The ⊛-composition of fuzzy implications: Closures with respect to properties, powers and families

Recently, Vemuri and Jayaram proposed a novel method of generating fuzzy implications from a given pair of fuzzy implications. Viewing this as a binary operation ⊛ on the set II of fuzzy implications they obtained, for the first time, a monoid structure (I,⊛)(I,⊛) on the set II. Some algebraic aspects of (I,⊛)(I,⊛) had already been explored and hitherto unknown representation results for the Yager's families of fuzzy implications were obtained in [53] (N.R. Vemuri and B. Jayaram, Representations through a monoid on the set of fuzzy implications, fuzzy sets and systems, 247 (2014) 51–67). However, the properties of fuzzy implications generated or obtained using the ⊛-composition have not been explored. In this work, the preservation of the basic properties like neutrality, ordering and exchange principles , the functional equations that the obtained fuzzy implications satisfy, the powers w.r.t. ⊛ and their convergence, and the closures of some families of fuzzy implications w.r.t. the operation ⊛, specifically the families of (S,N)(S,N)-, R-, f- and g-implications, are studied. This study shows that the ⊛-composition carries over many of the desirable properties of the original fuzzy implications to the generated fuzzy implications and further, due to the associativity of the ⊛-composition one can obtain, often, infinitely many new fuzzy implications from a single fuzzy implication through self-composition w.r.t. the ⊛-composition

Homomorphisms on the monoid of fuzzy implications and the iterative functional equation I(x,I(x,y))=I(x,y)

Recently, Vemuri and Jayaram proposed a novel method of generating fuzzy implications, called the ⊛⊛-composition, from a given pair of fuzzy implications [Representations through a Monoid on the set of Fuzzy Implications, Fuzzy Sets and Systems, 247, 51-67]. However, as with any generation process, the ⊛⊛-composition does not always generate new fuzzy implications. In this work, we study the generative power of the ⊛⊛-composition. Towards this end, we study some specific functional equations all of which lead to the solutions of the iterative functional equation I(x,I(x,y))=I(x,y)I(x,I(x,y))=I(x,y) involving fuzzy implications which has been studied extensively for different families of fuzzy implications in this very journal, see [Information Sciences 177, 2954–2970 (2007); 180, 2487–2497 (2010); 186, 209–221 (2012)]. In this work, unlike in other existing works, we do not restrict the solutions to a particular family of fuzzy implications. Thus we take an algebraic approach towards solving these functional equations. Viewing the ⊛⊛-composition as a binary operation ⊛⊛ on the set II of all fuzzy implications one obtains a monoid structure (I,⊛)(I,⊛) on the set II. From the Cayley’s theorem for monoids, we know that any monoid is isomorphic to the set of all right translations. We determine the complete set KK of fuzzy implications w.r.t. which the right translations also become semigroup homomorphisms on the monoid (I,⊛I,⊛) and show that KK not only answers our questions regarding the generative power of the ⊛⊛-composition but also contains many as yet unknown solutions of the iterative functional equation I(x,I(x,y))=I(x,y)I(x,I(x,y))=I(x,y)

Conjugacy Relations via Group Action on the set of Fuzzy Implications

Let denote the set of all increasing bijections on [0 ; 1] and I the set of fuzzy implications. In [1], the authors proposed a new way of generating fuzzy implications from fuzzy....

Bijective transformations of fuzzy implications – An algebraic perspective

Bijective transformations play an important role in generating fuzzy implications from fuzzy implications. In [Representations through a Monoid on the set of Fuzzy Implications, Fuzzy Sets and Systems, 247, 51–67], Vemuri and Jayaram proposed a monoid structure on the set of fuzzy implications, which is denoted by II, and using the largest subgroup SS of this monoid discussed some group actions on the set II. In this context, they obtained a bijective transformation which ultimately led to hitherto unknown representations of the Yager's families of fuzzy implications, viz., f-, g -implications. This motivates us to consider whether the bijective transformations proposed by Baczyński & Drewniak and Jayaram & Mesiar, in different but purely analytic contexts, also possess any algebraic connotations. In this work, we show that these two bijective transformations can also be seen as being obtained from some group actions of SS on II. Further, we consider the most general bijective transformation that generates fuzzy implications from fuzzy implications and show that it can also be obtained as a composition of group actions of SS on II. Thus this work tries to position such bijective transformations from an algebraic perspective

Lattice operations on fuzzy implications and the preservation of the exchange principle

In this work, we solve an open problem related to the preservation of the exchange principle (EP) of fuzzy implications under lattice operations ([3], Problem 3.1.). We show that generalizations of the commutativity of antecedents (CA) to a pair of fuzzy implications (I,J)(I,J), viz., the generalized exchange principle and the mutual exchangeability are sufficient conditions for the solution of the problem. Further, we determine conditions under which these become necessary too. Finally, we investigate the pairs of fuzzy implications from different families such that (EP) is preserved by the join and meet operations

Representations through a monoid on the set of fuzzy implications

Fuzzy implications are one of the most important fuzzy logic connectives. In this work, we conduct a systematic algebraic study on the set II of all fuzzy implications. To this end, we propose a binary operation, denoted by ⊛, which makes (I,⊛I,⊛) a non-idempotent monoid. While this operation does not give a group structure, we determine the largest subgroup SS of this monoid and using its representation define a group action of SS that partitions II into equivalence classes. Based on these equivalence classes, we obtain a hitherto unknown representations of the two main families of fuzzy implications, viz., the f- and g-implications

The *-composition -A Novel Generating Method of Fuzzy Implications: An Algebraic Study

Fuzzy implications are one of the two most important fuzzy logic connectives, the other being t-norms. They are a generalisation of the classical implication from two-valued logic to the multivalued setting. A binary operation I on [0; 1] is called a fuzzy implication if (i) I is decreasing in the first variable, (ii) I is increasing in the second variable, (iii) I(0; 0) = I(1; 1) = 1 and I(1; 0) = 0. The set of all fuzzy implications defined on [0; 1] is denoted by I. Fuzzy implications have many applications in fields like fuzzy control, approximate reasoning, decision making, multivalued logic, fuzzy image processing, etc. Their applicational value necessitates new ways of generating fuzzy implications that are fit for a specific task. The generating methods of fuzzy implications can be broadly categorised as in the following: (M1): From binary functions on [0; 1], typically other fuzzy logic connectives, viz., (S;N)-, R-, QL- implications, (M2): From unary functions on [0,1], typically monotonic functions, for instance, Yager’s f-, g- implications, or from fuzzy negations, (M3): From existing fuzzy implications

Homomorphisms on the Monoid of Fuzzy Implications (II, *) - A Complete Characterization

In [4], we had proposed a novel generating methods of fuzzy implications and investigated algebraic structures on the set of all fuzzy implications, which is denoted by II. Again in [5], we had defined a particular function gK on the monoid (II, *) (See Def. 16) and characterised the function K for which gK is a semigroup homomorphism (s.g.h) in two special cases, i.e., K is with trivial range and K (1, y ) = y for all y ∈ [0, 1](neutrality property). In this work we characterise the nontrivial range non neutral implications K such that gK is an s.g.h. and also present their representation

PRESERVATION OF THE EXCHANGE PRINCIPLE UNDER LATTICE OPERATIONS ON FUZZY IMPLICATIONS

In this work, we solve an open problem related to the exchange principle of fuzzy implications [Problem 3.1, Fuzzy Sets and Systems 261(2015) 112-123]. We show that two important generalizations of the exchange principle, namely, the generalized exchange principle(GEP) and the mutual exchangeability(ME) are sufficient conditions for the solution of the problem. We also show that, under some conditions, these are necessary too. Finally, we investigate the pairs (I, J) from different families of fuzzy implications such that the exchange principle is preserved under the join and meet operations

Molecular mechanisms of intestinal inflammation leading to colorectal cancer

Inflammatory bowel disease (IBD) is recognized as a leading cause in the development of colorectal cancer (CRC). Inflammatory bowel disease associated colorectal cancer (IBD-CRC) is a growing healthcare burden, causing significant morbidity and mortality world-wide. In the present review, relevant preclinical models of IBD-CRC, a concise overview of potential molecular mechanisms that are involved and responsible for IBD associated colonic tumorigenesis along with the current and the future therapeutic approaches including the role of novel natural and synthetic compounds for the prevention and treatment of IBD-CRC are outlined. This review will benefit various clinicians and translational researchers working in the area of IBD-CRC to have a quick snap-shot of the ongoing trends in IBD-CRC research and discoveries