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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., Comment: arXiv admin note: substantial text overlap with arXiv:1905.09617

In this paper we state and prove ad hoc "Separation Theorems" of the so-called Smooth Commutative Algebra, the Commutative Algebra of \(\mathcal{C}^{\infty}-\)rings. These results are formally similar to the ones we find in (ordinary) Commutative Algebra. However, their proof is not so straightforward, since it depends on the introduction of the concept of "smooth saturation". As an application of these theorems we present an interesting result that sheds light on the connections between the smooth Zariski spectrum and the real smooth spectrum of a \(\mathcal{C}^{\infty}-\)ring., Comment: 42 pages. arXiv admin note: substantial text overlap with arXiv:1904.02725

In the present work we carry on the study of the order theory for ($\mathcal{C}^{\infty}-$-reduced) $\mathcal{C}^{\infty}-$-rings initiated in \cite{rings1} (see also \cite{BM2}). In particular, we apply some results of the order theory of $\mathcal{C}^{\infty}-$-fields (e.g., every such field is real closed) to present another approach to the order theory of general $\mathcal{C}^{\infty}-$-rings: "smooth real spectra" (see \cite{separation}). This suggests that a model-theoretic investigation of the class of $\mathcal{C}^{\infty}-$-fields could be interesting and also useful to provide the first steps towards the development of the "Real Algebraic Geometry" of $\mathcal{C}^{\infty}-$-rings., Comment: 32 pages

In this paper we present the notion of a von Neumann regular $\mathcal{C}^{\infty}-$ring, we prove some results about them and we describe some of their properties. We prove, using two different methods, that the category of von Neumann regular $\mathcal{C}^{\infty}-$rings is a reflective subcategory of $\mathcal{C}^{\infty}{\rm \bf Rng}$. We prove that every homomorphism between Boolean algebras can be represented by a $\mathcal{C}^{\infty}-$ring homomorphism between von Neumann regular $\mathcal{C}^{\infty}-$rings., Comment: 72 pages

In this paper we present some basic results of the Universal Algebra of $\mathcal{C}^\infty$-rings which were nowhere to be found in the current literature. The outstanding book of I. Moerdijk and G. Reyes,[24], presents the basic (and advanced) facts about $\mathcal{C}^\infty$-rings, however such a presentation has no universal algebraic "flavour". We have been inspired to describe $\mathcal{C}^\infty$-rings through this viewpoint by D. Joyce in [15]. Our main goal here is to provide a comprehensive material with detailed proofs of many known "taken for granted" results and constructions used in the literature about $\mathcal{C}^\infty$-rings and their applications - such proofs either could not be found or were merely sketched. We present, in detail, the main constructions one can perform within this category, such as limits, products, homomorphic images, quotients, directed colimits, free objects and others, providing a "propaedeutic exposition" for the reader's benefit.

We present, in the same vein as in [20] and [21], some results of the so-called "Smooth (or $\mathcal{C}^\infty$) Commutative Algebra", a version of Commutative Algebra of $\mathcal{C}^{\infty}-$rings instead of ordinary commutative unital rings, looking for similar results to those one finds in the latter, and expanding some others presented in [20]. We give an explicit description of an adjunction between the categories $\mathcal{C}^\infty{\rm Rng}$ and ${\rm CRing}$, in order to study this "bridge". We present and prove many properties of the analog of the radical of an ideal of a ring (namely, the $\mathcal{C}^\infty$-radical of an ideal), saturation (which we define as "smooth saturation", inspired by [13]), rings of fractions ($\mathcal{C}^\infty$-rings of fractions, defined first by I. Moerdijk and G. Reyes in [20]), local rings (local $\mathcal{C}^\infty$-rings), reduced rings ($\mathcal{C}^\infty$-reduced $\mathcal{C}^\infty$-rings) and others. We also state and prove new results, such as ad hoc "Separation Theorems", similar to the ones we find in Commutative Algebra, and a stronger version (Theorem 6) of the Theorem 1.4 of [20], characterizing every $\mathcal{C}^\infty$-ring of fractions. We describe the fundamental concepts of Order Theory for $\mathcal{C}^\infty$-rings, proving that every $\mathcal{C}^\infty$-ring is semi-real, and we prove an important result on the strong interplay between the smooth Zariski spectrum and the real smooth spectrum of a $\mathcal{C}^\infty$-ring.

In this paper we present classifying toposes for the following theories: the theory of $\mathcal{C}^{\infty}-$rings, the theory of local $\mathcal{C}^{\infty}-$rings and the theory of von Neumann regular $\mathcal{C}^{\infty}-$rings. The classifying toposes for the first two theories were stated without proof by Ieke Moerdijk and Gonzalo Reyes on the page 366 of \cite{MSIA}, where they assert that the topos ${\rm \bf Set}^{\mathcal{C}^{\infty}{\rm \bf Rng}_{\rm fp}}$ classifies the theory of $\mathcal{C}^{\infty}-$rings and that the smooth Zariski topos classifies the theory of local $\mathcal{C}^{\infty}-$rings.