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

In this paper, we introduce the concept of a “von Neumann regular ${\mathcal{C}}^{\mathrm{\infty }}$-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 ${\mathcal{C}}^{\mathrm{\infty }}$-ring through a specific functor. Additionally, we establish that every homomorphism between Boolean algebras can be expressed through a ${\mathcal{C}}^{\mathrm{\infty }}$-ring homomorphism between von Neumann regular ${\mathcal{C}}^{\mathrm{\infty }}$-rings.