Quadrics via Semigroups
V. N. Krishnachandran

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 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 , the cone emerges as the set of nilpotent elements and the hyperbolic paraboloid as the set of semigroup-theoretic inverses of a singular element.
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Taxonomy
TopicsRings, Modules, and Algebras · semigroups and automata theory · Coding theory and cryptography
