Coherent Conditions: Algebraic Geometry for Arbitrary Classes of Algebras

TL;DR

This paper extends universal algebraic geometry to arbitrary classes of algebras, exploring spectra, topologies, and properties like Noetherianity, connecting algebraic and geometric perspectives.

Contribution

It introduces a generalized framework for algebraic geometry over classes of algebras, including spectra and topological structures, with new results on Noetherian classes and irreducibility.

Findings

Established a correspondence between quantifier-free propositions and closed sets in the Zariski topology.

Explored properties of spectra and their connection to current universal algebraic geometry.

Analyzed conditions for classes of algebras to be equationally Noetherian.

Abstract

Universal algebraic geometry is generalised from solutions of equations in a single algebra to the study of $\varphi$- or $K$-spectra, akin to the prime spectrum of a ring. We explore their basic properties and constructions, give a correspondence between certain quantifier-free propositions and closed sets in the Zariski topology of a free algebra, and show the connection with current UAG. Lastly, equationally Noetherian classes and irreducible spectra are explored.