Fuzzy topologies and topological space objects in a topos
Fuzzy Sets and Systems
Topology via logic
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
Necessity of non-stratified and anti-stratified spaces in lattice-valued topology
Fuzzy Sets and Systems
Variable-basis topological systems versus variable-basis topological spaces
Soft Computing - A Fusion of Foundations, Methodologies and Applications - Special Issue on Fuzzy Set Theory and Applications; Guest Editors: Ferdinand Chovanec, Olga Nánásiová, Alexander Šostak
Generalized fuzzy topology versus non-commutative topology
Fuzzy Sets and Systems
On limits and colimits of variety-based topological systems
Fuzzy Sets and Systems
Interweaving algebra and topology: Lattice-valued topological systems
Fuzzy Sets and Systems
Categorical foundations of variety-based topology and topological systems
Fuzzy Sets and Systems
Sobriety and spatiality in categories of lattice-valued algebras
Fuzzy Sets and Systems
Category-theoretic fuzzy topological spaces and their dualities
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Categorically algebraic topology versus universal topology
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On fuzzification of topological categories
Fuzzy Sets and Systems
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This paper studies localic products, traditional topological products, and L-topological products, and gives a complete outline of the localic product. Comparisons of localic and L-topological products are generally absent in the literature, and this paper answers longstanding open questions in that area as well as provides a complete proof of the classical comparison theorem for localic and traditional topological products. This paper contributes several L-valued comparison theorems, one of which states: the localic and L-topological products of L-topologies are order isomorphic if and only if the localic product is L-spatial, providing L is itself spatial and the family of L-topological spaces is ''prime separated''. These last two conditions always hold in the traditional setting, capturing the traditional comparison theorem as a special case, and the prime separation condition is satisfied by important lattice-valued examples that include the fuzzy real line and the fuzzy unit interval for L any complete Boolean algebra and the alternative fuzzy real line and fuzzy unit interval for L any (semi)frame. Separation conditions help control the ''sloppy'' behavior of the L-topological product when |L|2, and several separation conditions are studied in this context; and it should be noted that localic products have a point-free version of the ''product'' separation condition considered in this paper. The traditional comparison theorem is carefully proved both to fill gaps in the extant literature and to motivate the L-valued comparison theorem quoted above and reveal the special role played by cross sums of prime (L-)open subsets. En route, characterizations are given of prime L-open subsets of certain L-products, which in turn yield characterizations of prime open and irreducible closed subsets of traditional product spaces.