A unified theory of perfect and related functions

Abstract

A unified theory has been developed on the basis of the similarity in properties of perfect and allied types of functions. The theory intromites as a starting point a certain subset of $\mathcal{P}\mathcal{(}\mathcal{X}\mathcal{)}$, the power set of a nonvoid set $X$, and an operator $\alpha$ on $\mathcal{P}\mathcal{(}\mathcal{X}\mathcal{)}$; a second operator $\beta$ is also brought into action. This theory of $\beta$-perfect functions includes the theories of perfect, $\theta$-perfect and $\delta$-perfect functions and is seen to generate many new types of functions when different pairs of operators take the roles of the pair $(\alpha ,\beta )$.