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

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 ιLιL-sobriety. The obtained results suggest that modified L-sobriety is the most fruitful notion between the above concepts of lattice-valued sobriety.

► This paper gives a survey of fuzzifications of frames, the Papert–Papert–Isbell adjunction and sobriety. ► It is shown that Yao-L-frames, Zhang–Liu-L-frames and L-algebras are categorically isomorphic. ► Relations between L-sobriety, modified L-sobriety and ιL-sobriety are investigated. ► Results indicate that modified L-sobriety is the most reasonable fuzzy sobriety.