Abstract :
Abstract. The paper sets forth in detail categorically-algebraic or catalg
foundations for the operations of taking the image and preimage of (fuzzy)
sets called forward and backward powerset operators. Motivated by an open
question of S. E. Rodabaugh, we construct a monad on the category of sets,
the algebras of which generate the xed-basis forward powerset operator of
L. A. Zadeh. On the next step, we provide a direct lift of the backward powerset
operator using the notion of categorical biproduct. The obtained framework
is readily extended to the variable-basis case, justifying the powerset theories
currently popular in the fuzzy community. At the end of the paper, our general
variety-based setting postulates the requirements, under which a convenient
variety-based powerset theory can be developed, suitable for employment in
all areas of fuzzy mathematics dealing with fuzzy powersets, including fuzzy
algebra, logic and topology.
