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

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.