A unified approach to the theory of normed structures - Part I: The single-sorted case

Abstract

We introduce the concept of a prenormed model of a particular kind of finitary single-sorted first-order theories, interpreted over a category with finite products. These are referred to as prealgebraic theories, for the fact that their signature comprises, together with arbitrary function symbols (of finite ariety), only relation symbols whose interpretation, in any possible model, is a reflexive and transitive binary relation, namely a preorder. The result is an abstract approach to the very concept of norm and, consequently, to the theory of normed structures.