Some combinatorial characteristics of closure operations Nguyen Hoang Son and Vu Duc Thi

Abstract

Functional dependencies (FDs) play an important role the relational database theory. The equivalence of the family of FDs is one of the hottest topics that get a lot of attention and interest currently. There are many equivalent descriptions of the family of FDs. Based on the equivalent descriptions, we can obtain many important properties of the family of FDs. The closure operation is an equivalent description of family of FDs ([1]). A closure operation here is a map between the elements of a partial ordered set that verifies three axioms: extension, order-preservation and idempotence. In recent years, the closure operations have been widely studied (e.g. see [2, 7–9]). Closed set, minimal key and antikey of closure operations are the interestring concepts and significant. Such as the family of closed sets of a closure operation forms a closure system (or meet- semlattice). Recently the closure operations have also been applied in data mining (e.g. see [5, 6]). This paper investigates some characteristics of minimal key and antikey of closure operations as well as the closeness of closure operations class under some basic operations. The paper is organized as follows. After an introduction section, in Section 2, we introduce the notions of closure oper- ation, minimal key and antikey of closure operation. Section 3 we present some characteristics of minimal key and antikey of closure operation. The algorithms for finding all minimal key and antikey of closure operations are studied in Section 4 and 5. The closeness of closure operations class under the union and direct product operations is studied in Section 6.