Algebras with the same (algebraic) geometry

TL;DR

This paper explores the extension of classical algebraic geometry concepts to arbitrary varieties of algebras, focusing on when different algebras share the same geometric properties and relations.

Contribution

It introduces a framework for algebraic geometry in arbitrary algebraic varieties and investigates conditions under which different algebras exhibit identical geometric properties.

Findings

Examples of geometric properties of algebras in various varieties

Conditions for different algebras to have the same geometry

Analysis of geometric relations between different algebras

Abstract

Some basic notions of classical algebraic geometry can be defined in arbitrary varieties of algebras $\Theta.$ For every algebra $H$ in $\Theta$ one can consider algebraic geometry in $\Theta$ over $ H.$ Correspondingly, algebras in $\Theta$ are considered with the emphasis on equations and geometry. We give examples of geometric properties of algebras in $\Theta$ and of geometric relations between them. The main problem considered in the paper is when different $H_1$ and $H_2$ have the same geometry.