Geometrical isomorphisms between categories of fuzzy coverings and fuzzy partitions

Abstract

Let Covering be the category of the category of fuzzy coverings, and Partition, the category of fuzzy partitions. We geometrically construct an isomorphism of categories between Partition and a full subcategory of Covering, which can be used to derive bijections between fuzzy partitions and fuzzy coverings with finitely many fuzzy sets. Also, we establish an isomorphism between Covering[n], the category of coverings with n fuzzy sets, and a subcategory of Partition, whose objects are partitions with n fuzzy sets which satisfy certain conditions.