A survey of fuzzifications of frames, the Papert--Papert--Isbell adjunction and sobriety

Authors:
Wei Yao
Affiliations:
Department of Mathematics, Hebei University of Science and Technology, Shijiazhuang 050018, PR China
Venue:
Fuzzy Sets and Systems
Year:
2012

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Abstract

This paper gives a survey of fuzzifications of frames, the Papert-Papert-Isbell adjunction between the categories of topological spaces and locales as well as the related concept of sobriety. It then establishes categorical isomorphisms between Yao-L-frames, Zhang-Liu-L-frames and L-algebras. Lastly, it studies the relationships between L-sobriety, modified L-sobriety and @i"L-sobriety. The obtained results suggest that modified L-sobriety is the most fruitful notion between the above concepts of lattice-valued sobriety.