If \(G\) is a real semigroup (RS), write \(G^\times = \{ x \in G : x^2 = 1\}\) for the group of units in \(G\) and
\(Id(G) = \{ e \in G : e^2 = e \}\) for the distributive lattice of idempotents in \(G\)

Our purpose here is fourfold: firstly to give, employing the languages of special groups (SG) and real semigroups, new, oftentimes conceptually different and clearer, proofs of the characterization of RSs whose space of characters is Boolean in the natural Harrison (or spectral) topology, originally appearing in section 7.6 and 8.9 of Marshall (1996), therein treated as zero-dimensional abstract real spectra and here called named Boolean Real Semigroups. Secondly, to give a natural {\em Horn-geometric} axiomatization of Boolean RSs (in the language of RSs) and establish the closure of this class by certain important constructions: Boolean powers, arbitrary filtered colimits, products, reduced products and RS-sums and by surjective RS-morphisms (in particular, quotients). Thirdly, to characterize morphisms between Boolean RSs: if \(G\), \(H\) are Boolean RSs, there is a natural bijective correspondence between \(Mor_{RS}(G, H)\) and the set of pair of morphisms, \(\langle f, h \rangle\), where \(f\) is an RSG-morphism from \(G^\times\) to \(H^\times\) and \(h\) is a lattice morphism from \(Id(G)\) to \(Id(H)\), satisfying a certain compatibility condition. Fourthly, to give a characterization of quotients of Boolean RSs. Hence, the present work considerably extends the one by which it was motivated, namely the references in Marshall (1996) mentioned above.

The aim of this talk is to present, firstly, a number of results on the relationship between the ring \(A = F[[G]]\) of formal power series with coefficients in a formally real (i.e., orderable) field \(F\) and exponents in the positive cone of a totally ordered abelian group \(G\), and the real semigroup \(G_{\!A}\) associated to the ring \(A\). One of the main results shows that the real semigroup \(G_{\!A}\) is a fan in the category of real semigroups if and only if the preorder \(\Sigma F^{2}\) of \(F\) is a fan in the sense of fields. On the other hand, in the general case (i.e., for an arbitrary formally real field of coefficients), the real semigroup \(G_{\!A}\) satisfies certain fundamental conditions formulated in terms of the specialization partial order \(\leadsto\) defined in the real spectrum \({\mathrm{Sper}}(A)\) of the ring \(A\). These properties led us to introduce a new class of real semigroups which we baptized symmetric real semigroups.

Next, we investigate the theory of symmetric real semigroups and prove several results on their structure, leading to the following: I) Every finite symmetric real semigroup is realizable by a ring of formal power series.
II) There exists an infinite symmetric real semigroup (in fact, a fan) which is not realizable by any ring of formal power series.

The theory of Real Semigroups (RS) was created by M. Dickmann and A. Petrovich in the 2000's as a first order theory of real spectra of rings. It extends the concept of Reduced Special Group by using representation (or transversal representation) of dimension 2 form as primitive concept.

In this talk we will build the von Neumann Hull of a RS and describe its mains algebraic and categorical properties. As an application, we will prove a version of Marshall conjecture for real semigroups associated with (semi-real) rings.

The representation problem for special groups, asking whether every reduced special group is isomorphic to the special group of a field, is an outstanding open problem in the realm of quadratic forms. The same is true for the corresponding problem about real semigroups, as well as Efrat's question which \(\kappa\)-structures arise as the Milnor \(K\)-theory of a field, are outstanding open problems in the realm of quadratic forms. All of these problems allow variants where one replaces isomorphism by elementary equivalence, e.g. by the question whether every reduced special group is elementarily equivalent to one coming from a field.

Motivated by these problems, we study the general question when the essential image of a functor can be axiomatized by \(\kappa\)-geometric sequents -a certain fragment of formulas of infinitary first order logic.

We observe that one can study this question via topos theory: Under mild hypotheses, functors between accessible categories (such as categories of models of first order theories) can be assumed to be induced by a \(\kappa\)-geometric morphism between classifying \(\kappa\)-toposes, notions first stdied by Espíndola. We show how this morphism can be factorized into a surjection, followed by a dense inclusion, followed by a closed inclusion, and explain what that means in terms of the involved theories. We arrive at very concrete axiomatizability criteria using this factorization.

All results and involved notions will be explained and backed up with examples.

The relationship between Galois groups of fields with orderings and quadratic forms, established by the works of Artin-Schreier (1920's) and Witt (late 1930's) are reinforced by a seminal paper of John Milnor (1971) through the definition of a (mod 2) k-theory graded ring that "interpolates" the graded Witt ring and the cohomology ring of fields: the three graded rings constructions determine functors from the category of fields where 2 is invertible that, almost tree decades later, are proved to be naturally isomorphic by the work of Voevodsky with co-authors.

Since the 1980's, have appeared many abstract approaches to the algebraic theory of quadratic forms over fields that are essentially equivalent (or dually equivalent): between them we emphasize the (first-order) theory of special groups developed by Dickmann-Miraglia. The notions of (graded) Witt rings and k-theory are extended to the category of Special groups with remarkable pay-offs on questions on quadratic forms over fields.

In this talk we extended to (well-behaved) Special Groups the work of J. Minác and Spira that describes a (pro-2)-group of a field extension that encodes the quadratic form theory of a given field \(F\): Adem, Karagueuzian, J. Minác (1999) it is shown that its associated cohomology ring is contains a copy of the cohomology ring of the field \(F\). Our construction, a contravariant functor \(G \in SG \ \mapsto\ Gal(G)\in Pro-2-groups\), encodes the space of orders of the special group \(G\) and provides a criteria to detect when \(G\) is formally real or not. This motivate us to consider tree categories which are endowed with a underlying functor into the category of "pointed" groups of exponent 2: the category of pre-special groups, a category formed by certain pointed graded rings and a category given by some pairs of profinite 2-groups and a clopen subgroup of index at most 2 and with arrows the continuous homomorphisms compatible with this additional data. We establish precise (and canonical) functorial relationship between them and explore some of its model-theoretical aspects.

The uses of K-theoretic (and Boolean) methods in abstract theories of quadratic forms has been proved a very successful method, see for instance, these two papers of M. Dickmann and F. Miraglia: (1998) where they give an affirmative answer to Marshall's Conjecture, and (2003), where they give an affirmative answer to Lam's Conjecture.

These two central papers makes us take a deeper look at the theory of Special Groups by itself. This is not mere exercise in abstraction: from Marshall's and Lam's Conjecture many questions arise in the abstract and concrete context of quadratic forms.

There are some generalizations of Milnor's K-theory. In the quadratic forms context, the most significant one is the Dickmann-Miraglia's K-theory of Special Groups. It is a main tool in the proof of Marshall's and Lam's Conjecture.

In Marshall's paper (2006), he propose a new abstract theory of quadratic forms based on what he called a ``real reduced multiring'' and ``real reduced hyperfield''. This new theory has the advantage of brings new analogies with commutative algebra, and we developed and expand the details in Roberto, Ribeiro, Mariano (2020). It is even possible rewrite the axioms of special groups in a sort of "geometric manner" via hyperfields (Roberto, Ribeiro, Mariano (2021)).

Now, we will give another step in the ``marriage of multi structures and quadratic forms'' developing an appropriate K-theory for hyperfields. This new category generalizes simutaneously both Milnor's reduced and non-reduced K-theories and Dickmann-Miraglia's K-theory for special groups.

With these three K-theories on hands, it is desirable (or, at least, suggestive) the rise of an abstract enviroment that encapsule all them, and of course, provide an axiomatic approach to guide new extensions of the concept of K-theory in the context of the algebraic and abstract theories of quadratic forms. The inductive graded rings, introduced by M. Dickmann and F. Miraglia (2000) fits this purpose, and we finish this work showing that the K-theory of pre-special hyperfields is some kind of free inductive graded ring.