Properties of composite of closure operations and choice functions

Abstract

The equivalence of the family of FDs is among many hottest topics that get a lot of attention and consideration currently. There are many equivalent descriptions of the family of FDs. The closure operation and choice function are two of them. Major results of this paper are the properties of the composite function of the choice functions and closure operations. The first parts of this paper address the theories of the composite function of two choice functions and the sufficient and necessary condition of a composite function of two choice functions to be a choice function. Rest of the paper addresses the sufficient and necessary condition of a composite function of more than two choice functions to be a choice function and a composite function of more than two closure operations to be a closure operation.