L-interval spaces and some of their properties

In this paper, notions of L-interval spaces and L-2-arity convex spaces are introduced. It is showed that there is a Galois’s connection between the category of L-convex spaces and the category of L-interval spaces. In particular, the category of L-2-arity convex spaces can be embedded in the category of L-interval spaces as a coreflective subcategory. Further, some properties of L-interval spaces are introduced including L-geometric (resp. L-Peano, L-Pasch and L-sand-glass) property. It is proved that an L-2-arity convex space is an L-JHC convex space iff its segment operator has L-Peano property. It is also proved that an L-JHC convex space with an L-idempotent segment operator has L-sand-glass property. Further, it is also proved that an L-idempotent interval space having L-Peano+L-Pasch property has L-geometric property and L-sand-glass property.

The categorical relationships between neighborhood spaces, ┬-neighborhood spaces and stratified L-neighborhood spaces

In this paper, for a complete residuated lattice L, we present the categorical properties of ?-neighborhood spaces and their categorical relationships to neighborhood spaces and stratified L-neighborhood spaces. The main results are: (1) the category of ?-neighborhood spaces is a topological category; (2) neighborhood spaces can be embedded in ?-neighborhood spaces as a reflective subcategory, and when L is a meet-continuous complete residuated lattice, ?-neighborhood spaces can be embedded in stratified L-neighborhood spaces as a reflective subcategory; (3) when L is a continuous complete residuated lattice, neighborhood spaces (resp., ?-neighborhood spaces) can be embedded in ?-neighborhood spaces (resp., stratified L-neighborhood spaces) as a simultaneously reflective and coreflective subcategory.

On fuzzification of topological categories

This talk provides a fuzzification procedure for topological categories, i.e., given a topological category A, there exists a topological category B, which contains A as a full concretely coreflective subcategory, and which can be considered as a fuzzification of A.

Closure theories of powerset theories

A notion of a closure theory of a powerset theory in a ground category is introduced as a generalization of a topology theory of a powerset theory. Using examples of powerset theories in the category Set of sets and in the category of sets with similarity relations, it is proved that these theories can be used as ground theories for closure theories of powerset theories in these two categories. Moreover, it is proved that all these closure theories of powerset theories are topological constructs. A notion of a closure operator which preserves a canonical form of fuzzy objects in these categories is introduced, and it is proved that a closure theory of a powerset theory in the ground category Set is a coreflective subcategory of the closure theory of (Zadeh’s) powerset theory, which preserves canonical forms of fuzzy sets.

Completeness and cocompleteness of the categories of basic pairs and concrete spaces

We show that the category of basic pairs (BP) and the category of concrete spaces (CSpa) are both small-complete and small-cocomplete in the framework of constructive Zermelo–Frankel set theory extended with the set generation axiom. We also show thatCSpais a coreflective subcategory ofBP.

On classes of T0 spaces admitting completions

<p>For a given class X of T0 spaces the existence of a subclass C, having the same properties that the class of complete metric spaces has in the class of all metric spaces and non-expansive maps, is investigated. A positive example is the class of all T0 spaces, with C the class of sober T0 spaces, and a negative example is the class of Tychonoff spaces. We prove that X has the previous property (i.e., admits completions) whenever it is the class of T0 spaces of an hereditary coreflective subcategory of a suitable supercategory of the category Top of topological spaces. Two classes of examples are provided.</p>

Comparing Transition Systems with Independence and Asynchronous Transition Systems

Transition systems with independence and asynchronous transition systems are non-interleaving models for concurrency arising from the same simple idea of decorating transitions with events. They differ for the choice of a derived versus a primitive notion of event which induces considerable differences and makes the two models suitable for different purposes. This opens the problem of investigating their mutual relationships,<br />to which this paper gives a fully comprehensive answer.<br />In details, we characterise the category of extensional asynchronous transitions systems as the largest full subcategory of the category of (labelled) asynchronous transition systems which admits TSI, the category of transition systems with independence, as a coreflective subcategory. In addition, we introduce event-maximal asynchronous transitions systems and we show that their category is equivalent to TSI, so providing an exhaustive characterisation of transition systems with independence in terms of asynchronous transition systems.

On projective lift and orbit spaces

By constructing the projective lift of a dp-epimorphism, we find a covariant functor E from the category Cd of regular Hausdorff spaces and continuous dp-epimorphisms to its coreflective subcategory εd consisting of projective objects of Cd We use E to show that E(X/G) is homeomorphic to EX/G whenever G is a properly discontinuous group of homeomorphisms of a locally compact Hausdorff space X and X/G is an object of Cd.