Rigid and Non-Rigid Mathematical Theories: the Ring $\mathbb{Z}$ Is Nearly Rigid

TL;DR

This paper classifies mathematical theories into rigid and non-rigid, demonstrating that the ring structure on integers is nearly rigid, which refines the understanding of algebraic structures' rigidity.

Contribution

It introduces the concept of nearly rigidity in mathematical theories and applies it to the ring structure on integers, providing new insights into their foundational properties.

Findings

Ring theory is non-rigid but nearly rigid.

The integers' ring structure exhibits near rigidity.

Classification of theories enhances understanding of mathematical foundations.

Abstract

Mathematical theories are classified in two distinct classes : {\it rigid}, and on the other hand, {\it non-rigid} ones. Rigid theories, like group theory, topology, category theory, etc., have a basic concept - given for instance by a set of axioms - from which all the other concepts are defined in a unique way. Non-rigid theories, like ring theory, certain general enough pseudo-topologies, etc., have a number of their concepts defined in a more free or relatively independent manner of one another, namely, with {\it compatibility} conditions between them only. As an example, it is shown that the usual ring structure on the integers $\mathbb{Z}$ is not rigid, however, it is nearly rigid.