Axiomatic of lattice-valued F-transform

Axioms for upper and lower fuzzy transformation systems are introduced, and it is proved that a transformation function satisfies axioms for upper (or lower, respectively) transformation systems if and only if it is an upper (or lower, respectively) F-transform. Categories of upper (lower, respectively) transformation systems are introduced, and it is proved that both categories are isomorphic to the category of spaces with fuzzy partitions. We prove that lower and upper F-transforms preserve extensional fuzzy sets associated with spaces with fuzzy partitions.