Ideal Databases

TL;DR

This paper explores the algebraic and geometric foundations of databases, connecting finite varieties, polynomial ideals, and lattice structures, and extends the framework to linear algebra and quantum theory.

Contribution

It introduces a novel algebraic framework for databases using algebraic geometry, proving Heath's theorem and linking relational algebra to lattice and quantum structures.

Findings

Established that algebra of ideals forms a relational lattice.

Provided an analytic proof of Heath's theorem.

Connected database theory with linear algebra and quantum mechanics.

Abstract

From algebraic geometry perspective database relations are succinctly defined as Finite Varieties. After establishing basic framework, we give analytic proof of Heath theorem from Database Dependency theory. Next, we leverage Algebra/Geometry dictionary and focus on algebraic counterparts of finite varieties, polynomial ideals. It is well known that intersection and sum of ideals are lattice operations. We generalize this fact to ideals from different rings, therefore establishing that algebra of ideals is Relational Lattice. The final stop is casting the framework into Linear Algebra, and traversing to Quantum Theory.