Abstract
For general powerset theories in categories, new notions of relational, closure, or partition powerset theories in these categories are introduced. These new types of powerset theories are defined as categories whose objects are the original powerset objects with relational, closure, or partition structures defined on these objects. This construction generalizes classical constructions realized on powerset objects of all subsets of a given set, or all fuzzy sets on a given set, and it enables to define these structures on more general powerset objects. Examples of these new powerset objects are also shown.
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