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.
