Homomorphism and Isomorphism Theorems Generalized from a Relational Perspective Gunther Schmidt Institute for Software Technology, Department of Computing Science Universit¨at der Bundeswehr M¨unchen, 85577 Neubiberg, Germany
[email protected] Abstract. The homomorphism and isomorphism theorems traditionally taught to students in a group theory or linear algebra lecture are by no means theorems of group theory. They are for a long time seen as general concepts of universal algebra. This article goes even further and identifies them as relational properties which to study does not even require the concept of an algebra. In addition it is shown how the homomorphism and isomorphism theorems generalize to not necessarily algebraic and thus relational structures. Keywords: homomorphism theorem, isomorphism theorem, relation al- gebra, congruence, multi-covering. 1 Introduction Relation algebra has received increasing interest during the last years. Many areas have been reconsidered from the relational point of view, which often pro- vided additional insight. Here, the classical homomorphism and isomorphism theorems (see [1], e.g.) are reviewed from a relational perspective, thereby sim- plifying and slightly generalizing them. The paper is organized as follows. First we recall the notion of a heterogeneous relation algebra and some of the very basic rules governing work with relations. With these, function and equivalence properties may be formulated concisely. The relational concept of homomorphism is defined as well as the concept of a congruence which is related with the concept of a multi-covering, which have connections with topology, complex analysis, and with the equivalence problem for flow-chart programs. We deal with the relationship between mappings and equivalence relations.