The objective of this paper is to investigate the role of the set of irrational numbers as the codomain of order-preserving functions defined on topological totally preordered sets. We will show that although the set of irrational numbers does not satisfy the Debreu property it is still nonetheless true that any lower (respectively, upper) semicontinuous total preorder representable by a real-v...

Motivated by ideas from the study of abstract data types, we show how to interpret non-well-founded sets as tixed points of continuous transformations of an initial continuous algebra. We conisder a preordered structure closely related to the set HF of well-founded, hereditarily finite sets. By taking its ideal completion, we obtain an initial continuous algebra in which we are able to solve al...

The paper presents the essential connections between modal-like operators, topologies and fuzzy sets. We show, for example, that each fuzzy set determines a preorder and an Alexandrov topology, and that similar correspondences hold also for the other direction. Further, a category for preorder-based fuzzy sets is defined, and it is shown that its equivalent subcategory of representatives is iso...

Let X, X′ be two locally finite, preordered sets and let R be any indecomposable commutative ring. The incidence algebra I(X,R), in a sense, represents X, because of the wellknown result that if the rings I(X,R) and I(X′,R) are isomorphic, then X and X′ are isomorphic. In this paper, we consider a preordered set X that need not be locally finite but has the property that each of its equivalence...

in this note by considering a complete lattice l, we define thenotion of an l-fuzzy hyperrelation on a given non-empty set x. then wedefine the concepts of (pom)l-fuzzy graph, hypergraph and subhypergroupand obtain some related results. in particular we construct the categories ofthe above mentioned notions, and give a (full and faithful) functor form thecategory of (pom)l-fuzzy subhypergroups ...

We propose a convenient category for directed homotopy consisting of preordered topological spaces generated by cubes. Its main advantage is that, like the category of topological spaces generated by simplices suggested by J. H. Smith, it is locally presentable.

In this work, we focus on adjunctions (also named isotone Galois connections) between fuzzy preordered sets; specifically, we study necessary and sufficient conditions that have to be fulfilled in order such an adjunction to exist.

We prove versions of Ekeland, Takahashi and Caristi principles in preordered quasi-metric spaces, the equivalence between these principles, as well their to some completeness results for underlying space. These extend proved S.~Cobza\c{s}, Topology Appl. \textbf{265} (2019), 106831, 22, spaces. The key tools are Picard sequences special set-valued mappings on a space $X$, defined terms preorder...