Coherent Conditions: Algebraic Geometry for Arbitrary Classes of Algebras
K. R. van Nispen
TL;DR
This work generalizes universal algebraic geometry to arbitrary coherent classes of algebras by introducing coherent conditions $\varphi=\Phi_V(K)$ and their $\varphi$-spectra. It develops a Zariski-type topology on spectra, via $V_\varphi(S)$ and the antitone Galois connection between congruences and closed sets, and ties these geometric objects to equational logic through a generalized Nullstellensatz. A central theme is the role of radical congruences: $\operatorname{rad}^\varphi$, nilradical, and the radical $\sqrt{\varphi}$ yield a reduction theory and connect to $ISP(K)$ when $\varphi$ is radical. The paper also establishes a Noetherian framework for equationally Noetherian classes, generalizes Unification Theorems, and analyzes irreducible closed sets and spectra of $\varphi$-reduced algebras, culminating in a robust structure for studying algebraic geometry over broad algebraic classes. These results extend classical UAG by replacing prime/coordinate-geometry with $K$-spectra and coherent-condition-driven closures, enabling systematic analysis across diverse algebraic varieties.
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.