Enriched lower separation axioms and the principle of enriched continuous extension

[EN] This paper presents a version of the lower separation axioms and the principle of enriched continuous extension for quantale-enriched topological spaces. As a remarkable result, among other things, we point out that in the case of commutative Girard quantales the principle of continuous extension holds for projective modules in Sup.The authors acknowledge support from the Basque Government (grant IT1483-22). The first named author also acknowledges support from a postdoctoral fellowship of the Basque Government (grant POS-2022-1-0015)

Lattice-valued convergence spaces and regularity

We define a regularity axiom for lattice-valued convergence spaces where the lattice is a complete Heyting algebra. To this end, we generalize the characterization of regularity by a ”dual form” of a diagonal condition. We show that our axiom ensures that a regular T1-space is separated and that regularity is preserved under initial constructions. Further we present an extension theorem for a continuous mapping from a subspace to a regular space. A characterization in the restricted case that the lattice is a complete Boolean algebra in terms of the closure of an L-filter is given

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

Dense Subdifferentiability and Trustworthiness for Arbitrary Subdifferentials

2000 Mathematics Subject Classification: 49J52, 49J50, 58C20, 26B09.We show that the properties of dense subdifferentiability and of trustworthiness are equivalent for any subdifferential satisfying a small set of natural axioms. The proof relies on a remarkable property of the subdifferential of the inf-convolution of two (non necessarily convex) functions. We also show the equivalence of the dense subdifferentiability property with other basic properties of subdifferentials such as a weak* Lipschitz Separation property, a strong Compact Separation property and a Minimal property for the analytic approximate subdifferential of Ioffe

Representation of maxitive measures: an overview

Idempotent integration is an analogue of Lebesgue integration where $\sigma$-maxitive measures replace $\sigma$-additive measures. In addition to reviewing and unifying several Radon--Nikodym like theorems proven in the literature for the idempotent integral, we also prove new results of the same kind.Comment: 40 page

Sandwich-type characterization of completely regular spaces

[EN] All the higher separation axioms in topology, except for complete regularity, are known to have sandwich-type characterizations. This note provides a characterization of complete regularity in terms of inserting a continuous real-valued function. The known fact that each continuous real valued function on a compact subset o fa Tychonoff space has a continuous extension to the whole space is obtained as a corollary.This research was supported by the Ministry of Education and Science of Spain and FEDER under grant MTM2006-14925-C02-02.Gutiérrez García, J.; Kubiak, T. (2007). Sandwich-type characterization of completely regular spaces. Applied General Topology. 8(2):239-242. https://doi.org/10.4995/agt.2007.1887SWORD2392428

Quotient on some Generalizations of topological group

في هذا البحث ، تم تعريف بعض التعميمات للمجموعة التبولوجية وهي المجموعة التبولوجية - α ، والمجموعة التبولوجية - ب ، والمجموعة التبولوجية - β  مع أمثلة توضيحية. بالإضافة إلى ذلك ، تم تعريف المجموعة التبولوجية للشواء فيما يتعلق بالشواية. فيما بعد ، تم تداول حاصل قسمة تعميمات المجموعة التبولوجية في مجموعة تبولوجية - p معينة. علاوة على ذلك ، تمت مناقشة نموذج النظام الروبوتي الذي يعتمد على حاصل المجموعة التبولوجية – p.In this paper, we define some generalizations of topological group namely -topological group, -topological group and -topological group with illustrative examples. Also, we define grill topological group with respect to a grill. Later, we deliberate the quotient on generalizations of topological group in particular -topological group. Moreover, we model a robotic system which relays on the quotient of -topological group

On Internal Characterizations of CompletelyL-Regular Spaces

AbstractCompleteL-regularity is internally characterized in terms of separating chains of openL-sets. A possible characterization in terms of normal and separating families of closedL-sets is discussed and it is shown that spaces admitting such families are completelyL-regular. The question of whether the converse holds true remains open. Some partial solutions are however given, e.g. in the class of countably compact spaces