Algebraic Implementation of Objects over Objects

TLDR: The main technical result is a normal form theorem saying that any regular implementation, i.e. one composed of any number of extensions and encapsulations, in any order, can be done in just two steps: first an extension, and then an encapsulation.

Abstract: This paper gives semantic foundations of (correct) implementation as a relationship between an "abstract" object and a community of "base" objects. In our aproach, an object is an "observed process". Objects and object morphisms constitute a category OB in which colimits reflect object aggregation and interaction between objects. Our concept of implementation allows for composition, i.e. by composing any number of (correct) implementation steps, a (correct) entire implementation is obtained. We study two specific kinds of implementation, extension and encapsulation, in more detail and investigate their close relationship to object morphisms. Our main technical result is a normal form theorem saying that any regular implementation, i.e. one composed of any number of extensions and encapsulations, in any order, can be done in just two steps: first an extension, and then an encapsulation.