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

Developing a Climate for Continuous Development

This poster describes the progress and lessons learned as a result of newly implemented Faculty Mentoring Program in the IUPUI School of Science (SOS)

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....

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

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

Beyond just bacteria: Functional biomes in the gut ecosystem including virome, mycobiome, archaeome and helminths

Gut microbiota refers to a complex network of microbes, which exerts a marked influenceon the host’s health. It is composed of bacteria, fungi, viruses, and helminths. Bacteria, or collectively,the bacteriome, comprises a significant proportion of the well-characterized microbiome. However,the other communities referred to as ‘dark matter’ of microbiomes such as viruses (virome), fungi(mycobiome), archaea (archaeome), and helminths have not been completely elucidated. Developmentof new and improved metagenomics methods has allowed the identification of complete genomesfrom the genetic material in the human gut, opening new perspectives on the understanding ofthe gut microbiome composition, their importance, and potential clinical applications. Here, wereview the recent evidence on the viruses, fungi, archaea, and helminths found in the mammalian gut,detailing their interactions with the resident bacterial microbiota and the host, to explore the potentialimpact of the microbiome on host’s health. The role of fecal virome transplantations, pre-, pro-, andsyn-biotic interventions in modulating the microbiome and their related concerns are also discussed