  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.
