powerset theory, F-transform, residuated lattices, spaces with fuzzy partitions

A relationship between powerset theories and F-transforms are investigated. Both these methods represent very strong tools in thetheory and applications. Although both methods deal also with fuzzy object, both these methods use different tools and, so far, the relationship between the two methods has not been investigated. The aim of thispaper is to show that there is a very strong relationship between the two methods. Namely, arbitrary lower or upper F-transform of lattice-valued fuzzy sets can be derived from a special powerset theory and, conversely,there exists a special class of powerset theories, such that maps definedby these powerset theories are lower or upper F-transforms. This result allows, among other things, to extend the range of methods and tools that are used in both theories.