Sergey A. Solovyov

Sobriety and spatiality in varieties of algebras

The paper considers a generalization of the classical Papert-Papert-Isbell adjunction between the categories of topological spaces and locales to an arbitrary variety of algebras and illustrates the obtained results by the category of algebras over a given unital commutative quantale.

Variable-basis topological systems versus variable-basis topological spaces

The paper introduces a variable-basis generalization of the notion of topological system of Vickers and considers functorial relationships between the categories of variable-basis topological systems and variable-basis fuzzy topological spaces in the sense of Rodabaugh.

Categorical foundations of variety-based topology and topological systems

The paper considers a new approach to fuzzy topology based on the concept of variety and developed in the framework of topological theories resembling those of Rodabaugh. As a result, a categorical generalization of the notion of topological system of Vickers is obtained, and its theory unfolded, which clarifies the relations between algebra and topology. We also justify the use of semi-quantales as the basic underlying structure for doing lattice-valued topology as well as provide a categorical framework incorporating the theory of bitopological spaces.

From quantale algebroids to topological spaces: Fixed- and variable-basis approaches

Using the category of quantale algebroids the paper considers a generalization of the classical Papert-Papert-Isbell adjunction between the categories of topological spaces and locales to partial algebraic structures. It also provides a single framework in which to treat the concepts of quasi, standard and stratified fuzzy topology.

Localification of variable-basis topological systems

Abstract The paper provides another approach to the notion of variable-basis topological system generalizing the fixed-basis concept of S. Vickers, considers functorial relationships between the categories of modified variable-basis topological systems and variable-basis fuzzy topological spaces in the sense of S.E. Rodabaugh and shows that the procedure of localification is possible in the new setting.

Fuzzy algebras as a framework for fuzzy topology

The paper introduces a variety-based version of the notion of the (L,M)-fuzzy topological space of Kubiak and Sostak and embeds the respective category into a suitable modification of the category of topological systems of Vickers. The new concepts provide a common framework for different approaches to fuzzy topology and topological systems existing in the literature, paving the way for studying the problem of interweaving algebra and topology in mathematics, which was raised by Denniston, Melton and Rodabaugh in their recent research on variable-basis topological systems over the category of locales.

On a generalization of the concept of state property system

Based in the notions of topological system of S. Vickers and lattice-valued topological space of S.E. Rodabaugh, the paper introduces a generalization of the concepts of state property system of D. Aerts and closure space (used by many authors in the literature), showing that the categories of the new structures are equivalent.

Categories of lattice-valued sets as categories of arrows

In this paper we introduce a category X(A) which is a generalization of the category of lattice-valued subsets of sets Set(JCPos) introduced by us earlier. We show the necessary and sufficient conditions for X(A) to be topological over XxA.

Hypergraph functor and attachment

Using an arbitrary variety of algebras, the paper introduces a fuzzified version of the notion of attachment in a complete lattice of Guido, to provide a common framework for the concept of hypergraph functor considered by different authors in the literature. The new notion also gives rise to a category of variable-basis topological spaces which is a proper supercategory of the respective category of Rodabaugh.

Lattice-valued bornological systems

Motivated by the concept of lattice-valued topological system of J.T. Denniston, A. Melton, and S.E. Rodabaugh, which extends lattice-valued topological spaces, this paper introduces the notion of lattice-valued bornological system as a generalization of lattice-valued bornological spaces of M. Abel and A. Sostak. We aim at (and make the first steps towards) the theory, which will provide a common setting for both lattice-valued point-set and point-free bornology. In particular, we show the algebraic structure of the latter.