Quadrics via Semigroups

TL;DR

This paper explores classical quadric surfaces through the lens of semigroup theory, revealing geometric structures like hyperboloids and cones as algebraic sets of elements in the semigroup of 2x2 real matrices.

Contribution

It connects classical three-dimensional geometry of quadrics with algebraic structures in semigroups, offering a novel algebraic perspective on geometric surfaces.

Findings

Hyperboloid of one sheet as set of idempotents

Cone as set of nilpotent elements

Hyperbolic paraboloid as inverses of singular elements

Abstract

This is the story of the rediscovery of classical three-dimensional geometry, especially the geometry of quadric surfaces, while studying the semigroup $M_2(\mathbb R)$ of linear endomorphisms of a real plane. One of the surfaces that appears prominently in this context is the hyperboloid of one sheet, referred to as {\em spaghetti bundle} in \cite{Samu:88}. In this story the spaghetti presents itself as the set of idempotents in $M_2(\mathbb R)$, the cone emerges as the set of nilpotent elements and the hyperbolic paraboloid as the set of semigroup-theoretic inverses of a singular element.