Fuzzy Sets and Systems
Abstract and concrete categories
Abstract and concrete categories
Point-set lattice-theoretic topology
Fuzzy Sets and Systems - Special memorial volume on mathematical aspects of fuzzy set theory
Fuzzy Sets and Systems - Special issue on fuzzy topology
Some remarks on fuzzy powerset operators
Fuzzy Sets and Systems
Fuzzy Sets and Systems
An approach to fuzzy frames via fuzzy posets
Fuzzy Sets and Systems
Nuclei and conuclei on residuated lattices
Fuzzy Sets and Systems
A survey of fuzzifications of frames, the Papert--Papert--Isbell adjunction and sobriety
Fuzzy Sets and Systems
Categorical foundations of variety-based topology and topological systems
Fuzzy Sets and Systems
Composite variety-based topological theories
Fuzzy Sets and Systems
Category-theoretic fuzzy topological spaces and their dualities
Fuzzy Sets and Systems
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In a way more general than variable-basis approach to lattice-valued topological spaces, the present paper introduces an alternative approach to lattice-valued topological spaces-direct product representation spaces extending the notion of quantal spaces in the sense of Mulvey and Pelletier to semi-quantales recently proposed by Rodabaugh. This paper aims to give an answer to the main question whether there exists a categorical connection, possibly a categorical equivalence, between direct product representation spaces and variable-basis lattice-valued topological spaces. Small sources in the category of semi-quantales which are called pointed semi-quantales can be identified with direct product representation spaces. For this reason, the main problem will be handled in terms of pointed semi-quantales. Generalized quasi-lattice-valued topological spaces extending variable-basis quasi-topological spaces into the present setting are introduced to be a suitable topological counterpart of pointed semi-quantales. To formalize and to solve the main problem, categories of pointed semi-quantales and of generalized quasi-lattice-valued topological spaces are constructed, and the relations between these categories, providing a satisfactory answer to the main problem, are established in this paper.