A category approach to relation preserving functions in rough set theory

• A textural counterpart diRel of the category Rel is presented.
• diRel is a topological category over dfTex and Rel is a full subcategory of diRel.
• The preimage of a definable set with respect to a relation preserving function is also definable.
• Information systems and information system homomorphisms form a category denoted by IS.
• The category ISO of object-irreducible information systems is embeddable into Rel.

The category Rel whose objects are all pairs (U,r)(U,r), where r is a relation on a universe U, and whose morphisms are relation-preserving mappings is a canonical example in category theory. One of the convenient categories for rough set systems on a single universe is Rel since the objects of Rel are approximation spaces. The morphisms of a ground category dfTex whose objects are textures can be characterized by definability. Therefore, we particularly investigate a textural counterpart of the category Rel denoted by diRel of textural approximation spaces and direlation preserving difunctions. In this respect, we prove that diRel is a topological category over dfTex and Rel is a full subcategory of diRel. In view of the textural arguments, we show that the preimage of a definable subset of an approximation space with respect to a relation preserving function is also definable in the category Rere of reflexive relations. Furthermore, we denote the category of all information system homomorphisms and all information systems by IS and we show that the category ISO of all information system homomorphisms and all object-irreducible information systems where the attribute functions are surjective is embeddable into Rel.