A generalized setting for fixpoint theory

The mathematical semantics of programming languages is based largely on certain algebraic structures, usually complete lattices or complete partial orders. The usefulness of these structures is based on the existence of fixpoints of functions defined on the structures, and the fact that these classes of structures are closed under such operations as taking cross-products, disjoint unions or function spaces.This paper proposes more general versions of these structures which still retain the above desirable properties. Thus the techniques of mathematical semantics should become applicable in a wider context than heretofore.One important application is given, which in fact motivated the whole development. It is shown that in the generalized setting the existence of unique minimal solutions for recursive definitions of functions are guaranteed without having to resort to informal arguments of any sort.