We generalize the notions of quietness and semisymmetry defined by Doitchinov (1991) and Deák (1991) and we study the role of these extended notions on the coincidence of some well-known quasi-uniform completeness. In particular, it is shown that the bicompletion coincides (up to quasi-uniformism) with the standardD-completion in quiet -weakly pair Cauchy bounded quasi-uniform spaces and it coi...

In the practice of information extraction, the input data are usually arranged into pattern matrices, and analyzed by the methods of linear algebra and statistics, such as principal component analysis. In some applications, the tacit assumptions of these methods lead to wrong results. The usual reason is that the matrix composition of linear algebra presents information as flowing in waves, whe...

In this article we introduce and investigate the concept of a partial quasi-metric and some of its applications. We show that many important constructions studied in Matthews’s theory of partial metrics can still be used successfully in this more general setting. In particular we consider the bicompletion of the quasi-metric space that is associated with a partial quasi-metric space and study i...

In the practice of information extraction, the input data are usually arranged into pattern matrices, and analyzed by the methods of linear algebra and statistics, such as principal component analysis. In some applications, the tacit assumptions of these methods lead to wrong results. The usual reason is that the matrix composition of linear algebra presents information as flowing in waves, whe...

Let Top0 be the category of topological T0-spaces, QU0 the category of quasi-uniform T0-spaces, T : QU0 → Top0 the usual forgetful functor and K : QU0 → QU0 the bicompletion reflector with unit k : 1 → K. Any T -section F : Top0 → QU0 is called K-true if KF = FTKF, and upper (lower) K-true if KF is finer (coarser) than FTKF . The literature considers important T -sections F that enjoy all three...

Based on previous results that study the completion of fuzzy metric spaces, we show that every intuitionistic fuzzy quasi-metric space, using the notion of fuzzy metric space in the sense of Kramosil and Michalek to obtain a generalization to the quasi-metric setting, has a bicompletion which is unique up to isometry.

In this paper we survey some recent developments in duality for lattices with additional operations paying special attention to Heyting algebras and the connections to Esakia’s work in this area. In the process we analyse the Heyting implication in the setting of canonical extensions both as a property of the lattice and as an additional operation. We describe Stone duality as derived from cano...

This paper considers totally bounded quasi-uniformities and quasi-proximities for frames and shows that for a given quasi-proximity ⊳ on a frame L there is a totally bounded quasi-uniformity on L that is the coarsest quasi-uniformity, and the only totally bounded quasi-uniformity, that determines ⊳. The constructions due to B. Banaschewski and A. Pultr of the Cauchy spectrum ψL and the compacti...