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

TL;DR

This paper introduces a unified, abstract framework for understanding normed structures through prenormed models of prealgebraic theories, focusing on single-sorted first-order theories interpreted over categories with finite products.

Contribution

It develops a novel, categorical approach to normed structures by defining prenormed models of prealgebraic theories, expanding the theoretical foundation of normed spaces.

Findings

Defines prenormed models for prealgebraic theories

Provides an abstract categorical framework for normed structures

Lays groundwork for further generalizations in normed theory

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.