A construction of a fuzzy topology from a strong fuzzy metric

[EN] After the inception of the concept of a fuzzy metric by I. Kramosil and J. Michalek, and especially after its revision by A. George and G. Veeramani, the attention of many researches was attracted to the topology induced by a fuzzy metric. In most of the works devoted to this subject the resulting topology is an ordinary, that is a crisp one. Recently some researchers showed interest in the fuzzy-type topologies induced by fuzzy metrics. In particular, in the paper  (J.J. Mi\~{n}ana, A. \v{S}ostak, {\it Fuzzifying topology induced by a strong fuzzy metric}, Fuzzy Sets and Systems,  6938 DOI information: 10.1016/j.fss.2015.11.005.) a fuzzifying topology ${\mathcal T}:2^X \to [0,1]$ induced by a fuzzy metric  $m: X\times X \times [0,\infty)$ was constructed. In this paper we extend  this construction to get the fuzzy topology ${\mathcal T}: [0,1]^X \to [0,1]$ and study some properties of this fuzzy topology.54AGrecova, S.; Sostak, A.; Uljane, I. (2016). 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Fuzzifying completeness and compactness in fuzzifying bornological linear spaces

The notions of completeness and compactness play important role in classical functional analysis. The main purpose of this paper is to generalize these notions to the setting of fuzzifying bornological linear spaces. At first, the concepts of fuzzifying Cauchy sequences and fuzzifying completeness are introduced and some interesting properties of them are studied. The relationships among fuzzifying completeness, separation axiom and fuzzifying bornological closed set are discussed. Then the notions of fuzzifying compactness and precompactness are presented, several properties of them are discussed. Particularly, it is demonstrated that a subset is fuzzifying bornological compact if and only if it is fuzzifying bornological precompact and bornological complete

LL-fuzzy ideal degrees in effect algebras

summary:In this paper, considering $L$ being a completely distributive lattice, we first introduce the concept of $L$-fuzzy ideal degrees in an effect algebra $E$, in symbol $\mathfrak{D}_{ei}$. Further, we characterize $L$-fuzzy ideal degrees by cut sets. Then it is shown that an $L$-fuzzy subset $A$ in $E$ is an $L$-fuzzy ideal if and only if $\mathfrak{D}_{ei}(A)=\top,$ which can be seen as a generalization of fuzzy ideals. Later, we discuss the relations between $L$-fuzzy ideals and cut sets ($L_{\beta}$-nested sets and $L_{\alpha}$-nested sets). Finally, we obtain that the $L$-fuzzy ideal degree is an $(L,L)$-fuzzy convexity. The morphism between two effect algebras is an $(L,L)$-fuzzy convexity-preserving mapping

Enriched Topology and Asymmetry

Mathematically modeling the question of how to satisfactorily compare, in many-valued ways, both bitstrings and the predicates which they might satisfy-a surprisingly intricate question when the conjunction of predicates need not be commutative-applies notions of enriched categories and enriched functors. Particularly relevant is the notion of a set enriched by a po-groupoid, which turns out to be a many-valued preordered set, along with enriched functors extended as to be variable-basis . This positions us to model the above question by constructing the notion of topological systems enriched by many-valued preorders, systems whose associated extent spaces motivate the notion of topological spaces enriched by many-valued preorders, spaces which are non-commutative when the underlying lattice-theoretic base is equipped with a non-commutative (semi-)tensor product. Of special interest are crisp and many-valued specialization preorders generated by many-valued topological spaces, orders having these consequences for many-valued spaces: they characterize the well-established L-T0 separation axiom, define the L-T1(1) separation axiom-logically equivalent under appropriate lattice-theoretic conditions to the L-T1 axiom of T. Kubiak, and define an apparently new L-T1(2) separation axiom. Along with the consequences of such ideas for many-valued spectra, these orders show that asymmetry has a home in many-valued topology comparable in at least some respects to its home in traditional topology

A framework for fuzzy topology with particular reference to sequentiality and countability

Pu and Liu's Q-theory is combined with Lowen's goodness criterion for fuzzy extensions to provide a framework for fuzzifying topology. This framework is used for the study of fuzzy countability properties and for the fuzzification of classical sequentiality. In extending classical notions to fuzzy theory care is taken to ensure that they are a special case of the emerging fuzzy concepts. An examination of convergence in the sense of Pu and Liu in special fuzzy topological spaces demonstrates the advantage of Chang's definition of fuzzy topology, which is therefore adopted. A new criterion (called excellence) for the suitability of the fuzzy extensions of classical topological properties is introduced. In addition to passing Lowen's goodness test, an excellent property is expected to behave, under fuzzy extensions of induction and coinduction, in a way resembling that of the original classical property under these constructions. Fuzzy second countability, quasi-first countability and fuzzy sequentiality are found to be excellent extensions of classical second countability, first countability and sequentiality respectively

Dual attachment pairs in categorically-algebraic topology

[EN] The paper is a continuation of our study on developing a new approach to (lattice-valued) topological structures, which relies on category theory and universal algebra, and which is called categorically-algebraic (catalg) topology. The new framework is used to build a topological setting, based in a catalg extension of the set-theoretic membership relation "e" called dual attachment, thereby dualizing the notion of attachment introduced by the authors earlier. Following the recent interest of the fuzzy community in topological systems of S. Vickers, we clarify completely relationships between these structures and (dual) attachment, showing that unlike the former, the latter have no inherent topology, but are capable of providing a natural transformation between two topological theories. We also outline a more general setting for developing the attachment theory, motivated by the concept of (L,M)-fuzzy topological space of T. Kubiak and A. Sostak.This research was partially supported by the ESF Project of the University of Latvia No. 2009/0223/1DP/1.1.1.2.0/09/APIA/VIAA/008.Frascella, A.; Guido, C.; Solovyov, SA. (2011). Dual attachment pairs in categorically-algebraic topology. Applied General Topology. 12(2):101-134. doi:10.4995/agt.2011.1646.SWORD10113412

Fuzzy Sets, Fuzzy Logic and Their Applications

The present book contains 20 articles collected from amongst the 53 total submitted manuscripts for the Special Issue “Fuzzy Sets, Fuzzy Loigic and Their Applications” of the MDPI journal Mathematics. The articles, which appear in the book in the series in which they were accepted, published in Volumes 7 (2019) and 8 (2020) of the journal, cover a wide range of topics connected to the theory and applications of fuzzy systems and their extensions and generalizations. This range includes, among others, management of the uncertainty in a fuzzy environment; fuzzy assessment methods of human-machine performance; fuzzy graphs; fuzzy topological and convergence spaces; bipolar fuzzy relations; type-2 fuzzy; and intuitionistic, interval-valued, complex, picture, and Pythagorean fuzzy sets, soft sets and algebras, etc. The applications presented are oriented to finance, fuzzy analytic hierarchy, green supply chain industries, smart health practice, and hotel selection. This wide range of topics makes the book interesting for all those working in the wider area of Fuzzy sets and systems and of fuzzy logic and for those who have the proper mathematical background who wish to become familiar with recent advances in fuzzy mathematics, which has entered to almost all sectors of human life and activity

Fifty years of similarity relations: a survey of foundations and applications

On the occasion of the 50th anniversary of the publication of Zadeh's significant paper Similarity Relations and Fuzzy Orderings, an account of the development of similarity relations during this time will be given. Moreover, the main topics related to these fuzzy relations will be reviewed.Peer ReviewedPostprint (author's final draft

Best effort QoS support routing in mobile ad hoc networks

In the past decades, mobile traffic generated by devices such as smartphones, iphones, laptops and mobile gateways has been growing rapidly. While traditional direct connection techniques evolve to provide better access to the Internet, a new type of wireless network, mobile ad hoc network (MANET), has emerged. A MANET differs from a direct connection network in the way that it is multi-hopping and self-organizing and thus able to operate without the help of prefixed infrastructures. However, challenges such dynamic topology, unreliable wireless links and resource constraints impede the wide applications of MANETs. Routing in a MANET is complex because it has to react efficiently to unfavourable conditions and support traditional IP services. In addition, Quality of Service (QoS) provision is required to support the rapid growth of video in mobile traffic. As a consequence, tremendous efforts have been devoted to the design of QoS routing in MANETs, leading to the emergence of a number of QoS support techniques. However, the application independent nature of QoS routing protocols results in the absence of a one-for-all solution for MANETs. Meanwhile, the relative importance of QoS metrics in real applications is not considered in many studies. A Best Effort QoS support (BEQoS) routing model which evaluates and ranks alternative routing protocols by considering the relative importance of multiple QoS metrics is proposed in this thesis. BEQoS has two algorithms, SAW-AHP and FPP for different scenarios. The former is suitable for cases where uncertainty factors such as standard deviation can be neglected while the latter considers uncertainty of the problems. SAW-AHP is a combination of Simple Additive Weighting and Analytic Hierarchical Process in which the decision maker or network operator is firstly required to assign his/her preference of metrics with a specific number according to given rules. The comparison matrices are composed accordingly, based on which the synthetic weights for alternatives are gained. The one with the highest weight is the optimal protocol among all alternatives. The reliability and efficiency of SAW-AHP are validated through simulations. An integrated architecture, using evaluation results of SAW-AHP is proposed which incorporates the ad hoc technology into the existing WLAN and therefore provides a solution for the last mile access problems. The protocol selection induced cost and gains are also discussed. The thesis concludes by describing the potential application area of the proposed method. Fuzzy SAW-AHP is extended to accommodate the vagueness of the decision maker and complexity of problems such as standard deviation in simulations. The fuzzy triangular numbers are used to substitute the crisp numbers in comparison matrices in traditional AHP. Fuzzy Preference Programming (FPP) is employed to obtain the crisp synthetic weight for alternatives based on which they are ranked. The reliability and efficiency of SAW-FPP are demonstrated by simulations