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.
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.
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.
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.
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.
Wu, C., Hu, X. and Yang, J. (2008), "On blocks and coverings based on semi‐equivalence relation in pansystems methodology", Kybernetes, Vol. 37 No. 6, pp. 739-748. https://doi.org/10.1108/03684920810876945
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