An Equational Approach to Enriched Distrubitivity
The familiar adjunction between ordered sets and completely distributive lattices can be extended to generalised metric spaces, that is, categories enriched over a quantale (a lattice of \truth values"), via an appropriate distributive law between the \down-set" monad and the \up-set" monad on the category of quantale-enriched categories. If the underlying lattice of the quantale is completely distributive, and if powers distribute over non-empty joins in the quantale, then this distributive law can be concretely formulated in terms of operations, equations and choice functions, similar to the familiar distributive law of lattices.
A. Balan and A. Kurz, An equational approach to enriched distrubitivity. Rev. Roumaine Math. Pures Appl. 66 (2021), 3-4, 577-596. http://imar.ro/journals/Revue_Mathematique/pdfs/2021/3-4/4.pdf
EDITURA ACADEMIEI ROMÂNE