, have garnered a lot of attention for their important role in decision-making and penalty-based data aggregation. One of the major challenges herein is that of obtaining monometrics on a given betweenness set. Our second contribution in this work is in establishing the existence of pseudo-monometrics using dI, from whence it appears that fuzzy implications are a natural choice for obtaining pseudo-monometrics on a given betweenness set.

Algebraic structures are often converted to ordered structures to gain information about the algebra using the properties of partially ordered sets. Such studies have been predominantly undertaken for semigroups, using various proposed relations. This has led to a spate of works dealing with associative fuzzy logic connectives (FLCs) and the orders that they generate. One such relation, proposed by Clifford, is employed both for its generality as well as utility. In a recent work, Gupta and Jayaram classified the semigroups that yield a partial order through the relation. In this work, we characterise groupoids that would give a partial order by introducing a property called the Generalised Quasi-Projectivity. Further, for the groupoids that lead to an ordered set, we explore the monotonicity of the underlying groupoid operation on the obtained poset. Finally, in light of the above results, we explore the major non-associative fuzzy logic connectives along these lines, thus complementing and augmenting, already existing works in the literature. Our work also shows when an FLC from a given class of operations remains one even w.r.to the order generated from it.

It is well known that the usual point-wise ordering over the set T of t-norms makes it a poset but not a lattice, i.e., the point-wise maximum or minimum of two t-norms need not always be a t-norm again. In this work, we propose, two binary operations ▪ on the set T CA of continuous Archimedean t-norms and obtain, via these binary operations, a partial order relation ⊑, different from the usual point-wise order ≤, on the set T CA. As an interesting outcome of this structure, some stronger versions of some existing results dealing with the upper and lower bounds of two continuous Archimedean t-norms with respect to the point-wise order ≤ are also obtained. Finally, with the help of the operations ▪ on the set T CA, two binary operations ⊕ , ⊗ on the set T C of continuous t-norms are proposed and showed that ( T C , ⊕ , ⊗ ) is a lattice. Thus we have both a way of generating continuous t-norms from continuous t-norms and also obtain an order on them.