Power domain constructions

Abstract

The variety of power domain constructions proposed in the literature is put into a general algebraic framework. Power constructions are considered algebras on a higher level: for every ground domain, there is a power domain whose algebraic structure is specified by means of axioms concerning the algebraic properties of the basic operations empty set, union, singleton, and extension of functions. A host of derived operations is introduced and investigated algebraically. Every power construction is shown to be equipped with a characteristic semiring such that the resulting power domains become semiring modules. Power homomorphisms are introduced as a means to relate different power construction. They also allow for defining the notion of initial and final constructions for a fixed characteristic semiring. Such initial and final constructions are shown to exist for every semiring, and their basic properties are derived. Finally, the known power constructions are put into the general framework of this paper.