Algebraic geometry over Boolean algebras in the language with constants

TL;DR

This paper investigates algebraic properties of boolean algebras with constants, providing criteria for various algebraic conditions and solving the problem of geometric equivalence within this class.

Contribution

It introduces criteria for when boolean algebras with constants are equationally Noetherian, weakly equationally Noetherian, and other compactness properties, and addresses geometric equivalence.

Findings

Criteria for equationally Noetherian boolean algebras

Criteria for weakly equationally Noetherian boolean algebras

Solution to the geometric equivalence problem

Abstract

We study equations over boolean algebras with distinguished elements. We prove the criteria, when a boolean algebra is equationally Noetherian, weakly equationally Noetherian, $\mathbf{q}_\omega$-compact or $\mathbf{u}_\omega$-compact. Also we solve the problem of geometric equivalence in the class of boolean algebras with distinguished elements.