L-subquantales and L-filters of quantales based on quasi-coincidence of fuzzy points with parameters

The paper investigates fuzziness of quantales by means of quasi-coincidence of fuzzy points with two parameters based on L-sets and developes two more generalized fuzzy structures, called (?g,?g Vqh)-L-subquantale and (?g,?g Vqh)-L-filter. Some intrinsic connections between (?g,?g Vqh)-L-subquantales and crisp subquantales are established, and relationships between (?g,?g Vqh)-L-filters of quantales and their extensions (especially the essential connections between (?g,?g Vqh)-L-subquantales and (?g,?g Vqh)-Lfilters of quantales) are studied by employing the new characterizations of (?g,?g Vqh)-L-filters of quantales. Also, sufficient conditions for the extension of an (?g,?g Vqh)-L-filter to be an (?g,?g Vqh)-L-filter of a quantale are also o ered. In particular, it is proved that the category GLFquant (resp., GFFQant) of (?g,?g Vqh) Lsubquantales (resp., L-filters) is of a topological construct on Quant and posses equalizers and pullbacks.

Categorical Properties of Soft Sets

The present study investigates some novel categorical properties of soft sets. By combining categorical theory with soft set theory, a categorical framework of soft set theory is established. It is proved that the categorySFunof soft sets and soft functions has equalizers, finite products, pullbacks, and exponential properties. It is worth mentioning that we find thatSFunis both a topological construct and Cartesian closed. The categorySRelof soft sets andZ-soft set relations is also characterized, which shows the existence of the zero objects, biproducts, additive identities, injective objects, projective objects, injective hulls, and projective covers. Finally, by constructing proper adjoint situations, some intrinsic connections betweenSFunandSRelare established.

A better framework for first countable spaces

<p>In the realm of semiuniform convergence spaces first countability is divisible and leads to a well-behaved topological construct with natural function spaces and one-point extensions such that countable products of quotients are quotients. Every semiuniform convergence space (e.g. symmetric topological space, uniform space, filter space, etc.) has an underlying first countable space. Several applications of first countability in a broader context than the usual one of topological spaces are studied.</p>

The quasitopos hull of the construct of closure spaces

<p>In the list of convenience properties for topological constructs the property of being a quasitopos is one of the most interesting ones for investigations in function spaces, differential calculus, functional analysis, homotopy theory, etc. The topological construct Cls of closure spaces and continuous maps is not a quasitopos. In this article we give an explicit description of the quasitopos topological hull of Cls using a method of F. Schwarz: we first describe the extensional topological hull of Cls and of this hull we construct the cartesian closed topological hull.</p>

Functorial approach structures

<p>We show that there exists at least a proper class of functorial approach structures, i.e., right inverses to the forgetful functor T : AP→ Top (where AP denotes the topological construct of approach spaces and contractions as introduced by R. Lowen). There is however a great difference in nature of these functorial approach structures when compared to the quasi-uniform paradigm which has been extensively studied by the first author: whereas it is well-known from [2] that a large class of epireflective subcategories of Top₀ can be “parametrized” using the interaction of functorial quasi-uniformities with the quasi-uniform bicompletion, we show that using functorial approach structures together with the approach bicompletion developed in [10], only Top₀ itself can be retrieved in this way.</p>