Real structures. An introduction to a general approach

Abstract

In this paper we attempt to present a very general approach to the study of structures (somehow) defined on a set $X$ by a family of maps $d: X \times X \mapsto \mathbb{R}^+$. It will be shown how the assignment of a preorder $\prec_{\Pi}$ on a set $\Pi$ of families of maps from $X \times X$ into $\mathbb{R}^+$ defines a structure on $X$. The structures obtained in this way will be called \emph{real structures}. For real structures on two different sets we study when they are \emph{of the same type}. An answer to this question will allow to introduce the notion of \emph{morphism}, and then to give the definition of the initial real structure with respect to a family of maps, and so also the definition of product real structure. Various examples of preorders, and hence of real structures, will be exhibited and discussed. A few examples of morphisms will be proposed.