Classification of Boolean Algebras through von Neumann regular ${\mathcal{C}}^{\infty }-$Rings

Abstract

In this paper, we introduce the concept of a "von Neumann regular ${\mathcal{C}}^{\infty }$-ring", which is a model for a specific equational theory. We delve into the characteristics of these rings and demonstrate that each Boolean space can be effectively represented as the image of a von Neumann regular ${\mathcal{C}}^{\infty }$-ring through a specific functor. Additionally, we establish that every homomorphism between Boolean algebras can be expressed through a ${\mathcal{C}}^{\infty }$-ring homomorphism between von Neumann regular ${\mathcal{C}}^{\infty }$-rings.

arXiv admin note: substantial text overlap with arXiv:1905.09617