On induced congruences

In this note we present a proof of a theorem to be found in Ehrig and Mahr [3). The theorem states that a relation constructed from a given function is a congruence relation if that function is a homomorphism: we go on to generalise this result to the relational homomorphisms treated by the first author in [1]. Specifically, we prove that. a congruence relation can be constructed from a relational homomorphism. Our construction generalises that of Ehrig and Mahr. The significance of this note lies both in the economy of our calculations and in the novel use we make of weakest prespecifications. We have been unable to extend the theorem to an equivalence: we offer the remaining half of the equivalence as a challenge to the reader. We consider a type T defined according to the paradigm T = p(r : F). where F is a relator (such types are special cases of Hagino types’. details of which may be found in Hagino 14]). For the purposes of this note, the important properties of a relator F are the following.