Amalgamation, Epimorphisms and Definability Properties in Algebraic Logic

Here we investigate algebras associated with algebraizations of (variants of) first-order logic and of multi-modal logics. We will use the notation of the fundamental textbook Henkin Monk Tarski [8] (Parts I, II) of algebraic logic, but we will try to keep the present note self-contained. In a way, this work is a continuation of [23] but we do not assume familiarity with that work. For a class K of algebras we understand the amalgamation property (AP), embedding property (EP), and surjectiveness of epimorphisms (ES) in the standard algebraic sense, see e.g. Kiss at. al. [12], Bergman-McKenzie [3], and Pigozzi [24]. In particular, EP is defined to hold in a class K if for all A,B ∈ K having a common subalgebra D, both A and B are embeddable into a single C ∈ K. (Note that this might yield two different images of D.) Let A,B ∈ K and f : A → B a homomorphism. Now, f is an epimorphism (epi for short), of K iff for all C ∈ K and all h, g ∈ Hom(B,C) we have (h◦f = g◦f ⇒ h = g). In other words, f is an epi iff it is right cancellative. Now, K has ES iff every epimorphism f as above is onto B.