The aim of this paper is to establish a fuzzy version of the dualitybetween domains and completely distributive lattices. All values aretaken in a fixed frame $L$. A definition of (strongly) completelydistributive $L$-ordered sets is introduced. The main result inthis paper is that the category of fuzzy domains is dually equivalentto the category of strongly completely distributive $L$-ordereds...

The paper is devoted to a concept of a fuzzy ordered set and partially ordered set category fuzzyfication. The aim of this work is to define a fuzzy order relation on a usual set, on an L-valued set and on an L-set. Using the fuzzy order relation we construct fuzzy POS category and study the connection between this categories and SET(L).

We study the homotopy theory of locally ordered spaces, that is manifolds with boundary whose charts are partially ordered in a compatible way. Their category is not particularly well-behaved with respect to colimits. However, this category turns out to be a certain full subcategory of a topos of sheaves over a simpler site. The ambient topos makes available some general homotopical machinery.

We show that the category of principal ordered face structures pFs is equivalent to the category of multitopes Mlt.

Directed Algebraic Topology is a recent field, deeply linked with ordinary and higher dimensional Category Theory. A ‘directed space’, e.g. an ordered topological space, has directed homotopies (which are generally non reversible) and a fundamental category (replacing the fundamental groupoid of the classical case). Finding a simple possibly finite model of the latter is a non-trivial problem, ...

Ordered sheaves on a small quantaloid Q have been defined in terms of Qenriched categorical structures; they form a locally ordered category Ord(Q). The free-cocompletion KZ-doctrine on Ord(Q) has Mod(Q), the quantaloid of Q-modules, as category of Eilenberg-Moore algebras. In this paper we give an intrinsic description of the Kleisli algebras: we call them the locally principally generated Q-m...

Ordered sheaves on a small quantaloid Q have been defined in terms of Q-enriched categorical structures; they form a locally ordered category Ord(Q). It has previously been shown by the second author that the " free-cocompletion " KZ-doctrine on Ord(Q) has Mod(Q), the quantaloid of Q-modules, as category of Eilenberg-Moore algebras. In the first part of this paper we apply Q-enriched category t...

The 2-category of constructively completely distributive lattices is shown to be bidual to a 2-category of generalized orders that admits a monadic schizophrenic object biadjunction over the 2-category of ordered sets.

(1) some basic features to treat partially ordered sets; (2) the ability to enumerate, create, and display monotone and regular partitions of partially ordered sets; (3) the capability of constructing the lattices of partitions of a poset, and of doing some useful computations on these structures; (4) the possibility of computing products and coproducts in the category of partially ordered sets...

5 Goguen’s category of V-sets whose objects are functions with values in a pre-ordered set and morphisms are suitable maps is shown to be a topological construct if and only if the pre-ordered set is a complete lattice; in particular the category Set(L) of L-sets, 7 also considered by Goguen, is topological over Set. The special case when the considered pre-ordered set is L with the “mapping to...