Characterizations of M-fuzzifying convex spaces via Galois connections

This paper introduces a special Galois connection combined with the wedge-below relation. Furthermore, by using this tool, it is shown that the category of M-fuzzifying betweenness spaces and the category of M-fuzzifying convex spaces are isomorphic and the category of arity-2 M-fuzzifying convex spaces can be embedded in the category of M-fuzzifying interval spaces as a reflective subcategory.

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.

A categorical approach to abstract convex spaces and interval spaces

In this paper, we establish the axiomatic conditions of hull operators and introduce the category of interval spaces. We also investigate their relations with convex spaces from a categorical sense. It is shown that the category CS of convex spaces is isomorphic to the category HS of hull spaces, and they are all topological over Set. Also, it is proved that there is an adjunction between the category IS of interval spaces and the category CS of convex spaces. In particular, the category CS(2) of arity 2 convex spaces can be embedded in IS as a reflective subcategory.

Bounded sobriety and k-bounded sobriety of Q-cotopological spaces

In this paper, we extend bounded sobriety and k-bounded sobriety to the setting of Q-cotopological spaces, whereQis a commutative and integral quantale. The main results are: (1) The category BSobQ-CTop of all bounded sober Q-cotopological spaces is a full reflective subcategory of the category SQ-CTop of all stratified Q-cotopological spaces; (2) We present the relationships among Hausdorff, T1, sobriety, bounded sobriety and k-bounded sobriety in the setting ofQ-cotopological spaces; (3) For a linearly ordered quantale Q, a topological space X is bounded (resp., k-bounded) sober if and only if the corresponding Q-cotopological space ?Q(X) is bounded (resp., k-bounded) sober, where ?Q : Top ? SQ-CTop is the well-known Lowen functor in fuzzy topology.

How to centralize and normalize quandle extensions

We show that quandle coverings in the sense of Eisermann form a (regular epi)-reflective subcategory of the category of surjective quandle homomorphisms, both by using arguments coming from categorical Galois theory and by constructing concretely a centralization congruence. Moreover, we show that a similar result holds for normal quandle extensions.

A completion-invariant extension of the concept of meet continuous lattices

In this paper, the concept of meet F-continuous posets is introduced. The main results are: (1) A poset P is meet F-continuous iff its normal completion is a meet continuous lattice iff a certain system γ(P) which is, in the case of complete lattices, the lattice of all Scott closed sets is a complete Heyting algebra; (2) A poset P is precontinuous iff P is meet F-continuous and quasiprecontinuous; (3) The category of meet continuous lattices with complete homomorphisms is a full reflective subcategory of the category of meet F-continuous posets with cut-stable maps.

Cubical Sets and Trace Monoid Actions

This paper is devoted to connections between trace monoids and cubical sets. We prove that the category of trace monoids is isomorphic to the category of generalized tori and it is a reflective subcategory of the category of cubical sets. Adjoint functors between the categories of cubical sets and trace monoid actions are constructed. These functors carry independence preserving morphisms in the independence preserving morphisms. This allows us to build adjoint functors between the category of weak asynchronous systems and the category of higher dimensional automata.

ON THE SPECTRALIFICATION OF A HEMISPECTRAL SPACE

An open subset U of a topological space X is called intersection compact open, or ICO, if for every compact open set Q of X, U ∩ Q is compact. A continuous map f of topological spaces will be called spectral if f-1 carries ICO sets to ICO sets. Call a topological space Xhemispectral, if the intersection of two ICO sets of X is an ICO. Let HSPEC be the category whose objects are hemispectral spaces and arrows spectral maps. Let SPEC be the full subcategory of HSPEC whose objects are spectral spaces. The main result of this paper proves that SPEC is a reflective subcategory of HSPEC. This gives a complete answer to Problem BST1 of "O. Echi, H. Marzougui and E. Salhi, Problems from the Bizerte–Sfax–Tunis seminar, in Open Problems in Topology II, ed. E. Pearl (Elsevier, 2007), pp. 669–674."

On universal algebra over nominal sets

We investigate universal algebra over the category Nom of nominal sets. Using the fact that Nom is a full reflective subcategory of a monadic category, we obtain an HSP-like theorem for algebras over nominal sets. We isolate a ‘uniform’ fragment of our equational logic, which corresponds to the nominal logics present in the literature. We give semantically invariant translations of theories for nominal algebra and NEL into ‘uniform’ theories, and systematically prove HSP theorems for models of these theories.