44 research outputs found

    Quantale-valued Cauchy tower spaces and completeness

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    [EN] Generalizing the concept of a probabilistic Cauchy space, we introduce quantale-valued Cauchy tower spaces. These spaces encompass quantale-valued metric spaces, quantale-valued uniform (convergence) tower spaces and quantale-valued convergence tower groups. For special choices of the quantale, classical and probabilistic metric spaces are covered and probabilistic and approach Cauchy spaces arise. We also study completeness and completion in this setting and establish a connection to the Cauchy completeness of a quantale-valued metric space.Jäger, G.; Ahsanullah, TMG. (2021). Quantale-valued Cauchy tower spaces and completeness. Applied General Topology. 22(2):461-481. https://doi.org/10.4995/agt.2021.15610OJS461481222J. Adámek, H. Herrlich and G. E. Strecker, Abstract and Concrete Categories, Wiley, New York, 1989.T. M. G. Ahsanullah and G. Jäger, Probabilistic uniform convergence spaces redefined, Acta Math. 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    Characterization of fuzzy neighborhood commutative division rings II

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    In [4] we produced a characterization of fuzzy neighborhood commutative division rings; here we present another characterization of it in a sense that we minimize the conditions so that a fuzzy neighborhood system is compatible with the commutative division ring structure. As an additional result, we show that Chadwick [5] relatively compact fuzzy set is bounded in a fuzzy neighborhood commutative division ring

    A characterization of Fuzzy neighborhood commutative division rings

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    We give a characterization of fuzzy neighborhood commutative division ring; and present an alternative formulation of boundedness introduced in fuzzy neighborhood rings. The notion of β-restricted fuzzy set is considered

    Characterization of Transitivity in L-Tolerance Spaces by Convergence and Closure

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    We show that the category of quantale-valued tolerance spaces is isomorphic to a category of quantale-valued convergence spaces. We define suitable quantale-valued closure functions and use them to characterize transitivity axioms. Furthermore, transitivity is characterized by convergence and diagonal axioms. Quantale-valued tolerance relations compatible with group structures are also characterized by convergence and it is shown that they are transitive

    Characterization of Transitivity in L-Tolerance Spaces by Convergence and Closure

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    We show that the category of quantale-valued tolerance spaces is isomorphic to a category of quantale-valued convergence spaces. We define suitable quantale-valued closure functions and use them to characterize transitivity axioms. Furthermore, transitivity is characterized by convergence and diagonal axioms. Quantale-valued tolerance relations compatible with group structures are also characterized by convergence and it is shown that they are transitive

    Fuzzy congruences on groups and rings

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    We prove that the lattice of fuzzy congruences of a group G (resp. ring R) is isomorphic to the lattice of fuzzy normal subgroups of G (resp. fuzzy ideals of R)

    T-neighborhood groups

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    We generalize min-neighborhood groups to arbitrary T-neighborhood groups, where T is a continuous triangular norm. In particular, we point out that our results accommodate the previous theory on min-neighborhood groups due to T. M. G. Ahsanullah. We show that every T-neighborhood group is T-uniformizable, therefore, T-completely regular. We also present several results of T-neighborhood groups in conjunction with TI-groups due to J. N. Mordeson
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