  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.
