Universal geometrical equivalence of the algebraic structures of common signature

This article is a part of our effort to explain the foundations of algebraic geometry over arbitrary algebraic structures [1–8]. We introduce the concept of universal geometrical equivalence of two algebraic structures $\mathscr A$ and $\mathscr B$ of a common language {\tt L} which strengthens the available concept of geometrical equivalence and expresses the maximal affinity between $\mathscr A$ and $\mathscr B$ from the viewpoint of their algebraic geometries. We establish a connection between universal geometrical equivalence and universal equivalence in the sense of equality of universal theories.