Purpose – The purpose of this paper is to construct diverse granules and coverings derived from a semi‐equivalence class and then to reveal the relationships between granules and coverings and even relations, and to help one to study pansystems relations in a trans‐cross view. Design/methodology/approach – Forms regarding semi‐equivalence classes as primitives, blocks or granules such as tolerance class, join class, meet class, optimist/pessimist selected compatible class, and tolerantly kernel class in an incomplete information system are defined and compared. Different approximations are also introduced. Furthermore, diverse coverings are also suggested. Findings – A tolerance class of an object is proved to be the join of semi‐equivalence classes containing it, i.e. a result of acting union operation on some primitive classes. A compatibly kernel class of an object is a meet of semi‐equivalence classes including it, i.e. a result of acting intersection operation on some primitive classes. Related coverings can also be regarded as coverings derived from the covering consisting of primitive granules. Several necessary and/or sufficient conditions for a general covering to become a semi‐equivalence or complete covering are obtained. Meaningful property and relationship results are also exploited. Practical implications – Constructing diverse granules naturally from an incomplete information system to form a different knowledge expression system looks promising for data mining in the information society. It widens the approach and schema. Originality/value – The paper shows that the formation process of granules is natural, newly defined, and not similar and theoretic to those existing in a neighborhood system. The relationship between diverse granules and coverings is described by mathematical theorems in sufficient or necessary condition form.
Kybernetes – Emerald Publishing
Published: Jun 17, 2008
Keywords: Cybernetics; Systems theory; Set theory