# K-theories and Free Inductive Graded Rings in Abstract Quadratic Forms
Theories

Kaique Matias de Andrade Roberto111Instituto de Matemática e Estatística,
Universidade de São Paulo, Brazil.
Hugo Luiz Mariano222Emails<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We build on previous work on multirings ([3]) that provides generalizations of
the available abstract quadratic forms theories (special groups and real
semigroups) to the context of multirings ([17], [7]). Here we raise one step
in this generalization, introducing the concept of pre-special hyperfields and
expand a fundamental tool in quadratic forms theory to the more general
multivalued setting: the K-theory. We introduce and develop the K-theory of
hyperbolic hyperfields that generalize simultaneously Milnor’s K-theory ([18])
and Special Groups K-theory, developed by Dickmann-Miraglia ([12]). We develop
some properties of this generalized K-theory, that can be seen as a free
inductive graded ring, a concept introduced in [9] in order to provide a
solution of Marshall’s Signature Conjecture.

Keywords: quadratic forms, special groups, K-theory, multirings, hyperfields.

##  1 Introduction

Concerning Abstract Theories of Quadratic forms (in particular special groups
and real semigroups), the references [10], [13] and [14] are central. The
theory of special groups deals simultaneously reduced and non-reduced theories
but focuses on rings with an “expressive amount” of invertible coefficients to
quadratic forms and the theory of real semigroups consider general
coefficients of a ring, but only addresses the reduced case. Both are first-
order theory, thus they allow the use of model theoretic methods.

M. Marshall in [17] introduced an approach to (reduced) theory of quadratic
forms through the concept of multiring (roughly, a ring with a multi valued
sum) 333The main terminology in the literature is ”hyperring”. Moreover, M.
Marshall makes a distinction between ”multiring” and ”hyperrings” which is
important in the context of quadratic forms. But throughout this entire work,
we deal essentially with multifields/hyperfields and then, the main
terminology here will be ”hyperfield”.: this seems more intuitive for an
algebraist since it encompasses (generalizes, in fact) some techniques of
ordinary Commutative Algebra. Moreover, the multirings encode copies of
special groups and real semigroups (see [7]) and still allows the use of
model-theoretic tools, since multirings (hyperrings) endowed with convenient
notion of morphisms constitutes a category that is isomorphic to a category of
appropriate first-order structures.

In the recent work [3]: (i) we have considered interesting pairs $(A,T)$ where
$A$ is a multiring and $T\subseteq A$ is a certain multiplicative subset in
such a way to obtain models of abstract theories of quadratic forms (special
groups and real semigroups) via natural quotients - Marshall’s quotient
construction and (ii) we have used this new setting to motivate a "non
reduced" expansion of the theory of real semigroups to deal the formally real
case, isolating axioms over pairs involving multirings and a multiplicative
subset with some properties.

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 Dickmann and Miraglia: [9] where they give an affirmative
answer to Marshall’s Conjecture, and [11], where they give an affirmative
answer to Lam’s Conjecture (previously both conjecture have kept open for
almost three decades). These two central papers makes us take a deeper look at
the theory of special groups (and hence, hyperbolic/pre-special hyperfields)
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.

In the present paper we provide some new steps towards the development of
tools of algebraic theory of quadratic forms in this multiring setting: we
have defined and explored K-theory and graded rings in the context of
hyperfields that, in particular, provides a generalization and unification of
Milnor’s K-theory ([18]) and special groups K-theory ([12]). We develop some
properties of this generalized K-theory, that can be seen as a free inductive
graded ring. This is a preprint version of [4].

Outline: In Section 2 we: recall the basic notions of K-theory in quadratic
forms theories (Milnor’s K-theory, Special Groups K-theory); provide the basic
definitions, constructions and results on multirings and hyperfields;
introduce the concept of pre-special hyperfield. In the third section we
introduce the notion of K-theory of hyperfields and establish its main
properties and some technical results. In Section 4 we recognize the K-theory
of special hyperfields as a free construction in the category of inductive
graded rings ([10]) and prove that this concept, encompasses both Milnor’s
K-theory and Special Groups K-theory. We finish the work indicating some
themes of research motivated by the present paper.

Throughout this entire paper, we adopt the following convention: let $A$ be an
arbitrary structure (field, graded ring and so on) and let $T\subseteq A$ be a
subset such that the quotient $A/T$ is defined. We usually denote the elements
of $A/T$ by $\overline{a}\in A/T$. This notation is overwhelming here (as you
will see rapidly). So, we will omit the overline symbol, and an equation
“$\overline{a}=\overline{b}$” that took place in $A/T$ will be denoted simply
by “$a=b$ in $A/T$”. In other words, we will add “in $A/T$” or “in $A$” to
make distinctions between some set and its quotients.

##  2 Preliminaries

This section contains, basically, the fundamental definitions and results
included for the convenience of the reader such as multirings, hyperfields,
special groups, Minor’s K-theory of fields and Dickmann-Miraglia K-theory of
(pre) special groups. For more details, consult [18], [10], [12], [17], [7] or
[3].

###  2.1 Milnor’s K-theory

Here we get some definitions and results about Milnor’s K-theory, as developed
in [18].

######  Definition 2.1 (The Milnor’s K-theory of a Field [18]).

For a field $F$ (of characteristic not 2), $K_{*}F$ is the graded ring

$K_{*}F=(K_{0}F,K_{1}F,K_{2}F,...)$

defined by the following rules: $K_{0}F:=\mathbb{Z}$. $K_{1}F$ is the
multiplicative group $\dot{F}$ written additively. With this purpose, we fix
the canonical “logarithm” isomorphism

$l:\dot{F}\rightarrow K_{1}F,$

where $l(ab)=l(a)+l(b)$. Then $K_{n}F$ is defined to be the quotient of the
tensor algebra

$K_{1}F\otimes K_{1}F\otimes...\otimes K_{1}F\,(n\mbox{ times})$

by the (homogeneous) ideal generated by all $l(a)\otimes l(1-a)$, with $a\neq
0,1$. We also have the reduced K-theory graded ring
$k_{*}F=(k_{0}F,k_{1}F,...,k_{n}F,...),$ which is defined by the rule
$k_{n}F:=K_{n}F/2K_{n}F$ for all $n\geq 0$.

######  Theorem 2.2 (Theorem 4.1 of [18]).

There is only one morphism

$s_{n}:k_{n}F\rightarrow I^{n}F/I^{n+1}F$

which carries each product $l(a_{1})...l(a_{n})$ in $K_{n}F/2K_{n}F$ to the
product $(\langle a_{1}\rangle-\langle 1\rangle)...(\langle
a_{n}\rangle-\langle 1\rangle)$ modulo $I^{n+1}F$.

These morphisms determines an epimorphism $s_{*}:k_{*}F\rightarrow W_{g}(F)$,
where

$W_{g}(F)=(WF/IF,IF/I^{2}F,...,I^{n}F/I^{n+1}F,...).$

For a field $F$, let $F_{s}$ be the a separable closure of $F$ and
$G_{F}=\mbox{Gal}(F_{s})$. Then, the exact sequence

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\\{\pm
1\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\dot{F}_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{2}$$\textstyle{\dot{F}_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$

is taken to the following exact sequence

$\textstyle{H^{0}(G_{F},\dot{F}_{s})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{2}$$\textstyle{H^{0}(G_{F},\dot{F}_{s})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{H^{1}(G_{F},\\{\pm
1\\})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(G_{F},\dot{F}_{s})}$

of cohomology groups. Identifying the two first groups with $\dot{F}$, and
$\\{\pm 1\\}$ with $\mathbb{Z}/2\mathbb{Z}$ and applying Hilbert’s 90, we have

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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The quotient $\dot{F}/\dot{F}^{2}$ is identified with $k_{1}F$.

######  Theorem 2.3 (Lemma 6.1 of [18]).

The isomorfism $l(a)\mapsto\delta(a)$ from $K_{1}F/2K_{1}F$ to
$H^{1}(G_{F},\mathbb{Z}/2\mathbb{Z})$ admits a unique extension to a graded
ring morphism

$h_{f}:k_{*}F\rightarrow H^{*}(G_{F},\mathbb{Z}/2\mathbb{Z}).$

The Milnor Conjecture consists of the assertion that $s$ and $h$ are graded
rings isomorfisms, which makes the fuctors
$K_{*}F/2K_{*}F,W_{g}(F),H^{*}(G,\mathbb{Z}/2\mathbb{Z})$ isomorphic.

###  2.2 Dickmann-Miraglia K-theory for Special Groups

There are some generalizations of Milnor’s K-theory. In the quadratic forms
context, maybe the most significant one is the Dickmann-Miraglia K-theory of
Special Groups. It is a main tool in the proof of Marshall’s and Lam’s
Conjecture. In this section, we get some definitions and results from [10] and
[12].

Firstly, we make a brief summary on special groups. Let $A$ be a set and
$\equiv$ a binary relation on $A\times A$. We extend $\equiv$ to a binary
relation $\equiv_{n}$ on $A^{n}$, by induction on $n\geq 1$, as follows:

  1. i -

$\equiv_{1}$ is the diagonal relation $\Delta_{A}\subseteq A\times A$.

  2. ii -

$\equiv_{2}=\equiv$.

  3. iii -

If $n\geq 3$, $\langle a_{1},...,a_{n}\rangle\equiv_{n}\langle
b_{1},...,b_{n}\rangle$ if and only there are $x,y,z_{3},...,z_{n}\in A$ such
that

$\langle a_{1},x\rangle\equiv\langle b_{1},y\rangle,\,\langle
a_{2},...,a_{n}\rangle\equiv_{n-1}\langle x,z_{3},...,z_{n}\rangle\mbox{ and
}\langle b_{2},...,b_{n}\rangle\equiv_{n-1}\langle y,z_{3},...,z_{n}\rangle.$

Whenever clear from the context, we frequently abuse notation and indicate the
afore-described extension $\equiv$ by the same symbol.

######  Definition 2.4 (Special Group, 1.2 of [10]).

A special group is a tuple $(G,-1,\equiv)$, where $G$ is a group of exponent
2, i.e, $g^{2}=1$ for all $g\in G$; $-1$ is a distinguished element of $G$,
and $\equiv\subseteq G\times G\times G\times G$ is a relation (the special
relation), satisfying the following axioms for all $a,b,c,d,x\in G$:

SG 0

$\equiv$ is an equivalence relation on $G^{2}$;

SG 1

$\langle a,b\rangle\equiv\langle b,a\rangle$;

SG 2

$\langle a,-a\rangle\equiv\langle 1,-1\rangle$;

SG 3

$\langle a,b\rangle\equiv\langle c,d\rangle\Rightarrow ab=cd$;

SG 4

$\langle a,b\rangle\equiv\langle c,d\rangle\Rightarrow\langle
a,-c\rangle\equiv\langle-b,d\rangle$;

SG 5

$\langle a,b\rangle\equiv\langle c,d\rangle\Rightarrow\langle
ga,gb\rangle\equiv\langle gc,gd\rangle,\,\mbox{for all }g\in G$.

SG 6 (3-transitivity)

the extension of $\equiv$ for a binary relation on $G^{3}$ is a transitive
relation.

A group of exponent 2, with a distinguished element $-1$, satisfying the
axioms SG0-SG3 and SG5 is called a proto special group; a pre special group is
a proto special group that also satisfy SG4. Thus a special group is a pre-
special group that satisfies SG6 (or, equivalently, for each $n\geq 1$,
$\equiv_{n}$ is an equivalence relation on $G^{n}$).

A $n$-form (or form of dimension $n\geq 1$) is an $n$-tuple of elements of a
pre-SG $G$. An element $b\in G$ is represented on $G$ by the form
$\varphi=\langle a_{1},...,a_{n}\rangle$, in symbols $b\in D_{G}(\varphi)$, if
there exists $b_{2},...,b_{n}\in G$ such that $\langle
b,b_{2},...,b_{n}\rangle\equiv\varphi$.

A pre-special group (or special group) $(G,-1,\equiv)$ is:
$\bullet$ formally real if $-1\notin\bigcup_{n\in\mathbb{N}}D_{G}(n\langle
1\rangle)$444Here the notation $n\langle 1\rangle$ means the form $\langle
a_{1},...,a_{n}\rangle$ where $a_{j}=1$ for all $j=1,...,n$. In other words,
$n\langle 1\rangle$ is the form $\langle 1,...,1\rangle$ with $n$ entries
equal to 1. ;
$\bullet$ reduced if it is formally real and, for each $a\in G$, $a\in
D_{G}(\langle 1,1\rangle)$ iff $a=1$.

######  Definition 2.5 (1.1 of [10]).

A map
$\textstyle{(G,\equiv_{G},-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{(H,\equiv_{H},-1)}$
between pre-special groups is a morphism of pre-special groups or PSG-morphism
if $f:G\rightarrow H$ is a homomorphism of groups, $f(-1)=-1$ and for all
$a,b,c,d\in G$

$\langle a,b\rangle\equiv_{G}\langle c,d\rangle\Rightarrow\langle
f(a),f(b)\rangle\equiv_{H}\langle f(c),f(d)\rangle$

A morphism of special groups or SG-morphism is a pSG-morphism between the
corresponding pre-special groups. $f$ will be an isomorphism if is bijective
and $f,f^{-1}$ are PSG-morphisms.

It can be verified that a special group $G$ is formally real iff it admits
some SG-morphism $f:G\to 2$. The category of special groups (respectively
reduced special groups) and their morphisms will be denoted by $\mathcal{SG}$
(respectively $\mathcal{RSG}$).

######  Definition 2.6 (The Dickmann-Miraglia K-theory [12]).

For each special group $G$ (written multiplicatively) we associate a graded
ring

$k_{*}G=(k_{0}G,k_{1}G,...,k_{n}G,...)$

as follows: $k_{0}G:=\mathbb{F}_{2}$ and $k_{1}G:=G$ written additively. With
this purpose, we fix the canonical “logarithm” isomorphism
$\lambda:G\rightarrow k_{1}G$, $\lambda(ab)=\lambda(a)+\lambda(b)$. Observe
that $\lambda(1)$ is the zero of $k_{1}G$ and $k_{1}G$ has exponent 2, i.e,
$\lambda(a)=-\lambda(a)$ for all $a\in G$. In the sequel, we define $k_{*}G$
by the quotient of the $\mathbb{F}_{2}$-graded algebra

$(\mathbb{F}_{2},k_{1}G,k_{1}G\otimes_{\mathbb{F}_{2}}k_{1}G,k_{1}G\otimes_{\mathbb{F}_{2}}k_{1}G\otimes_{\mathbb{F}_{2}}k_{1}G,...)$

by the (graded) ideal generated by $\\{\lambda(a)\otimes\lambda(ab),\,a\in
D_{G}(1,b)\\}$. In other words, for each $n\geq 2$,

$k_{n}G:=T^{n}(k_{1}G)/Q^{n}(G),$

where

$T^{n}(k_{1}G):=k_{1}G\otimes_{\mathbb{F}_{2}}k_{1}G\otimes_{\mathbb{F}_{2}}...\otimes_{\mathbb{F}_{2}}k_{1}G$

and $Q^{n}(G)$ is the subgroup generated by all expressions of type
$\lambda(a_{1})\otimes\lambda(a_{2})\otimes...\otimes\lambda(a_{n})$ such that
for some $i$ with $1\leq i<n$, there exist $b\in G$ such that $a_{i}\in
D_{G}(1,b)$ and $a_{i}=a_{i+1}b$, which in symbols, means

$\displaystyle Q^{n}(G)$
$\displaystyle:=\langle\\{\lambda(a_{1})\otimes\lambda(a_{2})\otimes...\otimes\lambda(a_{n}):\mbox{
exists }1\leq i<n\mbox{ and }\,b\in G$ $\displaystyle\mbox{such that
}a_{i}=a_{i+1}b\mbox{ and }a_{i}\in D_{G}(1,b)\\}\rangle.$

Since $\lambda(a)+\lambda(a)=0$ for all $a\in k_{1}G$, it follow that
$\eta+\eta=0$ for all $\eta\in k_{n}G$, so this is a group of exponent 2.

Moreover, we will denote
“$\lambda(a_{1})\otimes\lambda(a_{2})\otimes...\otimes\lambda(a_{n})$” simply
by “$\lambda(a_{1})\lambda(a_{2})...\lambda(a_{n})$”. Clearly, $k_{\ast}(G)$
is a graded ring, and in particular, a ring, so that we are able to multiply
elements $\eta\in k_{n}G$, $\tau\in k_{m}G$. Whenever we want to do this, we
will denote $\eta\cdot\tau$, in order to avoid confusion with the
simplifications described above.

Finally, since we only take tensorial products with parameters in
$\mathbb{F}_{2}$, we abbreviate “$A\otimes_{\mathbb{F}_{2}}B$” simply by
“$A\otimes B$”. In this way, $T^{n}(k_{1}G)$ we will be denoted simply by

$T^{n}(k_{1}G)=k_{1}G\otimes k_{1}G\otimes...\otimes k_{1}G.$

Next, we have a result that approximates Dickmann-Miraglia’s K-theory with the
Milnor’s reduced K-theory:

######  Proposition 2.7 (2.1 [12]).

Let $G$ be a special group, $x,y,a_{1},...,a_{n}\in G$ and $\sigma$ be a
permutation on $n$ elements.

  1. a -

In $k_{2}G$, $\lambda(a)^{2}=\lambda(a)\lambda(-1)$. Hence in $k_{m}G$,
$\lambda(a)^{m}=\lambda(a)\lambda(-1)^{m-1}$, $m\geq 2$;

  2. b -

In $k_{2}G$, $\lambda(a)\lambda(-a)=\lambda(a)^{2}=0$;

  3. c -

In $k_{n}G$, $\lambda(a_{1})\lambda(a_{2})...\lambda(a_{n})=\lambda(a_{\sigma
1})\lambda(a_{\sigma 2})...\lambda(a_{\sigma n})$;

  4. d -

For $n\geq 1$ and $\xi\in k_{n}G$, $\xi^{2}=\lambda(-1)^{n}\xi$;

  5. e -

If $G$ is a reduced special group, then $x\in D_{G}(1,y)$ and
$\lambda(y)\lambda(a_{1})...\lambda(a_{n})=0$ implies

$\lambda(x)\lambda(a_{1})\lambda(a_{2})...\lambda(a_{n})=0.$

######  Definition 2.8 (2.4 [12]).

  1. a -

A reduced special group is [MC] if for all $n\leq 1$ and all forms $\varphi$
over $G$,

$\mbox{For all }\sigma\in X_{G},\mbox{ if }\sigma(\varphi)\equiv 0\,\mbox{mod
}2^{n}\mbox{ then }\varphi\in I^{n}G.$

  2. b -

A reduced special group is [SMC] if for all $n\geq 1$, the multiplication by
$\lambda(-1)$ is an injection of $k_{n}G$ in $k_{n+1}G$.

######  Fact 2.9.

We summarize some properties of the Dickmann-Miraglia K-theory below:

  1. i -

An inductive system of special groups

$\mathcal{G}=(G_{i};\\{f_{ij}:i\leq j\in I\\}),$

provides an inductive system of graded ring, which nodes are $k_{*}G_{i}$ and
morphisms are

$(f_{ij})_{*}:k_{*}G_{i}\rightarrow k_{*}G_{j},\,\mbox{for }i\leq j\mbox{ in
}I.$

  2. ii -

(4.5 of [12]) Let $\mathcal{G}=(G_{i};\\{f_{ij}:i,j\in I,\,i\leq j\\})$ an
inductive system of special groups over a directed poset $I$ and
$(G;\\{f_{i}:i\in I\\})=\varinjlim\mathcal{G}$. Then

$k_{*}G\cong\varinjlim\limits_{i\in I}k_{*}G_{i}.$

  3. iii -

(4.6, 5.1, 5,7 and 6.8 of [12]) The inductive limit, finite products, SG-sum
and extension of SMC groups is a SMC group.

  4. iv -

(5.1 of [12]) Let $G_{1},...,G_{m}$ be special groups and
$\prod^{m}_{i=1}G_{i}$. Then there exists a graded isomorphism

$\gamma:k_{*}P\rightarrow\bigoplus^{m}_{i=1}k_{*}G_{i}.$

###  2.3 Multifields/Hyperfields

Roughly speaking, a multiring is a “ring” with a multivalued addition, a
notion introduced in the 1950s by Krasner’s works. The notion of multiring was
joined to the quadratic forms tools by the hands of M. Marshall in the last
decade ([17]). We gather the basic information about multirings/hyperfields
and expand some details that we use in the context of K-theories. For more
detailed calculations involving multirings/hyperfields and quadratic forms we
indicate to the reader the reference [7] (or even [15] and [3]). Of course,
multi-structures is an entire subject of research (which escapes from the
"quadratic context"), and in this sense, we indicate the references [19],
[20], [1].

######  Definition 2.10.

A multigroup is a quadruple $(G,\ast,r,1)$, where $G$ is a non-empty set,
$\ast:G\times G\rightarrow\mathcal{P}(G)\setminus\\{\emptyset\\}$ and
$r:G\rightarrow G$ are functions, and $1$ is an element of $G$ satisfying:

  1. i -

If $z\in x\ast y$ then $x\in z\ast r(y)$ and $y\in r(x)\ast z$.

  2. ii -

$y\in 1\ast x$ if and only if $x=y$.

  3. iii -

With the convention $x\ast(y\ast z)=\bigcup\limits_{w\in y\ast z}x\ast w$ and
$(x\ast y)\ast z=\bigcup\limits_{t\in x\ast y}t\ast z$,

$x\ast(y\ast z)=(x\ast y)\ast z\mbox{ for all }x,y,z\in G.$

A multigroup is said to be commutative if

  4. iv -

$x\ast y=y\ast x$ for all $x,y\in G$.

Observe that by (i) and (ii), $1\ast x=x\ast 1=\\{x\\}$ for all $x\in G$. When
$a\ast b=\\{x\\}$ be a unitary set, we just write $a\ast b=x$.

######  Definition 2.11 (Adapted from Definition 2.1 in [17]).

A multiring is a sextuple $(R,+,\cdot,-,0,1)$ where $R$ is a non-empty set,
$+:R\times R\rightarrow\mathcal{P}(R)\setminus\\{\emptyset\\}$, $\cdot:R\times
R\rightarrow R$ and $-:R\rightarrow R$ are functions, $0$ and $1$ are elements
of $R$ satisfying:

  1. i -

$(R,+,-,0)$ is a commutative multigroup;

  2. ii -

$(R,\cdot,1)$ is a commutative monoid;

  3. iii -

$a.0=0$ for all $a\in R$;

  4. iv -

If $c\in a+b$, then $c.d\in a.d+b.d$. Or equivalently, $(a+b).d\subseteq
a.d+b.d$.

Note that if $a\in R$, then $0=0.a\in(1+(-1)).a\subseteq 1.a+(-1).a$, thus
$(-1).a=-a$.

$R$ is said to be an hyperring if for $a,b,c\in R$, $a(b+c)=ab+ac$.

A multring (respectively, a hyperring) $R$ is said to be a multidomain
(hyperdomain) if it has no zero divisors. A multring $R$ will be a multifield
if every non-zero element of $R$ has multiplicative inverse; note that
hyperfields and multifields coincide. We will use "hyperfield" since this is
the prevailing terminology.

######  Definition 2.12.

Let $A$ and $B$ multirings. A map $f:A\rightarrow B$ is a morphism if for all
$a,b,c\in A$:

  1. i -

$c\in a+b\Rightarrow f(c)\in f(a)+f(b)$;

  2. ii -

$f(-a)=-f(a)$;

  3. iii -

$f(0)=0$;

  4. iv -

$f(ab)=f(a)f(b)$;

  5. v -

$f(1)=1$.

If $A$ and $B$ are multirings, a morphism $f\colon A\to B$ is a strong
morphism if for all $a,b,c\in A$, if $f(c)\in f(a)+f(b)$, then there are
$a^{\prime},b^{\prime},c^{\prime}\in A$ with
$f(a^{\prime})=f(a),f(b^{\prime})=f(b),f(c^{\prime})=f(c)$ such that
$c^{\prime}\in a^{\prime}+b^{\prime}$. In the quadratic context, there is a
more detailed analysis in Example 2.10 of [7].

######  Example 2.13.

  1. a -

Suppose that $(G,+,0)$ is an abelian group. Defining $a+b=\\{a+b\\}$ and
$r(g)=-g$, we have that $(G,+,r,0)$ is an abelian multigroup. In this way,
every ring, domain and field is a multiring, multidomain and hyperfield,
respectively.

  2. b -

$Q_{2}=\\{-1,0,1\\}$ is hyperfield with the usual product (in $\mathbb{Z}$)
and the multivalued sum defined by relations

$\begin{cases}0+x=x+0=x,\,\mbox{for every }x\in Q_{2}\\\
1+1=1,\,(-1)+(-1)=-1\\\ 1+(-1)=(-1)+1=\\{-1,0,1\\}\end{cases}$

  3. c -

Let $K=\\{0,1\\}$ with the usual product and the sum defined by relations
$x+0=0+x=x$, $x\in K$ and $1+1=\\{0,1\\}$. This is a hyperfield called
Krasner’s hyperfield [16].

Now, another example that generalizes $Q_{2}=\\{-1,0,1\\}$. Since this is a
new one, we will provide the entire verification that it is a multiring:

######  Example 2.14 (Kaleidoscope, Example 2.7 in [7]).

Let $n\in\mathbb{N}$ and define

$X_{n}=\\{-n,...,0,...,n\\}\subseteq\mathbb{Z}.$

We define the $n$-kaleidoscope multiring by $(X_{n},+,\cdot,-,0,1)$, where
$-:X_{n}\to X_{n}$ is restriction of the opposite map in $\mathbb{Z}$,
$+:X_{n}\times X_{n}\rightarrow\mathcal{P}(X_{n})\setminus\\{\emptyset\\}$ is
given by the rules:

$a+b=\begin{cases}\\{a\\},\,\mbox{ if }\,b\neq-a\mbox{ and }|b|\leq|a|\\\
\\{b\\},\,\mbox{ if }\,b\neq-a\mbox{ and }|a|\leq|b|\\\
\\{-a,...,0,...,a\\}\mbox{ if }b=-a\end{cases},$

and $\cdot:X_{n}\times X_{n}\rightarrow X_{n}$ is given by the rules:

$a\cdot b=\begin{cases}\mbox{sgn}(ab)\max\\{|a|,|b|\\}\mbox{ if }a,b\neq 0\\\
0\mbox{ if }a=0\mbox{ or }b=0\end{cases}.$

With the above rules we have that $(X_{n},+,\cdot,-,0,1)$ is a multiring.

Now, another example that generalizes $K=\\{0,1\\}$.

######  Example 2.15 (H-hyperfield, Example 2.8 in [7]).

Let $p\geq 1$ be a prime integer and
$H_{p}:=\\{0,1,...,p-1\\}\subseteq\mathbb{N}$. Now, define the binary
multioperation and operation in $H_{p}$ as follows:

$\displaystyle a+b$ $\displaystyle=\begin{cases}H_{p}\mbox{ if }a=b,\,a,b\neq
0\\\ \\{a,b\\}\mbox{ if }a\neq b,\,a,b\neq 0\\\ \\{a\\}\mbox{ if }b=0\\\
\\{b\\}\mbox{ if }a=0\end{cases}$ $\displaystyle a\cdot b$
$\displaystyle=k\mbox{ where }0\leq k<p\mbox{ and }k\equiv ab\mbox{ mod p}.$

$(H_{p},+,\cdot,-,0,1)$ is a hyperfield such that for all $a\in H_{p}$,
$-a=a$. In fact, these $H_{p}$ are a kind of generalization of $K$, in the
sense that $H_{2}=K$.

There are many natural constructions on the category of multrings as:
products, directed inductive limits, quotients by an ideal, localizations by
multiplicative subsets and quotients by ideals. Now, we present some
constructions that will be used further. For the first one, we need to
restrict our category:

######  Definition 2.16 (Definition 3.1 of [3]).

An hyperbolic multiring is a multiring $R$ such that $1-1=R$. The category of
hyperbolic multirings and hyperbolic hyperfields will be denoted by
$\mathcal{HMR}$ and $\mathcal{HMF}$ respectively.

Let $F_{1}$ and $F_{2}$ be two hyperbolic hyperfields. We define a new
hyperbolic hyperfield $(F_{1}\times_{h}F_{2},+,-,\cdot,(0,0),(1,1))$ by the
following: the adjacent set of this structure is

$F_{1}\times_{h}F_{2}:=(\dot{F}_{1}\times\dot{F}_{2})\cup\\{(0,0)\\}.$

For $(a,b),(c,d)\in F_{1}\times_{h}F_{2}$ we define

$\displaystyle-(a,b)$ $\displaystyle=(-a,-b),$ $\displaystyle(a,b)\cdot(c,d)$
$\displaystyle=(a\cdot c,b\cdot d),$ $\displaystyle(a,b)+(c,d)$
$\displaystyle=\\{(e,f)\in F_{1}\times F_{2}:e\in a+c\mbox{ and }f\in
b+d\\}\cap(F_{1}\times_{h}F_{2}).$ (1)

In other words, $(a,b)+(c,d)$ is defined in order to avoid elements of
$F_{1}\times F_{2}$ of type $(x,0),(0,y)$, $x,y\neq 0$.

######  Theorem 2.17 (Product of Hyperbolic Hyperfields).

Let $F_{1},F_{2}$ be hyperbolic hyperfields and $F_{1}\times_{h}F_{2}$ as
above. Then $F_{1}\times_{h}F_{2}$ is a hyperbolic hyperfield and satisfy the
Universal Property of product for $F_{1}$ and $F_{2}$.

###### Proof.

We will verify the conditions of definition 2.11 (in a very similar manner as
in Theorem 3.3 of [7]). Note that by the definition of multivalued sum in
$F_{1}\times_{h}F_{2}$ we have for all $a,c\in F_{1}$ and all $b,d\in F_{2}$
that $(a,b)+(c,d)\neq\emptyset$ and $(a,c)-(a,c)=F_{1}\times_{h}F_{2}$ if
$a,c\neq 0$.

  1. i -

In order to prove that $(F_{1}\times_{h}F_{2},+,-,(0,0))$ is a multigroup we
follow the steps below.

    1. a -

We will prove that if $(e,f)\in(a,b)+(c,d)$, then $(a,b)\in(c,d)+(-e,-f)$ and
$(c,d)\in(-a,-b)+(e,f)$.

If $(a,b)=(0,0)$ (or $(c,d)=(0,0)$) or $(a,b)=(-c,-d)$, then
$(e,f)\in(a,b)+(c,d)$ means $(e,f)=(c,d)$ or $e\in a-a$, $f\in b-b$. In both
cases we get $(a,b)\in(c,d)+(-e,-f)$ and $(c,d)\in(-a,-b)+(e,f)$.

Now suppose $a,b,c,d\neq 0$ with $a\neq-c$, $b\neq-d$. Let
$(e,f)\in(a,b)+(c,d)$. Then $(e,f)\in F_{1}\times_{h}F_{2}$ with $e\in a+c$
and $f\in c+d$. Moreover $a\in c-e$ and $b\in d-f$ with $(a,b)\in
F_{1}\times_{h}F_{2}$, implying $(a,b)\in(c,d)+(-e,-f)$. We prove
$(c,d)\in(-a,-b)+(e,f)$ by the very same argument.

    2. b -

Commutativity and $((a,b)\in(c,d)+(0,0))\Leftrightarrow((a=b)\wedge(c=d))$ are
direct consequence of the definition of multivaluated sum.

    3. c -

Now we prove the associativity, that is,

$[(a,b)+(c,d)]+(e,f)=(a,b)+[(c,d)+(e,f)].$

In fact (see the remarks after Lemma 2.4 of [7]), it is enough to show

$[(a,b)+(c,d)]+(e,f)\subseteq(a,b)+[(c,d)+(e,f)].$

If $(a,b)=0$ or $(c,d)=0$ or $(e,f)=0$ we are done. Now let $a,b,c,d,e,f\neq
0$ and $(v,w)\in[(a,b)+(c,d)]+(e,f)$. If $(c,d)=-(e,f)$, we have

$(a,b)+[(c,d)+(e,f)]=(a,b)+F_{1}\times_{h}F_{2}=F_{1}\times_{h}F_{2}\supseteq[(a,b)+(c,d)]+(e,f).$

If $-(e,f)\in(a,b)+(c,d)$ then $-(a,b)\in(c,d)+(e,f)$ and we have

$[(a,b)+(c,d)]+(e,f)=F_{1}\times_{h}F_{2}=(a,b)+[(c,d)+(e,f)]$

Now suppose $a,b,c,d,e,f\neq 0$, $(c,d)\neq-(e,f)$, $-(e,f)\notin(a,b)+(c,d)$.
Let $(x,y)\in[(a,b)+(c,d)]+(e,f)$. Then there exists $(v,w)\in
F_{1}\times_{h}F_{2}$ such that $(v,w)\in(a,b)+(c,d)$ and
$(x,y)\in(v,w)+(e,f)$. This imply $(v\in a+c)\wedge(x\in v+e)$ and $(w\in
b+y)\wedge(y\in d+f)$, so there exists $p\in F_{1}$, $q\in F_{2}$ such that
$(v\in a+p)\wedge(p\in c+e)$ and $(w\in b+q)\wedge(q\in d+f)$. If $p,q=0$ or
$p,q\neq 0$ we have $(p,q)\in F_{1}\times_{h}F_{2}$, which imply
$(v,w)\in[(a,b)+(c,d)]+(e,f)$. If $p=0$ and $q\neq 0$ (the case $q=0$ and
$p\neq 0$ is analogous), then $v=a$ and $c=-e$. Since $a,c\neq 0$ and $F_{1}$
is hyperbolic, we have $a-a=c-c=F_{1}$. Then $(v\in a-a)\wedge(-a\in c-c)$ and
$(w\in b+q)\wedge(q\in d+f)$, with $(-a,q)\in F_{1}\times_{h}F_{2}$ and again,
we get $(v,w)\in[(a,b)+(c,d)]+(e,f)$.

  2. ii -

Since $(F_{1}\times_{h}F_{2},\cdot,(1,1))$ is an abelian group, we conclude
that $(F_{1}\times_{h}F_{2},\cdot,(1,1))$ is a commutative monoid. Beyond
this, every nonzero element of $F_{1}\times_{h}F_{2}$ has an inverse.

  3. iii -

$(a,b)\cdot(0,0)=(0,0)$ for all $(a,b)\in F_{1}\times_{h}F_{2}$ is direct from
definition.

  4. iv -

For the distributive property, let $(a,b),(c,d),(e,f)\in F_{1}\times_{h}F_{2}$
and consider $(x,y)\in(e,f)[(a,b)+(c,d)]$. We need to prove that

$\displaystyle(x,y)\in(e,f)\cdot(a,b)+(e,f)\cdot(c,d).$ (*)

It is the case if $(a,b)=(0,0)$, or $(c,d)=(0,0)$ or $(e,f)=(0,0)$. Moreover
(*) also holds if $a,b,c,d,e,f\neq 0$ and $(c,d)=-(a,b)$.

Now suppose $a,b,c,d,e,f\neq 0$ and $(c,d)\neq-(a,b)$. Then $(x,y)=(ev,fw)$
for some $(v,w)\in(a,b)+(c,d)$. Since $(e,f)\in F_{1}\times_{h}F_{2}$, $e=0$
iff $f=0$, which imply $(ev,fw)\in F_{1}\times_{h}F_{2}$, with $ev\in ea+ec$
and $fw\in fb+fd$. Therefore
$(x,y)=(ev,fw)\in(e,f)\cdot(a,b)+(e,f)\cdot(c,d)$, as desired.

Then $(F_{1}\times_{h}F_{2},+,-,\cdot,(0,0),(1,1))$ is a hyperbolic
hyperfield. Moreover we have projections
$\pi_{1}:F_{1}\times_{h}F_{2}\rightarrow F_{1}$,
$\pi_{2}:F_{1}\times_{h}F_{2}\rightarrow F_{2}$ given respectively by the
rules $\pi_{1}(x,y)=x$, $\pi_{2}(x,y)=y$.

Finally, suppose that $F$ is another hyperfield with morphisms
$p_{1}:F\rightarrow F_{1}$, $p_{2}:F\rightarrow F_{2}$. Consider
$(p_{1},p_{2}):F\rightarrow F_{1}\times_{h}F_{2}$ given by the rule
$(p_{1},p_{2})(x)=(p_{1}(x),p_{2}(x))$. It is immediate that $(p_{1},p_{2})$
is the unique morphism making the following diagram commute

$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$$\scriptstyle{(p_{1},p_{2})}$$\textstyle{F_{1}}$$\textstyle{F_{1}\times_{h}F_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$$\textstyle{F_{2}}$

so $F_{1}\times_{h}F_{2}$ is the product in the category of hyperbolic
hyperfields, completing the proof. ∎

In order to avoid confusion and mistakes, we denote the binary product in
$\mathcal{HMF}$ by $F_{1}\times_{h}F_{2}$. For hyperfields $\\{F_{i}\\}_{i\in
I}$, we denote the product of this family by

$\prod^{h}_{i\in I}F_{i},$

with underlying set defined by

$\prod^{h}_{i\in I}F_{i}:=\left(\prod_{i\in
I}\dot{F}_{i}\right)\cup\\{(0_{i})_{i\in I}\\}$

and operations defined by rules similar to the ones defined in 2.3. If
$I=\\{1,...n\\}$, we denote

$\prod^{h}_{i\in I}F_{i}=\prod^{n}_{\begin{subarray}{c}i=1\\\
[h]\end{subarray}}F_{i}.$

######  Example 2.18.

Note that if $F_{1}$ (or $F_{2}$) is not hyperbolic, then
$F_{1}\times_{h}F_{2}$ is not a hyperfield. Let $F_{1}$ be a field (considered
as a hyperfield), for example $F_{1}=\mathbb{R}$ and $F_{2}$ be another
hyperfield. Then if $a,b\in F_{2}$, we have $1-1=\\{0\\}$, so
$(1,a)+(-1,b)=\\{0\\}\times(a-b)$, and

$[\\{0\\}\times(a-b)]\cap(F_{1}\times_{h}F_{2})=\emptyset.$

######  Proposition 2.19 (Example 2.6 in [17]).

Fix a multiring $A$ and a multiplicative subset $S$ of $A$ such that $1\in S$.
Define an equivalence relation $\sim$ on $A$ by $a\sim b$ if and only if
$as=bt$ for some $s,t\in S$. Denote by $\overline{a}$ the equivalence class of
$a$ and set $A/_{m}S=\\{\overline{a}:a\in A\\}$. Then, we define in agreement
with Marshall’s notation, $\overline{a}+\overline{b}=\\{\overline{c}:cv\in
as+bt,\,\mbox{for some }s,t,v\in S\\}$, $-\overline{a}=\overline{-a}$, and
$\overline{a}\overline{b}=\overline{ab}$.

Then $A/_{m}S$ are multirings. Moreover, if $A$ is a hyperring, the same holds
for $A/_{m}S$. The canonical projection $\pi:A\rightarrow A/_{m}S$ is a
morphism.

######  Proposition 2.20 (2.19 in [7]).

Let $A,B$ be a multiring and $S\subseteq A$ a multiplicative subset of $A$.
Then for every morphism $f:A\rightarrow B$ such that $f[S]=\\{1\\}$, there
exist a unique morphism $\tilde{f}:A/_{m}S\rightarrow B$ such that the
following diagram commute:

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{f}$$\textstyle{A/_{m}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{!\tilde{f}}$$\textstyle{B}$

where $\pi:A\rightarrow A/_{m}S$ is the canonical projection
$\pi(a)=\overline{a}$.

######  Proposition 2.21 (3.13 of [7]).

Let $(G,\equiv,-1)$ be a special group and define $M(G)=G\cup\\{0\\}$ where
$0:=\\{G\\}$555Here, the choice of the zero element was ad hoc. Indeed, we can
define $0:=\\{x\\}$ for any $x\notin G$.. Then $(M(G),+,-,\cdot,0,1)$ is a
hyperfield, where

  * •

$a\cdot b=\begin{cases}0\,\mbox{if }a=0\mbox{ or }b=0\\\ a\cdot
b\,\mbox{otherwise}\end{cases}$

  * •

$-(a)=(-1)\cdot a$

  * •

$a+b=\begin{cases}\\{b\\}\,\mbox{if }a=0\\\ \\{a\\}\,\mbox{if }b=0\\\
M(G)\,\mbox{if }a=-b,\,\mbox{and }a\neq 0\\\
D_{G}(a,b)\,\mbox{otherwise}\end{cases}$

######  Corollary 2.22 (3.14 of [7]).

The correspondence $G\mapsto M(G)$ extends to a faithful functor
$M:\mathcal{SG}\rightarrow MField$.

######  Definition 2.23 (3.15-3.19 of [7]).

A hyperfield $F$ is a special multifield if there exist a special group $G$
such that $F=M(G)$. The category of special multifields will be denoted by
$\mathcal{SMF}$.

######  Definition 2.24 (Definition 3.2 of [3]).

A Dickmann-Miraglia multiring (or DM-multiring for short) 666The name
“Dickmann-Miraglia” is given in honor to professors Maximo Dickmann and
Francisco Miraglia, the creators of the special group theory. is a pair
$(R,T)$ such that $R$ is a multiring, $T\subseteq R$ is a multiplicative
subset of $R\setminus\\{0\\}$, and $(R,T)$ satisfies the following properties:

DM0

$R/_{m}T$ is hyperbolic.

DM1

If $\overline{a}\neq 0$ in $R/_{m}T$, then $\overline{a}^{2}=\overline{1}$ in
$R/_{m}T$. In other words, for all $a\in R\setminus\\{0\\}$, there are $r,s\in
T$ such that $ar=s$.

DM2

For all $a\in R$,
$(\overline{1}-\overline{a})(\overline{1}-\overline{a})\subseteq(\overline{1}-\overline{a})$
in $R/_{m}T$.

DM3

For all $a,b,x,y,z\in R\setminus\\{0\\}$, if

$\begin{cases}\overline{a}\in\overline{x}+\overline{b}\\\
\overline{b}\in\overline{y}+\overline{z}\end{cases}\mbox{ in }R/_{m}T,$

then exist $\overline{v}\in\overline{x}+\overline{z}$ such that
$\overline{a}\in\overline{y}+\overline{v}$ and
$\overline{vb}\in\overline{xy}+\overline{az}$ in $R/_{m}T$.

If $R$ is a ring, we just say that $(R,T)$ is a DM-ring, or $R$ is a DM-ring.
A Dickmann-Miraglia hyperfield (or DM-hyperfield) $F$ is a hyperfield such
that $(F,\\{1\\})$ is a DM-multiring (satisfies DM0-DM3). In other words, $F$
is a DM-hyperfield if $F$ is hyperbolic and for all $a,b,v,x,y,z\in F^{*}$,

  1. i -

$a^{2}=1$.

  2. ii -

$(1-a)(1-a)\subseteq(1-a)$.

  3. iii -

$\mbox{If }\begin{cases}a\in x+b\\\ b\in y+z\end{cases}\mbox{ then there
exists }v\in x+z\mbox{ such that }a\in y+v\mbox{ and }vb\in xy+az$.

######  Theorem 2.25 (Theorem 3.4 of [3]).

Let $(R,T)$ be a DM-multiring and denote

$Sm(R,T)=(R/_{m}T).$

Then $Sm(R)$ is a special hyperfield (thus $Sm(R,T)^{\times}$ is a special
group).

######  Theorem 2.26 (Theorem 3.9 of [3]).

Let $F$ be a hyperfield satisfying DM0-DM2. Then $F$ satisfies DM3 if and only
if satisfies SMF4. In other words, $F$ is a DM-hyperfield if and only if it is
a special hyperfield.

In this sense, we define the following category:

######  Definition 2.27.

A pre-special hyperfield is a hyperfield satisfying DM0, DM1 and DM2. In other
words, a pre-special hyperfield is a hyperbolic hyperfield $F$ such that for
all $a\in\dot{F}$, $a^{2}=1$ and $(1-a)(1-a)\subseteq 1-a$.

The category of pre-special hyperfields will be denoted by $PSMF$.

######  Example 2.28 (An hyperfield satisfying DM3 but not DM2).

Let $H_{3}$ as in Example 2.15. We have that $H_{3}$ is hyperbolic and has
exponent 2 but does not satisfies $DM2$:

$1+2=\\{1,2\\}\mbox{ and }(1+2)(1+2)=1+2+2+1=(1+1)+(2+2)=H_{3}.$

Moreover, by a case analysis we have that $H_{3}$ satisfies DM3, and we prove
it by a case analysis: we have 27 possible choices for the triple $(x,y,z)$
with $x,y,z\in H_{3}$.

  1. i -

Let $(x,y,z)$ such that $z=0$. Then $b\in y+z$ means $b=y$ and $v\in x+z$
means $v=x$. Let $a\in x+b$ with $b\in y+z$. Then $a\in b+x=y+v$ and

$vb=xy\in xy+az.$

So we cover the cases

$(x,y,z)\in\\{(0,0,0),(0,1,0),(0,2,0),(1,0,0),(1,1,0),(1,2,0),(2,0,0),(2,1,0),(2,2,0)\\}.$

  2. ii -

Let $(x,y,z)$ such that $x=0$. Then $v\in x+z$ means $v=z$. Let $a\in
x+b=0+b=\\{b\\}$ with $b\in y+z$. Then $a\in b+z=y+z$, $az=bz$ and

$vb=bz\in xy+bz=xy+az.$

So we cover the cases

$(x,y,z)\in\\{(0,0,1),(0,0,2),(0,1,1),(0,1,2),(0,2,1),(0,2,2)\\}.$

  3. iii -

Let $(x,y,z)$ such that $y=0$ and $x,z\neq 0$. Then $\\{1,2\\}\subseteq x+z$,
$b\in y+z$ means $b=z$ and $a\in y+v$ means $a=v$. Let $a\in x+b=x+z$. Then
$vb=az$ and

$vb=az\in 0+az=xy+az.$

So we cover the cases

$(x,y,z)\in\\{(1,0,1),(1,0,2),(2,0,1),(2,0,2)\\}.$

  4. iv -

Let $(x,y,z)$ such that $x,y,z\neq 0$ and $x+z=H_{3}$. Let $a\in x+b$ with
$b\in y+z$. If $a=0$ then $b=x$. Taking $v=y\in H_{3}=x+z$ we get $a\in
y+y=y+v$ and

$vb=xy\in xy+0=xy+az.$

If $b=0$ then $z=y$ $a=x$ (so $az=xy$). Taking $v=y\in H_{3}=x+z$ we get $a\in
y+y=y+v$ and

$vb=0\in xy+xy=xy+az.$

If $a,b\neq 0$, then there exist $v\in H_{3}=x+z$ such that $vb=xy$. Then
$a\in\\{1,2\\}\subseteq y+v$ (because $x,y,z,v\neq 0$) and

$vb=xy\in xy+az.$

So we cover the cases

$(x,y,z)\in\\{(1,1,1),(1,2,1),(2,1,2),(2,2,2)\\}.$

  5. v -

Let $(x,y,z)$ such that $x,y,z\neq 0$ and $y+z=H_{3}$. Let $a\in x+b$ with
$b\in y+z$. If $a=0$ then $b=x$. Taking $v=y\in\\{1,2\\}\subseteq x+z$ we get
$a\in y+y=y+v$ and

$vb=xy\in xy+0=xy+az.$

If $b=0$ then $z=y$ $a=x$ (so $az=xy$). Taking $v=y\in\\{1,2\\}\subseteq x+z$
we get $a\in y+y=y+v$ and

$vb=0\in xy+xy=xy+az.$

If $a,b\neq 0$, then there exist $v\in\\{1,2\\}\subseteq x+z$ such that
$vb=xy$. Then $a\in\\{1,2\\}\subseteq y+v$ (because $x,y,z,v\neq 0$) and

$vb=xy\in xy+az.$

So we cover the cases

$(x,y,z)\in\\{(1,1,2),(1,2,2),(2,1,1),(2,2,1)\\}.$

Example 2.28 is a big surprise in the sense that, in the context of special
groups, this example exhibits a group that it is not pre-special but satisfies
SG6.

##  3 The K-theory for Multifields/Hyperfields

In this section we introduce the notion of K-theory of a hyperfield
essentially repeating the construction in 2.1 replacing the word “field” by
“hyperfield” and explore some of this basic properties. In particular, Theorem
3.8 is an extension of a result [21], that gives us some evidence, that apart
from the obvious resemblance, more technical aspects of this new theory can be
developed (but with other proofs) in multistructure setting in parallel with
classical K-theory.

######  Definition 3.1 (The K-theory of a Hyperfield).

For a hyperfield $F$, $K_{*}F$ is the graded ring

$K_{*}F=(K_{0}F,K_{1}F,K_{2}F,...)$

defined by the following rules: $K_{0}F:=\mathbb{Z}$. $K_{1}F$ is the
multiplicative group $\dot{F}$ written additively. With this purpose, we fix
the canonical “logarithm” isomorphism

$\rho:\dot{F}\rightarrow K_{1}F,$

where $\rho(ab)=\rho(a)+\rho(b)$. Then $K_{n}F$ is defined to be the quotient
of the tensor algebra

$K_{1}F\otimes K_{1}F\otimes...\otimes K_{1}F\,(n\mbox{ times})$

by the (homogeneous) ideal generated by all $\rho(a)\otimes\rho(b)$, with
$a\neq 0,1$ and $b\in 1-a$.

In other words, for each $n\geq 2$,

$K_{n}F:=T^{n}(K_{1}F)/Q^{n}(K_{1}(F)),$

where

$T^{n}(K_{1}F):=K_{1}F\otimes_{\mathbb{Z}}K_{1}F\otimes_{\mathbb{Z}}...\otimes_{\mathbb{Z}}K_{1}F$

and $Q^{n}(K_{1}(F))$ is the subgroup generated by all expressions of type
$\rho(a_{1})\otimes\rho(a_{2})\otimes...\otimes\rho(a_{n})$ such that
$a_{i}\in 1-a_{j}$ for some $i,j$ with $1\leq i,j\leq n$.

To avoid carrying the overline symbol, we will adopt all the conventions used
in Dickmann-Miraglia’s K-theory (as explained in above definition 2.6). Just
as it happens with the previous K-theories, a generic element $\eta\in K_{n}F$
has the pattern

$\eta=\rho(a_{1})\otimes\rho(a_{2})\otimes...\otimes\rho(a_{n})$

for some $a_{1},...,a_{n}\in\dot{F}$, with $a_{i}\in 1-a_{j}$ for some $1\leq
i<j\leq n$. Note that if $F$ is a field, then “$b\in 1-a$” just means $b=1-a$,
and the hyperfield and Milnor’s K-theory for $F$ coincide.

The very first task, is to extend the basic properties valid in Milnor’s and
Dickmann-Miraglia’s K-theory to ours. Here we already need to restrict our
attention to hyperbolic hyperfields:

######  Lemma 3.2 (Basic Properties I).

Let $F$ be an hyperbolic hyperfield. Then

  1. a -

$\rho(1)=0$.

  2. b -

For all $a\in\dot{F}$, $\rho(a)\rho(-a)=0$ in $K_{2}F$.

  3. c -

For all $a,b\in\dot{F}$, $\rho(a)\rho(b)=-\rho(a)\rho(b)$ in $K_{2}F$.

  4. d -

For every $a_{1},...,a_{n}\in\dot{F}$ and every permutation $\sigma\in S_{n}$,

$\rho(a_{1})...\rho(a_{i})...\rho(a_{n})=\mbox{sgn}(\sigma)\rho(a_{1})...\rho(a_{n})\mbox{
in }K_{n}F.$

  5. e -

For every $\xi\in K_{m}F$ and $\eta\in K_{n}F$, $\eta\xi=(-1)^{mn}\xi\eta$ in
$K_{m+n}F$.

  6. f -

For all $a\in\dot{F}$, $\rho(a)^{2}=\rho(a)\rho(-1)$.

###### Proof.

  1. a -

Is an immediate consequence of the fact that $\rho$ is an isomorphism.

  2. b -

Since $F$ hiperbolic, $1-1=F$. Then $-a^{-1}\in 1-1$ for all $a\in\dot{F}$,
and hence, $-1\in-1+a^{-1}$. Multiplying this by $-a$, we get $a\in 1-a$. By
definition, this imply $\rho(a)\rho(-a)=0$.

  3. c -

By item (b), $\rho(ab)\rho(-ab)=0$ in $K_{2}F$. But

$\displaystyle\rho(ab)\rho(-ab)$
$\displaystyle=\rho(a)\rho((-a)b)+\rho(b)\rho((-b)a)$
$\displaystyle=\rho(a)\rho(-a)+\rho(a)\rho(b)+\rho(b)\rho(-b)+\rho(b)\rho(a)$
$\displaystyle=\rho(a)\rho(b)+\rho(b)\rho(a).$

From $\rho(a)\rho(b)+\rho(b)\rho(a)=\rho(ab)\rho(-ab)=0$, we get the desired
result $\rho(a)\rho(b)=-\rho(a)\rho(b)$ in $K_{2}F$.

  4. d -

This is a consequence of item (c) and an inductive argument.

  5. e -

This is a consequence of item (d) and an inductive argument, using the fact
that an element in $K_{n}F$ has pattern

$\eta=\rho(a_{1})\otimes\rho(a_{2})\otimes...\otimes\rho(a_{n})$

for some $a_{1},...,a_{n}\in\dot{F}$, with $a_{i}\in 1-a_{j}$ for some $1\leq
i<j\leq n$.

  6. f -

Follow from the fact that $F$ is hyperbolic i.e, for all $a\in\dot{F}$, $a\in
1-1$.

∎

An element $a\in\dot{F}$ induces a morphism of graded rings
$\omega^{a}=\\{\omega^{a}_{n}\\}_{n\geq 1}:K_{*}F\rightarrow K_{*}F$ of degree
1, where $\omega^{a}_{n}:K_{n}F\rightarrow K_{n+1}F$ is the multiplication by
$\lambda(-a)$. When $a=-1$, we write

$\omega=\\{\omega_{n}\\}_{n\geq 1}=\\{\omega^{-1}_{n}\\}_{n\geq
1}=\omega^{-1}.$

######  Proposition 3.3 (Adapted from 3.3 of [12]).

Let $F,K$ be hyperbolic hyperfields and $\varphi:F\rightarrow L$ be a
morphism. Then $\varphi$ induces a morphism of graded rings

$\varphi_{*}=\\{\varphi_{n}:n\geq 0\\}:K_{*}F\rightarrow K_{*}L,$

where $\varphi_{0}=Id_{\mathbb{Z}}$ and for all $n\geq 1$, $\varphi_{n}$ is
given by the following rule on generators

$\varphi_{n}(\rho(a_{1})...\rho(a_{n}))=\rho(\varphi(a_{1}))...\rho(\varphi(a_{n})).$

Moreover if $\varphi$ is surjective then $\varphi_{*}$ is also surjective, and
if $\psi:L\rightarrow M$ is another morphism then

  1. a -

$(\psi\circ\varphi)_{*}=\psi_{*}\circ\varphi_{*}$ and $Id_{*}=Id$.

  2. b -

For all $a\in\dot{F}$ the following diagram commute:

$\textstyle{K_{n}F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{n}}$$\scriptstyle{\omega^{a}_{n}}$$\textstyle{K_{n+1}F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{n+1}}$$\textstyle{K_{n}L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega^{\varphi(a)}_{n}}$$\textstyle{K_{n+1}L}$

  3. c -

If $\varphi(1)=1$ then for all $n\geq 1$ the following diagram commute:

$\textstyle{K_{n}F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{n}}$$\scriptstyle{\omega^{-1}_{n}}$$\textstyle{K_{n+1}F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{n+1}}$$\textstyle{K_{n}L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega^{-1}_{n}}$$\textstyle{K_{n+1}L}$

###### Proof.

Firstly, note that $\varphi$ extends to a function
$\varphi_{1}:K_{1}F\rightarrow K_{1}L$ given by the rule

$\varphi_{1}(\rho(a))=\rho(\varphi(a)).$

Certainly $\varphi_{1}$ is a morphism because

$\varphi_{1}(0)=\varphi_{1}(\rho(1))=\rho(\varphi(1))=\rho(1)=0,$

and for all $\rho(a),\rho(b)\in K_{1}F$,

$\varphi_{1}(\rho(a)+\rho(b))=\varphi_{1}(\rho(ab))=\rho(\varphi(ab))=\rho(\varphi(a)\varphi(b))=\rho(\varphi(a))+\rho(\varphi(b)).$

Proceeding inductively, for all $n\geq 1$ we extend $\varphi$ to a function
$\varphi_{n}:\prod^{n}_{i=1}K_{1}F\rightarrow K_{n}L$ given by the rule

$\varphi(\rho(a_{1}),...,\rho(a_{n})):=\varphi_{1}(\rho(a_{1}))...\varphi_{1}(\rho(a_{n}))=\rho(\varphi(a_{1}))...\rho(\varphi(a_{n})).$

Then if $i=1,...,n$ and $b_{i}\in k_{1}F$ we have

$\displaystyle\varphi_{n}(\rho(a_{1}),...,\rho(a_{i})+\rho(b_{i}),...,\rho(a_{n}))=\varphi_{n}(\rho(a_{1}),...,\rho(a_{i}b_{i}),...,\rho(a_{n}))=$
$\displaystyle\rho(\varphi(a_{1}))...\rho(\varphi(a_{i}b_{i})...\rho(\varphi(a_{n}))=\rho(\varphi(a_{1}))...\rho(\varphi(a_{i})\varphi(b_{i}))...\rho(\varphi(a_{n}))=$
$\displaystyle\rho(\varphi(a_{1})...[\rho(\varphi(a_{i})+\varphi(b_{i}))]...\rho(\varphi(a_{n}))=$
$\displaystyle\rho(\varphi(a_{1}))...\rho(\varphi(a_{i}))...\rho(\varphi(a_{n}))+\rho(\varphi(a_{1}))...\rho(\varphi(b_{i}))...\rho(\varphi(a_{n}))=$
$\displaystyle\varphi_{n}(\rho(a_{1}),...,\rho(a_{i}),...,\rho(a_{n}))+\varphi_{n}(\rho(a_{1}),...,\rho(b_{i}),...,\rho(a_{n})),$

then for each $n$, $\varphi_{n}:\prod^{n}_{i=1}K_{1}F\rightarrow K_{n}L$ is
multilinear and by the universal property of tensor product there is an unique
morphism

$\tilde{\varphi}_{n}:\bigotimes^{n}_{j=1}K_{1}F\rightarrow K_{n}L$

extending $\varphi_{n}$. By construction (and using the fact that $\varphi$ is
a morphism), $\mbox{Ker}(\tilde{\varphi}_{n})=Q^{n}(K_{1}F)$, which provides
an unique morphism
$\overline{\varphi}_{n}:T^{n}(K_{1}F)/Q^{n}(K_{1}(F)\rightarrow K_{n}L$ such
that $\tilde{\varphi}_{n}=\overline{\varphi}_{n}\circ\pi_{n}$, where $\pi_{n}$
is the canonical projection $T^{n}(K_{1}F)$ in $Q^{n}(k_{1}F)$. Then taking
$\varphi_{0}=Id_{\mathbb{Z}}$, we get a morhism $\varphi_{*}:K_{*}F\rightarrow
K_{*}L$, given by $\varphi_{*}=\\{\overline{\varphi}_{n}:n\geq 0\\}$.

For items (a) and (b), it is enough to note that these properties holds for
$\tilde{\varphi}_{n}$, $n\geq 0$, and after the application of projection, we
get the validity for $\overline{\varphi}_{n}=\pi_{n}\circ\tilde{\varphi}_{n}$.

Item (c) follows by the same argument of itens (a) and (b), noting that
$\varphi(1)=1$ imply $\varphi(-1)=-1$. By abuse of notation, we denote

$\varphi_{*}=\\{\overline{\varphi}_{n}:n\geq 0\\}=\\{\varphi_{n}:n\geq 0\\}.$

∎

We also have the reduced K-theory graded ring
$k_{*}F=(k_{0}F,k_{1}F,...,k_{n}F,...)$ in the hyperfield context, which is
defined by the rule $k_{n}F:=K_{n}F/2K_{n}F$ for all $n\geq 0$. Of course for
all $n\geq 0$ we have an epimorphism $q:K_{n}F\rightarrow k_{n}F$ simply
denoted by $q(a):=[a]$, $a\in K_{n}F$. It is immediate that $k_{n}F$ is
additively generated by
$\\{[\rho(a_{1})]..[\rho(a_{n})]:a_{1},...,a_{n}\in\dot{F}\\}$. We simply
denote such a generator by $\tilde{\rho}(a_{1})...\tilde{\rho}(a_{n})$ or even
$\rho(a_{1})...\rho(a_{n})$ whenever the context allows it.

We also have some basic properties of the reduced K-theory, which proof is
just a translation of 2.1 of [12]:

######  Lemma 3.4 (Adapted from 2.1 [12]).

Let $F$ be a hyperbolic hyperfield, $x,y,a_{1},...,a_{n}\in\dot{F}$ and
$\sigma$ be a permutation on $n$ elements.

  1. a -

In $k_{2}F$, $\rho(a)^{2}=\rho(a)\rho(-1)$. Hence in $k_{m}F$,
$\rho(a)^{m}=\rho(a)\rho(-1)^{m-1}$, $m\geq 2$;

  2. b -

In $k_{2}F$, $\rho(a)\rho(-a)=\rho(a)^{2}=0$;

  3. c -

In $k_{n}F$, $\rho(a_{1})\rho(a_{2})...\rho(a_{n})=\rho(a_{\sigma
1})\rho(a_{\sigma 2})...\rho(a_{\sigma n})$;

  4. d -

For $n\geq 1$ and $\xi\in k_{n}F$, $\xi^{2}=\rho(-1)^{n}\xi$;

  5. e -

If $F$ is a real reduced hyperfield, then $x\in 1+y$ and
$\rho(y)\rho(a_{1})...\rho(a_{n})=0$ implies

$\rho(x)\rho(a_{1})\rho(a_{2})...\rho(a_{n})=0.$

Moreover the results in Proposition 3.3 continue to hold if we took
$\varphi_{*}=\\{\varphi_{n}:n\geq 0\\}:k_{*}F\rightarrow k_{*}L$.

######  Proposition 3.5.

Let $F$ be a hyperfield and $T\subseteq F$ be a multiplicative subset such
that $F\subseteq T$. Then

$K(F/_{m}T^{*})\cong k(F/_{m}T^{*}).$

###### Proof.

Since $F^{2}\subseteq T$, for all $a\in F/_{m}T^{*}$ we have

$0=\rho(a^{2})=\rho(a)+\rho(a).$

Then $2K(F/_{m}T^{*})=0$ and we get $K(F/_{m}T^{*})\cong k(F/_{m}T^{*})$. ∎

######  Theorem 3.6.

Let $F$ be a hyperbolic hyperfield and $T\subseteq F$ be a multiplicative
subset such that $F\subseteq T$. Then there is a surjective morphism

$k(F)\rightarrow k(F/_{m}T^{*}).$

Moreover,

$k(F)\cong K(F/_{m}\dot{F}^{2})\cong k(F/_{m}\dot{F}^{2}).$

Berfore we prove it, we need a Lemma:

######  Lemma 3.7.

Let $F$ be a hyperfield and $n\geq 1$. Then

$2K_{n}(F)=\left\\{\sum^{p}_{j=1}\rho(a_{j1})...\rho(a_{jn}):\mbox{for all
}j\mbox{ there is an index }k\mbox{ such that
}a_{jk}=b_{i}^{2},\,b_{i}\in\dot{F}\right\\}.$

###### Proof.

Let $\eta\in 2K_{n}F$. Then

$\eta=\left(\sum^{p}_{j=1}\rho(a_{j1})...\rho(a_{jn})\right)+\left(\sum^{p}_{j=1}\rho(a_{j1})...\rho(a_{jn})\right),\,d_{ij}\in\dot{F}.$

By induction, we only need to consider the case $p=1$, so

$\displaystyle\rho(a_{1})...\rho(a_{n})+\rho(a_{1})...\rho(a_{n})$
$\displaystyle=\rho(a_{1}^{2})\rho(a_{2})...\rho(a_{n}).$

and we get $\subseteq$. The reverse inclusion follow by the same calculation.
∎

###### Proof of Theorem 3.6.

Let $\pi:F\rightarrow F/_{m}T^{*}$ denote the canonical projection. By
Proposition 3.3 there is a morphism $\pi_{*}:K(F)\rightarrow K(F/_{m}T^{*})$.
Since $\pi$ is surjective, $\pi_{*}$ is surjective.

Now, let $\pi:F\rightarrow F/_{m}\dot{F}^{2}$ and $q:K(F)\rightarrow k(F)$ the
canonical projections. Denote elements in $F/_{m}\dot{F}^{2}$ by $[a]\in
F/_{m}\dot{F}^{2}$, $a\in F$ and elements in $k_{n}(F)$ by
$\tilde{\rho}(a_{1})...\tilde{\rho}(a_{n})$. For all $n\geq 1$ we have an
induced morphism $\tilde{q}_{n}:K_{n}(F/_{m}\dot{F}^{2})\rightarrow k_{n}(F)$
given by the rule

$\tilde{q}_{n}(\rho([a_{1}])...\rho([a_{n}])):=\tilde{\rho}(a_{1})...\tilde{\rho}(a_{n}).$

This morphism $\tilde{\pi}_{n}$ makes the following diagram commute

---
$\textstyle{K_{n}(F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\scriptstyle{\pi_{n}}$$\textstyle{k_{n}(F)}$$\textstyle{K_{n}(F/_{m}\dot{F}^{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{q}_{n}}$

and then, $\tilde{q}_{n}$ is surjective. Finally, if
$\tilde{q}_{n}(\rho([a_{1}])...\rho([a_{n}]))=0$, then
$\tilde{\rho}(a_{1})...\tilde{\rho}(a_{n})=0$, and hence
$\rho(a_{1})...\rho(a_{n})\in 2K_{n}(F)$. By Lemma 3.7

$\rho(a_{1})...\rho(a_{n})=\sum^{p}_{j=1}\rho(d_{j1})...\rho(d_{jn}),\,d_{ij}\in\dot{F}$

and for all $i$ there is an index $k$ such that $a_{ik}=b_{i}^{2}$,
$b_{i}\in\dot{F}$. Therefore

$\displaystyle\pi_{n}(\rho(a_{1})...\rho(a_{n}))$
$\displaystyle=\pi_{n}\left(\sum^{p}_{j=1}\rho(d_{j1})...\rho(d_{jn})\right)$
$\displaystyle=\sum^{p}_{j=1}\pi_{n}(\rho(d_{j1})...\rho(d_{jn}))=\sum^{p}_{j=1}\rho([d_{j1}])...\rho([d_{jn})])$
$\displaystyle=\sum^{p}_{j=1}[d_{j1}])...\rho([1])...\rho([d_{jn})]=0.$

Then $\mbox{Ker}(\tilde{q}_{n})=[0]$, proving that $\tilde{q}_{n}$ is
injective. Then $\tilde{q}_{n}$ is an isomorphism, and composing all the
isomorphisms obtained here we get

$k(F)\cong K(F/_{m}\dot{F}^{2})\cong k(F/_{m}\dot{F}^{2}).$

∎

The Theorem 3.8 below generalizes Proposition 5.10 of [21]: this constitutes a
fundamental technical step to build profinite (Galois) groups associated to a
pre-special hyperfield in [2].

Lets establish some notation: for $n\geq 0$ we denote

$P(n)=\mathcal{P}(\\{0,...,n-1\\})\setminus\\{\emptyset\\}$

and for $0\leq i\leq n-1$, denote

$P(n,i)=\\{X\in P(n):i\in X\\}.$

For a be a pre-special hyperfield $F$ and $\\{a_{0},...,a_{n-1}\\}\subseteq
F^{*}$ $\mathbb{F}_{2}$-linearly independent, if $S\in P(n)$ we denote

$a_{S}:=a_{0}^{\varepsilon_{0}}...a_{n-1}^{\varepsilon_{n-1}},$

where $\varepsilon_{0}\in\\{0,1\\}$ for all $i=0,..,n-1$ and
$\varepsilon_{i}=1$ if and only if $i\in S$.

Remember that by the very definition of $k_{n}(F)$,

$k_{n}(F):=[k_{1}(G)\otimes k_{1}(G)]/M,$

where $M$ is the subgroup of $k_{1}(G)\otimes k_{1}(G)$ generated by

$\\{\rho(a)\rho(b):a\in D_{G}(1,b),\,a\\}.$

######  Theorem 3.8.

Let $F$ be a pre-special hyperfield and $\\{a_{0},...,a_{n-1}\\}\subseteq
F^{*}$ $\mathbb{F}_{2}$-linearly independent. The following conditions are
equivalent:

  1. i -

There exists $\\{b_{0},...,b_{n-1}\\}\subseteq F^{*}$ such that

$\sum_{k<n}\rho(a_{k})\rho(b_{k})=0\mbox{ in }k_{2}(F).$

  2. ii -

There exist subsets $\\{c_{0},...,c_{m-1}\\},\\{d_{0},...,d_{n-1}\\}$ of
$F^{*}$ with $m\geq n$ such that

    1. (a)

$\\{c_{0},...,c_{m-1}\\}$ is linearly independent and $c_{i}=a_{i}$ for all
$i<n$;

    2. (b)

$d_{i}=b_{i}$ for all $i<n$ and $d_{i}=1$ for $i=n,...,m-1$.

    3. (c)

For all $x\in C:=[c_{0},...,c_{m-1}]$, there is some $r_{x}\in
1+x\setminus\\{0\\}$ such that for each $i<m$

$d_{i}=\prod_{x\in C_{i}}r_{x}$

where

$C_{i}=\left\\{\prod_{k<m}c_{k}^{\varepsilon_{k}}:\varepsilon_{k}\in\\{0,1\\}\mbox{
and }\varepsilon_{i}=1\right\\}.$

In other words, $C_{i}$ is "counting" all products
$c_{0}^{r_{0}}...c_{i}^{1}...c_{m-1}^{r_{m-1}}$. Since for all $x\in
C:=[c_{0},...,c_{m-1}]$ there exist $S\in P(m)$ such that

$x=\prod_{i\in S}c_{i}:=c_{S}.$

Denoting $r_{x}$ by $r_{S}$ we can rewrite

$d_{i}=\prod_{x\in C_{i}}r_{x}=\prod_{S\in P(m)}r_{S}.$

###### Proof of Theorem 3.8.

(i) $\Rightarrow$ (ii). Let

$\sum_{k<n}\rho(a_{k})\rho(b_{k})=0\mbox{ in }k_{2}(F).$

Then there exist $u_{0},...,u_{p-1},v_{0},...,v_{p-1}\in F^{*}$ such that
$v_{i}\in 1+u_{i}$ for $i=0,...,p-1$ and

$\sum_{k<n}\rho(a_{k})\rho(b_{k})=\sum_{k<n}\rho(a_{k})\rho(b_{k})\mbox{ in
}k_{1}(F)\otimes k_{1}(F).$

Enlarge the set $\\{a_{0},...,a_{n-1}\\}$ to a base $\\{c_{0},...,c_{m-1}\\}$
of $[\\{a_{0},...,a_{n-1},u_{0},...,u_{p-1}\\}]$, with $c_{i}=a_{i}$ for all
$i<n$. For all $x\in C:=[c_{0},...,c_{m-1}]$ there exist $S\in P(m)$ such that

$x=\prod_{i\in S}c_{i}:=c_{S}.$

Moreover, since $\\{c_{0},...,c_{m-1}\\}$ is a basis, for each $i=0,...,p-1$
there is only one $S_{i}\in P(m)$ such that

$u_{i}=c_{S_{i}}.$

For each $S\in P(m)$, set

$r_{S}:=\prod_{\begin{subarray}{c}those\,j\,with\\\
S_{j}=S\end{subarray}}v_{j}.$

If no $S_{j}=S$, set $r_{S}=1$. Note that if there is an index $j$ with
$S=S_{j}$, this index must be unique (because the expression $u_{i}=c_{S_{i}}$
is unique). Then by construction $r_{S}\in 1+c_{S}\setminus\\{0\\}$ and in
$k_{2}(F)$ we get

$\displaystyle\sum_{k<m}\rho(a_{k})\rho(b_{k})$
$\displaystyle=\sum_{k<n}\rho(c_{k})\rho(d_{k})=\sum_{k<p}\rho(u_{k})\rho(v_{k})$
$\displaystyle=\sum_{S\in
P(m)}\rho\left(\prod_{\begin{subarray}{c}those\,j\,with\\\
S_{j}=S\end{subarray}}c_{j}\right)\rho(v_{j})$ $\displaystyle=\sum_{S\in
P(m)}\sum_{\begin{subarray}{c}those\,j\,with\\\
S_{j}=S\end{subarray}}\rho(c_{j})\rho(v_{j})$ $\displaystyle=\sum_{S\in
P(m)}\rho(c_{S})\rho\left(\prod_{\begin{subarray}{c}those\,j\,with\\\
S_{j}=S\end{subarray}}v_{S}\right)$ $\displaystyle=\sum_{S\in
P(m)}\rho(c_{S})\rho(r_{S})=\sum_{S\in P(m)}\sum_{k\in
S}\rho(c_{k})\rho(r_{S})$
$\displaystyle=\sum_{k<m}\rho(c_{k})\rho\left(\prod_{S\in P(n)}r_{S}\right).$

Since $\\{c_{0},...,c_{m-1}\\}$ is a basis, it follows that

$d_{i}=\prod_{S\in P(n)}r_{S}$

as desired.

(ii)$\Rightarrow$(i). Under the hypotheses of (ii) we get

$\displaystyle\sum_{k<n}\rho(a_{k})\rho(b_{k})$
$\displaystyle=\sum_{k<m}\rho(c_{k})\rho(d_{k})=\sum_{k<m}\rho(c_{k})\rho\left(\prod_{S\in
P(n)}r_{S}\right)$ $\displaystyle=\sum_{k<m}\sum_{S\in
P(m)}\rho(c_{k})\rho(r_{S})=\sum_{S\in P(m)}\sum_{k<m}\rho(c_{i})\rho(r_{S})$
$\displaystyle=\sum_{S\in P(m)}\rho(c_{S})\rho(r_{S})=0.$

∎

##  4 Inductive Graded Rings and Interchanging of K-theories

After the three K-theories defined in the above sections, it is desirable (or,
at least, suggestive) to build of an abstract environment that encapsulates
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 fit well to this
purpose777A categorical and systematic development of inductive graded rings
in connection with ”quadratic multirings” is carried out in [6]..

######  Definition 4.1 (Inductive Graded Rings First Version (Definition 9.7
of [10])).

An inductive graded ring (or Igr for short) is a structure $R=((R_{n})_{n\geq
0},(h_{n})_{n\geq 0},\ast_{nm})$ where

  1. i -

$R_{0}\cong\mathbb{F}_{2}$.

  2. ii -

$R_{n}$ is a group of exponent 2 with a distinguished element $\top_{n}$.

  3. iii -

$h_{n}:R_{n}\rightarrow R_{n+1}$ is a group homomorphism such that
$h_{n}(\top_{n})=\top_{n+1}$.

  4. iv -

For all $n\geq 0$, $h_{n}=\ast_{1n}(\top_{1},\\_)$.

  5. v -

The ring

$R=\bigoplus_{n\geq 0}R_{n}$

is a commutative graded ring.

  6. vi -

For $0\leq s\leq t$ define

$h^{t}_{s}=\begin{cases}Id_{R_{s}}\mbox{ if }s=t\\\ h_{t-1}\circ...\circ
h_{s+1}\circ h_{s}\mbox{ if }s<t.\end{cases}$

Then if $p\geq n$ and $q\geq m$, for all $x\in R_{n}$ and $y\in R_{m}$,

$h^{p}_{n}(x)\ast h^{q}_{m}(y)=h^{p+q}_{n+m}(x\ast y).$

A morphism between Igr’s $R$ and $S$ is a pair $f=(f,(f_{n})_{n\geq 0})$ where
$f_{n}:R_{n}\rightarrow S_{n}$ is a morphism of pointed groups and

$f=\bigoplus\limits_{n\geq 0}f_{n}:R\rightarrow S$

is a morphism of commutative rings with unity. The category of inductive
graded rings (in first version) and their morphisms will be denoted by Igr.

A first consequence of these definitions is that: if

$f:((R_{n})_{n\geq 0},(h_{n})_{n\geq 0},\ast_{nm})\rightarrow((S_{n})_{n\geq
0},(l_{n})_{n\geq 0},\ast_{nm})$

is a morphism of Igr’s then $f_{n+1}\circ h_{n}=l_{n}\circ f_{n}$.

$\textstyle{R_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{0}}$$\scriptstyle{f_{0}}$$\textstyle{R_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{1}}$$\scriptstyle{f_{1}}$$\textstyle{R_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{2}}$$\scriptstyle{f_{2}}$$\textstyle{...\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{n-1}}$$\textstyle{R_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{n}}$$\scriptstyle{f_{n}}$$\textstyle{R_{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{n+1}}$$\scriptstyle{f_{n+1}}$$\textstyle{...}$$\textstyle{S_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{l_{0}}$$\textstyle{S_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{l_{1}}$$\textstyle{S_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{l_{2}}$$\textstyle{...\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{l_{n-1}}$$\textstyle{S_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{l_{n}}$$\textstyle{S_{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{l_{n+1}}$$\textstyle{...}$

In fact, $f_{0}:R_{0}\rightarrow S_{0}$ is an isomorphism, so $f_{1}\circ
h_{0}=l_{0}\circ f_{0}$. If $n\geq 1$, for all $a_{n}\in R_{n}$ holds

$\displaystyle f_{n+1}\circ h_{n}(a_{n})$
$\displaystyle=f_{n+1}\circ(\ast_{1n}(\top_{1},a_{n}))=f_{1}(\top_{1})\ast_{1n}f_{n}(a_{n})$
$\displaystyle=\top_{1}\ast_{1n}f_{n}(a_{n})=l_{n}(f_{n}(a_{n}))=l_{n}\circ
f_{n}(a_{n}).$

######  Example 4.2.

  1. a -

Let $F$ be a field of characteristic not 2. The main actors here are $WF$, the
Witt ring of $F$ and $IF$, the fundamental ideal of $WF$. Is well know that
$I^{n}F$, the $n$-th power of $IF$ is additively generated by $n$-fold Pfister
forms over $F$. Now, let $R_{0}=WF/IF\cong\mathbb{F}_{2}$ and
$R_{n}=I^{n}F/I^{n+1}F$. Finally, let $h_{n}=\\_\otimes\langle 1,1\rangle$.
With these prescriptions we have an inductive graded ring $R$ associated to
$F$.

  2. b -

The previous example still works if we change the Witt ring of a field $F$ for
the Witt ring of a (formally real) special group $G$.

######  Definition 4.3.

We denote $\mbox{Igr}_{fin}$ the full subcategory of Igr such that

$\mbox{Obj}(\mbox{Igr}_{fin})=\left\\{R\in\mbox{Obj}(\mbox{Igr}):\left|R_{n}\right|<\omega\mbox{
for all }n\geq 1\right\\}.$

Of course,

$\left\\{R\in\mbox{Obj}(\mbox{Igr}):\left|\bigoplus\limits_{n\geq
1}R_{n}\right|<\omega\right\\}\neq\mbox{Obj}(\mbox{Igr}_{fin}),$

for example, in 4.2(a), $\mathbb{F}_{2}[x]\in\mbox{Obj}(\mbox{Igr}_{fin})$ and
$\mathbb{F}_{2}[x]$ is not finite.

We finally this section with an use of Igr’s to interchanging the three
K-theory notions presented before in a functorial fashion. Lets first, look
more carefully at theorem 4.4. We make the following distinctions between
K-theories:

$\displaystyle K^{mil}$ $\displaystyle\mbox{ will denote the Milnor's
K-theory},$ $\displaystyle K^{dm}$ $\displaystyle\mbox{ will denote the
Dickmann-Miraglia's K-theory},$ $\displaystyle K^{mult}$ $\displaystyle\mbox{
will denote the K-theory of Hyperfields}.$

We will deal with some different categories, so in order to ease the
exposition lets establish some dictionary:

  1. 1.

An object in $Igr$ will be denoted simply by $R=(R_{n},h_{n},\ast_{nm})$
(being implicit the identation $n\in\mathbb{N}$ or $n\geq 0$). We even denote
simply $R=(R_{n},h,\ast)$ if the context allows it.

  2. 2.

The distinguished elements $\top_{n}\in R_{n}$ will be denoted simply by
$\top$, whenever the context allows it, and we do the same for the symbols
$+_{n},0_{n}$.

  3. 3.

An IGR-morphism between $R$ and $S$ will be simply denoted by $f:R\rightarrow
S$, with the immediate convention that $f=(f_{n})_{n\geq 0}$, with
$f_{n}:R_{n}\rightarrow S_{n}$ being a morphism of pointed
$\mathbb{F}_{2}$-modules.

  4. 4.

Let $\mathcal{C}$ be a category. We use the notation "$A,B\in\mathcal{C}$" as
a synonym of "$A,B$ be objects of $\mathcal{C}$".

  5. 5.

Let $\mathcal{C}$ be a category, $A,B$ be objects of $\mathcal{C}$. We use the
notations "$f\in\mathcal{C}(A,B)$" and $f\in\mbox{Hom}_{\mathcal{C}}(A,B)$" as
a synonyms of "$f:A\rightarrow B$ is a morphism in $\mathcal{C}$".

  6. 6.

We will define various functors below. In order to facilitate the exposition,
we define a functor $F:\mathcal{C}\rightarrow\mathcal{D}$ by the sentence "for
$f:A\rightarrow B$, we define $F(A):=...$ and $F(f):=...$"; being implicit
that $A$ and $B$ are objects and $f:A\rightarrow B$ is a morphism in category
$\mathcal{C}$.

Of course, we need the following theorem:

######  Theorem 4.4.

  1. a -

Let $F$ be a field. Then $k^{mil}_{*}F$ (the reduced Milnor K-theory) is an
inductive graded ring.

  2. b -

Let $G$ be a special group. Then $k^{dm}_{*}G$ (the Dickmann-Miraglia K-theory
of $G$) is an inductive graded ring.

  3. c -

Let $F$ be a hyperfield. Then $k^{mult}_{*}F$ (our reduced K-theory) is an
inductive graded ring.

###### Proof.

Item (a) is the content of Lemma 9.11 in [10], and item (b) is the content of
Lemma 9.12 in [10]. We prove item (c) and itens (a) and (b) will proceed by
the same argument.

Let $k^{mult}_{*}F=(k_{0}F,k_{1}F,...,k_{n}F,...)$ be the reduced K-theory of
a hyperfield $F$. Let $\top_{0}=1$ and for each $n\geq 1$, we set
$\top_{n}=l(-1)^{n}$ as the distinguished element of $m-n$. For each $n\geq
0$, let
$\theta_{n}:\prod^{n}_{j=1}K^{mult}_{1}F\rightarrow\otimes^{n+1}_{j=1}K^{mult}_{n+1}F$
given by the rule

$\theta_{n}(\rho(a_{1}),...,\rho(a_{n})):=\rho(-1)\rho(a_{1})...\rho(a_{n}).$

We have for each $i\in\\{1,...,n\\}$ and each $a_{1},...,a_{n},b_{i}\in F^{*}$
that

$\displaystyle\theta_{n}(\rho(a_{1}),...\rho(a_{i})+\rho(b_{i}),...,\rho(a_{n}))=\theta_{n}(\rho(a_{1}),...\rho(a_{i}b_{i}),...,\rho(a_{n})):=$
$\displaystyle\rho(-1)\rho(a_{1})...\rho(a_{i}b_{i})...\rho(a_{n})=\rho(-1)\rho(a_{1})...[\rho(a_{i})+\rho(b_{i})]...\rho(a_{n})=$
$\displaystyle\rho(-1)\rho(a_{1})...\rho(a_{i})...\rho(a_{n})+\rho(-1)\rho(a_{1})...\rho(b_{i})...\rho(a_{n})=$
$\displaystyle\theta_{n}(\rho(a_{1}),...,\rho(a_{i}),...,\rho(a_{n}))+\theta_{n}(\rho(a_{1})...\rho(b_{i})...\rho(a_{n})),$

then $\theta_{n}$ is multilinear. By the universal property of tensor product,
we have a group homomorphism $\tilde{\theta}_{n}:K^{mult}_{n}F\rightarrow
K^{mult}_{n+1}F$ given by the rule888Remember that we are using the
simplificated notation for elements in $K^{mult}_{n}F$ (and all other
K-theories), which is
$\rho(a_{1})...\rho(a_{n}):=\rho(a_{1})\otimes...\otimes\rho(a_{n})$.

$\tilde{\theta}_{n}(\rho(a_{1})...\rho(a_{n}))=\rho(-1)\rho(a_{1})...\rho(a_{n}).$

In order to make distinctions between reduced and non-reduced K-theories, we
pontually denote an element in $k^{mult}_{n}F:=K^{mult}_{n}F/2K^{mult}_{n}F$
by $\tilde{\rho}(a_{1})...\tilde{\rho}(a_{n})$. Lets also denote the canonical
projection by $\pi_{n}:K^{mult}_{n}F\rightarrow k^{mult}_{n}F$. We define
$\omega_{n}:k^{mult}_{n}F\rightarrow k^{mult}_{n+1}F$ by the following rule
(on generators): for $a_{1},...,a_{n}\in\dot{F}$,

$\omega_{n}(\tilde{\rho}(a_{1})...\tilde{\rho}(a_{n}))=\tilde{\rho}(-1)\tilde{\rho}(a_{1})...\tilde{\rho}(a_{n}).$

In fact, if $\rho(a_{1})...\rho(b_{n})-\rho(b_{1})...\rho(b_{n})\in
2K^{mult}_{n}F$ then

$\displaystyle\rho(-1)\rho(a_{1})...\rho(b_{n})-\rho(-1)\rho(b_{1})...\rho(b_{n})=\rho(-1)[\rho(a_{1})...\rho(b_{n})-\rho(b_{1})...\rho(b_{n})]\in
2K^{mult}_{n+1}F,$

which proves that $\omega_{n}$ is in fact a group homomorphism making the
following diagram commute

$\textstyle{K^{mult}_{n}F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\theta}_{n}}$$\scriptstyle{\pi_{n}}$$\textstyle{K^{mult}_{n+1}F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{n+1}}$$\textstyle{k^{mult}_{n}F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega_{n}}$$\textstyle{k^{mult}_{n+1}F}$

With these rules, we already have the properties (i)-(iv) of Definition 4.1
holding in $k_{*}F$, remaining only property (v). Note that
$\omega^{t}_{s}=\tilde{\rho}(-1)^{t-s}$ for $0\leq s<t$.

Now let $m,n,p,q\in\mathbb{N}$, $p\geq n$, $q\geq m$ and consider $x\in
k^{mult}_{n}F$ and $y\in k^{mult}_{m}F$. Note that
$\omega^{p}_{n}(x)=\tilde{\rho}(-1)^{p-n}\cdot x$ and
$\omega^{q}_{m}(y)=\tilde{\rho}(-1)^{q-m}\cdot y$. Then

$\omega^{p}_{n}(x)\cdot\omega^{q}_{m}(y)=(\tilde{\rho}(-1)^{p-n}\cdot
x)(\tilde{\rho}(-1)^{q-m}\cdot y)=\tilde{\rho}(-1)^{p-n+q-m}\cdot(x\cdot
y)=\omega^{p+q}_{n-m}(x),$

completing the proof. ∎

Using this Theorem (in addition with the argument of Lemma 3.3 in [12]) we
obtain the following.

######  Corollary 4.5.

Let $F$ be a field. We have a functor and $k:Field_{2}\rightarrow\mbox{Igr}$
induced by K-theory and Milnor’s reduced K-theory.

######  Corollary 4.6.

Let $F$ be a hyperfield. We have a functor
$k^{mult}:MField_{2}\rightarrow\mbox{Igr}$ induced by our reduced K-theory.

######  Theorem 4.7 (Theorem 2.5 in [11]).

Let $F$ be a field. The functor $G:Field_{2}\rightarrow SG$ provides a functor
$k^{dm}_{*}:Field_{2}\rightarrow\mbox{Igr}$ (the special group K-theory
functor) given on the objects by $k^{dm}_{*}(F):k^{dm}_{*}(G(F))$ and on the
morphisms $f:F\rightarrow K$ by $k^{dm}_{*}(f):=G(f)_{*}$ (in the sense of
Lemma 3.3 of [12]). Moreover, this functor commutes with the functors $G$ and
$k$, i.e, for all $F\in Field$, $k^{dm}_{*}(G(F))\cong k_{*}(F)$.

######  Theorem 4.8.

Let $G$ be a special group. The equivalence of categories $M:SG\rightarrow
SMF$ induces a functor $k^{mult}_{*}:SG\rightarrow\mbox{Igr}$ given on the
objects by $k^{mult}_{*}(G):=k^{mult}_{*}(M(G))$ and on the morphisms
$f:G\rightarrow H$ by $k^{mult}_{*}(f):=k^{mult}_{*}(M(f))$. Moreover, this
functor commutes with $M$ and $k^{dm}$, i.e, for all $G\in SG$,
$k^{mult}_{*}(M(G))\cong k^{dm}_{*}(G)$.

###### Proof.

The only part requiring proof is that for all $G\in SG$,
$k^{mult}_{*}(M(G))\cong k^{dm}_{*}(G)$. The very first observation is that:
since $G$ is an exponent 2 group, the reduced and non-reduced
$K^{mult}$-theory of $M(G)$ coincide.

Following the argument of Theorem 2.5 in [11], it is enough to show the
following two statements:

  1. i -

For all $a,b\in G$, if $b\in 1-a$ in $M(G)$ then $\lambda(b)\lambda(a)=0$;

  2. ii -

For all $a,b\in G$, if $b\in D_{G}(1,a)$ then $\rho(b)\rho(a)=0$.

For (i), if $b\in 1-a$ in $M(G)$ then $b\in D_{G}(1,-a)$ and then,
$\lambda(b)\lambda(-a)=0$. Hence

$\lambda(b)^{2}=\lambda(b)\lambda(-a)=\lambda(b)\lambda(a)+\lambda(b)\lambda(-1).$

Since $\lambda(b)\lambda(-1)=\lambda(b)^{2}$, we get $\lambda(b)\lambda(a)=0$.

For (ii) we just use the same argument: if $b\in D_{G}(1,a)$ then $b\in 1+a$
in $M(G)$ and then, $\rho(b)\rho(-a)=0$. Hence

$\rho(b)^{2}=\rho(b)\rho(-a)=\rho(b)\rho(a)+\rho(b)\rho(-1).$

Since $\rho(b)\rho(-1)=\rho(b)^{2}$, we get $\rho(b)\rho(a)=0$. ∎

Combining Theorems 4.4, 4.7, 4.8, 3.6 and Corollaries 4.5 and 4.6 we obtain
the following Theorem, that unificate in some sense all three K-theories:

######  Theorem 4.9 (Interchanging K-theories Formulas).

Let $F\in Field_{2}$. Then

$k^{mil}(F)\cong k^{dm}(G(F))\cong k^{mult}(M(G(F))).$

If $F$ is formally real and $T$ is a preordering of $F$, then

$k^{dm}(G_{T}(F))\cong k^{mult}(M(G_{T}(F))).$

Moreover, since $M(G(F))\cong F/_{m}\dot{F}^{2}$ and $M(G_{T}(F))\cong
F/_{m}T^{*}$, we get

$\displaystyle k^{mil}(F)$ $\displaystyle\cong k^{dm}(G(F))\cong
k^{mult}(F/_{m}\dot{F}^{2})\mbox{ and }$ $\displaystyle k^{dm}(G_{T}(F))$
$\displaystyle\cong k^{mult}(F/_{m}T^{*}).$

######  Corollary 4.10.

Let $F$ be a field. Then

$k^{mil}(F)\cong k^{mult}(F/_{m}\dot{F}^{2}).$

###### Proof.

Using the previous Corollary, we already have

$k^{mil}(F)\cong k^{dm}(G(F))\cong k^{mult}(M(G(F))).$

Now, is enough to observe that $M(G(F))\cong F/_{m}\dot{F}^{2}$. ∎

Combining Theorem 4.9, Corollary 4.10 and Theorem 3.6 we get the following
Corollaries.

######  Corollary 4.11.

Let $F$ be a formally real field and $T$ be a preordering. Then we have a
surjective map

$k^{mil}(F)\rightarrow k^{mult}(F/_{m}T^{*}).$

######  Corollary 4.12.

Let $G$ be a pre-special group and $H\subseteq G$ be a subgroup of $G$. Let
$M(G)$ be the pre-special multifield associated to $G$ and
$M(H)=H\cup\\{0\\}\subseteq M(G)$. Then

$G/H\cong M(G)/_{m}M(H)^{*}.$

Moreover, $M(H)\subseteq M(G)$ is a preordering if and only if $H$ is
saturated.

######  Corollary 4.13.

Let $G$ be a special group and $H$ be a saturated subgroup. Then we have a
surjective map

$k^{dm}(G)\rightarrow k^{mult}(G/_{m}H)\cong k^{dm}(G/H).$

We close this paper "embedding" the three K-theories considered until now into
the framework of inductive graded rings, in such a way that this "embedding"
provides a sort of free objects in Igr. Lets firstly define functors that
formalize the construction of K-theory.

######  Definition 4.14.

We define the graded subring generated by the level 1 functor

$\mathbbm{1}:\mbox{Igr}\rightarrow\mbox{Igr}$

as follow: for an object $R=(R_{n},h_{n},\ast_{nm})$,

  1. i -

$\mathbbm{1}(R)_{0}:=R_{0}\cong\mathbb{F}_{2}$,

  2. ii -

$\mathbbm{1}(R)_{1}:=R_{1}$,

  3. iii -

for $n\geq 2$,

$\displaystyle\mathbbm{1}(R)_{n}$ $\displaystyle:=\\{x\in
R_{n}:x=\sum\limits^{r}_{j=1}a_{1j}\ast_{11}...\ast_{11}a_{nj},$
$\displaystyle\mbox{with }a_{ij}\in R_{1},\,1\leq i\leq n,\,1\leq j\leq
r\mbox{ for some }r\geq 1\\}.$

Of course, $\mathbbm{1}(R)$ provides an inclusion
$\iota_{\mathbbm{1}(R)}:\mathbbm{1}(R)\rightarrow R$ in the obvious way.

On the morphisms, for $f\in\mbox{Igr}(R,S)$, we define
$\mathbbm{1}(f)\in\mbox{Igr}(\mathbbm{1}(R),\mathbbm{1}(S))$ by the
restriction $\mathbbm{1}(f)=f\upharpoonleft_{\mathbbm{1}(R)}$. In other words,
$\mathbbm{1}(f)$ is the only Igr-morphisms that makes the following diagram
commute:

$\textstyle{\mathbbm{1}(R)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{\mathbbm{1}(R)}}$$\scriptstyle{\mathbbm{1}(f)}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\mathbbm{1}(S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{\mathbbm{1}(S)}}$$\textstyle{S}$

We denote $\mbox{Igr}_{\mathbbm{1}}$ the full subcategory of Igr such that

$\mbox{Obj}(\mbox{Igr}_{\mathbbm{1}})=\\{R\in\mbox{Igr}:\iota_{\mathbbm{1}(R)}:\mathbbm{1}(R)\rightarrow
R\mbox{ is an isomorphism}\\}.$

######  Definition 4.15.

We define the quotient graded ring functor

$\mathcal{Q}:\mbox{Igr}\rightarrow\mbox{Igr}$

as follow: for an object $R=(R_{n},h_{n},\ast_{nm})$, $\mathcal{Q}(R):=R/Q$,
where $Q=(Q_{n})_{n\geq 0}$ is the ideal generated by
$\\{(\top+a)\ast_{11}a\in R_{2}:a\in R_{1}\\}$. More explicit,

  1. i -

$Q_{0}:=\\{0\\}\subseteq R_{0}$,

  2. ii -

$Q_{1}:=\\{0\\}\subseteq R_{1}$,

  3. iii -

for $n\geq 2$, $Q_{n}\subseteq R_{n}$ is the pointed
$\mathbb{F}_{2}$-submodule generated by

$\displaystyle\\{x\in R_{n}:x$
$\displaystyle=y_{l}\ast_{l1}(\top+a_{1})\ast_{11}a_{1}\ast_{1r}z_{r},$
$\displaystyle\mbox{with }a_{1}\in R_{1},\,y_{l}\in R_{l},\,z_{r}\in
R_{r},\,1\leq r,l\leq n-2,\,l+r=n-2\\}.$

Of course, $\mathcal{Q}(R)$ provides a projection
$\pi_{\mathcal{Q}(R)}:R\rightarrow\mathcal{Q}(R)$ in the obvious way.

On the morphisms, for $f\in\mbox{Igr}(R,S)$, we define
$\mathcal{Q}(f)\in\mbox{Igr}(\mathcal{Q}(R),\mathcal{Q}(S))$ by the only Igr-
morphisms that makes the following diagram commute:

$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\mathcal{Q}(R)}}$$\scriptstyle{f}$$\textstyle{\mathcal{Q}(R)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{Q}(f)}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\mathcal{Q}(S)}}$$\textstyle{\mathcal{Q}(S)}$

We denote $\mbox{Igr}_{q}$ the full subcategory of Igr such that

$\mbox{Obj}(\mbox{Igr}_{q})=\\{R\in\mbox{Igr}:\pi_{\mathcal{Q}}:R\rightarrow\mathcal{Q}(R)\mbox{
is an isomorphism}\\}.$

######  Definition 4.16.

We denote by $\mbox{Igr}_{+}$ the full subcategory of Igr such that

$\mbox{Obj}(\mbox{Igr}_{+})=\mbox{Obj}(\mbox{Igr}_{\mathbbm{1}})\cap\mbox{Obj}(\mbox{Igr}_{q}).$

We denote by $j_{+}:\mbox{Igr}_{+}\rightarrow\mbox{Igr}$ the inclusion
functor.

By the very definition of the K-theory of hyperfields (with the notations in
Theorem 3.3) we define the following functor.

######  Definition 4.17 (K-theories Functors).

With the notations of Theorem 3.3 we have a functors
$k:\mathcal{HMF}\rightarrow\mbox{Igr}_{+}$,
$k:\mathcal{PSMF}\rightarrow\mbox{Igr}_{+}$ induced by the reduced K-theory
for hyperfields.

Now, let $R\in\mbox{Igr}$. We define a hyperfield $(\Gamma(R),+,-.\cdot,0,1)$
by the following: firstly, fix an isomorphism
$e_{R}:(R_{1},+_{1},0_{1},\top_{1})\rightarrow(G(R),\cdot,1,-1)$. This
isomorphism makes, for example, an element $a\ast_{11}(\top+b)\in R_{2}$,
$a,b\in R_{1}$ take the form $(e^{-1}_{R}(x))\ast_{11}(e^{-1}_{R}((-1)\cdot
y))\in R_{2}$, $x,y\in\Gamma(R)$. By an abuse of notation, we simply write
$x\ast_{11}(-y)\in R_{2}$, $x,y\in\Gamma(R)$. In this sense, an element in
$Q_{2}$ (see Definition 4.15) has the form $x\ast_{11}(-x)$, $x\in\Gamma(R)$,
and we can extend this terminology for all $Q_{n}$, $n\geq 2$.

Now, let $\Gamma(R):=G(R)\cup\\{0\\}$ and for $a,b\in\Gamma(R)$ we define

$\displaystyle-a$ $\displaystyle:=(-1)\cdot a,$ $\displaystyle a\cdot 0$
$\displaystyle=0\cdot a:=0,$ $\displaystyle a+0$ $\displaystyle=0+a=\\{a\\},$
$\displaystyle a+(-a)$ $\displaystyle=\Gamma(R),$ for $\displaystyle a,b\neq
0,a\neq-b\mbox{ define }$ $\displaystyle a+b$
$\displaystyle:=\\{c\in\Gamma(R):\mbox{there exist }d\in G(R)\mbox{ such that
}$ $\displaystyle a\cdot b=c\cdot d\mbox{ and }a\ast_{11}b=c\ast_{11}d\in
R_{2}\\}.$ (2)

######  Proposition 4.18.

With the above rules, $(\Gamma(R),+,-.\cdot,0,1)$ is a pre-special hyperfield.

###### Proof.

The proof is similar to Theorem 2.17: we will verify the conditions of
Definition 2.11. Note that by the definition of multivalued sum once we proof
that $\Gamma(R)$ is an hyperfield, it will be hyperbolic.

  1. i -

In order to prove that $(\Gamma(R),+,-.\cdot,0,1)$ is a multigroup we follow
the steps below.

    1. a -

Commutativity and $(a\in b+0)\Leftrightarrow(a=b)$ are direct consequence of
the definition of multivaluated sum and the fact that
$a\ast_{11}b=b\ast_{11}a$.

    2. b -

We will prove that if $c\in a+b$, then $a\in b-c$ and $b\in-c+a$.

If $a=0$ (or $b=0$) or $a=-b$, then $c\in a+b$ means $c=a$ or $c\in a-a$. In
both cases we get $a\in b-c$ and $b\in-c+a$.

Now suppose $a,b\neq 0$ with $a\neq-b$. Let $c\in a+b$. Then $a\cdot b=c\cdot
d$ and $a\ast_{11}b=c\ast_{11}d\in R_{2}$ for some $d\in G(R)$. Then
$b\cdot(-c)=a\cdot(-d)$ and

$\displaystyle a\ast_{11}b=c\ast_{11}d$ $\displaystyle\Rightarrow
a\ast_{11}b\ast_{11}[(-b)\ast_{11}(-d)]=c\ast_{11}d\ast_{11}[(-b)\ast_{11}(-d)]$
$\displaystyle\Rightarrow a\ast_{11}(-d)=b\ast_{11}(-c),$

proving that $a\in b-c$. Similarly we prove that $b\in-c+a$.

    3. c -

Now we prove the associativity, that is,

$(a+b)+c=a+(b+c).$

In fact (see the remarks after Lemma 2.4 of [7]), it is enough to show

$(a+b)+c\subseteq a+(b+c).$

If $0\in\\{a,b,c\\}$ we are done. Now let $a,b,c\neq 0$ and $x\in(a+b)+c$. If
$b=-c$ we have

$a+(b+c)=a+\Gamma(R)=\Gamma(R)\supseteq(a+b)+c.$

If $-c\in a+b$, then $-a\in b+c$ and we have

$(a+b)+c=\Gamma(R)=a+(b+c).$

Now suppose $a,b,c\neq 0$, $c\neq-b$, $-c\notin a+b$. Let $p\in(a+b)+c$. Then
$p\in x+c$ for some $x\in a+b$, and there exists $q,y\in G(R)$ such that

$\displaystyle p\cdot q=x\cdot c$ $\displaystyle\mbox{ and
}p\ast_{11}q=x\ast_{11}c$ $\displaystyle x\cdot y=a\cdot b$
$\displaystyle\mbox{ and }x\ast_{11}y=a\ast_{11}b.$

Now let $z,w\in G(R)$ defined by $z=xca$, $w=abx$. We have $p\cdot q=x\cdot
c=x\cdot c\cdot(a^{2})=a\cdot z$ and $z\cdot w=(xca)\cdot(abx)=b\cdot c$.
Moreover

$\displaystyle p\ast_{11}q$
$\displaystyle=x\ast_{11}c=x\ast_{11}c\ast_{11}(a^{2})=x\ast_{11}c\ast_{11}a\ast_{11}a$
$\displaystyle=[x\ast_{11}c\ast_{11}a]\ast_{11}a=(xca)\ast_{11}a=z\ast_{11}a$

and

$\displaystyle z\ast_{11}w$
$\displaystyle=(xca)\ast_{11}(abx)=(cxa)\ast_{11}(axb)=c\ast_{11}(xa)\ast_{11}(ax)\ast_{11}b=c\ast_{11}b.$

Then $p\in a+z$ with $z\in c+b=b+c$, and hence $p\in a+(b+c)$.

  2. ii -

Since $(G(R),\cdot,1)$ is an abelian group, we conclude that
$(\Gamma(R),\cdot,1)$ is a commutative monoid. Beyond this, every nonzero
element $a\in\Gamma(R)$ is such that $a^{2}=1$.

  3. iii -

$a\cdot 0=0$ for all $a\in\Gamma(R)$ is direct from definition.

  4. iv -

For the distributive property, let $a,b,d\in\Gamma(R)$ and consider $x\in
d(a+b)$. We need to prove that

$\displaystyle x\in d\cdot a+d\cdot b.$ (*)

It is the case if $0\in\\{a,b,d\\}$ or if $b=-a$.

Now suppose $a,b,d\neq 0$ with $b\neq-a$. Then there exist $y\in G(R)$ such
that $x=dy$ and $y\in a+b$. Moreover, there exist some $z\in G(R)$ such that
$y\cdot z=a\cdot b$ and $y\ast_{11}z=a\ast_{11}b$. Therefore
$(dy)\cdot(dz)=(da)\cdot(db)$ and $(dy)\ast_{11}(dz)=(da)\ast_{11}(db)$, and
$x=dy\in d\cdot a+d\cdot b$.

Then $(\Gamma(R),+,-,\cdot,0,1)$ is a hyperbolic hyperfield.

Finally, let $a\in\Gamma(R)$ and $x,y\in 1-a$. If $a=0$ or $a=1$ then we
automatically have $x\cdot y\in 1-a$, so let $a\neq 0$ and $a\neq 1$. Then
$x,y\in G(R)$ and there exist $p,q\in\Gamma(R)$ such that

$\displaystyle x\cdot p=1\cdot a$ $\displaystyle\mbox{ and
}x\ast_{11}p=1\ast_{11}a$ $\displaystyle y\cdot q=1\cdot a$
$\displaystyle\mbox{ and }y\ast_{11}q=1\ast_{11}a.$

Then $(xy)\cdot(pqa)=1\cdot a$ and

$\displaystyle(xy)\ast_{11}(pqa)$
$\displaystyle=x\ast_{11}y\ast_{11}p\ast_{11}q\ast_{11}a=[x\ast_{11}p]\ast_{11}a\ast_{11}[y\ast_{11}q]$
$\displaystyle=[1\ast_{11}a]\ast_{11}a\ast_{11}[1\ast_{11}a]=1\ast_{11}a,$

then $xy\in 1-a$, proving that $(1-a)(1-a)\subseteq(1-a)$. ∎

######  Definition 4.19.

With the notations of Proposition 4.18 we have a functor
$\Gamma:\mbox{Igr}_{+}\rightarrow\mbox{PSMF}$ defined by the following rules:
for $R\in\mbox{Igr}_{+}$, $\Gamma(R)$ is the special hyperfield obtained in
Proposition 4.18 and for $f\in\mbox{Igr}_{+}(R,S)$,
$\Gamma(f):\Gamma(R)\rightarrow\Gamma(S)$ is the unique morphism such that the
following diagram commute

$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}}$$\scriptstyle{e_{R}}$$\textstyle{\Gamma(R)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Gamma(f)}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{S}}$$\textstyle{\Gamma(S)}$

In other words, for $x\in R$ we have

$\Gamma(f)(x)=(e_{S}\circ f_{1}\circ
e^{-1}_{R})(x)=e_{S}(f_{1}(e_{R}^{-1}(x))).$

######  Theorem 4.20.

The functor $k:\mathcal{PSMF}\rightarrow\mbox{Igr}_{+}$ is the left adjoint of
$\Gamma:\mbox{Igr}_{+}\rightarrow\mathcal{PSMF}$. The unity of the adjoint is
the natural transformation $\phi:1_{\mathcal{PSMF}}\rightarrow\Gamma\circ k$
defined for $F\in\mathcal{PSMF}$ by $\phi_{F}=e_{k(F)}\circ\rho_{F}$.

###### Proof.

We show that for all $f\in\mathcal{PSMF}(F,\Gamma(R))$ there is an unique
$f^{\sharp}:\mbox{Igr}_{+}(k(F),R)$ such that
$\Gamma(f^{\sharp})\circ\phi_{F}=f$. Note that
$\phi_{F}=e_{k(F)}\circ\rho_{F}$ is an isomorphism (because $e_{k(F)}$ and
$\rho_{F}$ are isomorphisms).

Let
$f^{\sharp}_{0}:1_{\mathbb{F}_{2}}:\mathbb{F}_{2}\rightarrow\mathbb{F}_{2}$
and $f^{\sharp}_{1}:=e_{R}^{-1}\circ f\circ(\phi_{F})^{-1}\circ
e_{k(F)}:k_{1}(F)\rightarrow R_{1}$. For $n\geq 2$, define
$h_{n}:\prod^{n}_{i=1}k_{1}(F)\rightarrow R_{n}$ by the rule

$h_{n}(\rho(a_{1}),...,\rho(a_{n})):=e_{R}^{-1}(f(a_{1}))\ast...\ast
e_{R}^{-1}(f(a_{n})).$

We have that $h_{n}$ is multilinear and by the Universal Property of tensor
products we have an induced morphism $\bigotimes^{n}_{i=1}k_{n}(F)\rightarrow
R_{n}$ defined on the generators by

$h_{n}(\rho(a_{1})\otimes...\otimes\rho(a_{n})):=e_{R}^{-1}(f(a_{1}))\ast...\ast
e_{R}^{-1}(f(a_{n}).$

Now let $\eta\in Q_{n}(F)$. Suppose without loss of generalities that
$\eta=\rho(a_{1})\otimes...\otimes\rho(a_{n})$ with $a_{1}\in 1-a_{2}$. Then
$f(a_{1})\in 1-f(a_{2})$ which imply $e_{R}^{-1}(f(a_{1}))\in
1-e_{R}^{-1}(f(a_{2}))$. Since $R_{n}\in\mbox{Igr}_{+}$,

$\displaystyle
h_{n}(\eta):=h_{n}(\rho(a_{1})\otimes...\otimes\rho(a_{n}))=e_{R}^{-1}(f(a_{1}))\ast...\ast
e_{R}^{-1}(f(a_{n}))=0\in R_{n}.$

Then $h_{n}$ factors through $Q_{n}$, and we have an induced morphism
$\overline{h}_{n}:k_{n}(F)\rightarrow R_{n}$. We set
$f^{\sharp}_{n}:=\overline{h}_{n}$. In other words, $f^{\sharp}_{n}$ is
defined on the generators by

$f^{\sharp}_{n}(\rho(a_{1})...\rho(a_{n})):=e_{R}^{-1}(f(a_{1}))\ast...\ast
e_{R}^{-1}(f(a_{n}).$

Finally, we have

$\displaystyle\Gamma(f^{\sharp})\circ\phi_{F}$
$\displaystyle=[e_{R}\circ(f^{\sharp}_{1})\circ
e^{-1}_{k(F)}]\circ[e_{k(F)}\circ\rho_{F}]=e_{R}\circ(f^{\sharp}_{1})\circ\rho_{F}$
$\displaystyle=e_{R}\circ[e_{R}^{-1}\circ f\circ(\phi_{F})^{-1}\circ
e_{k(F)}]\circ\rho_{F}$
$\displaystyle=f\circ(\phi_{F})^{-1}\circ[e_{k(F)}\circ\rho_{F}]$
$\displaystyle=f\circ(\phi_{F})^{-1}\circ\phi_{F}=f.$

For the uniqueness, let $u,v\in\mbox{Igr}_{+}(k(F),R)$ such that
$\Gamma(u)\circ\phi_{F}=\Gamma(v)\circ\phi_{F}$. Since $\phi_{F}$ is an
isomorphism we have $u_{1}=v_{1}$ and since $k(F)\in\mbox{Igr}_{+}$ we have
$u=v$. ∎

##  5 Final Remarks and future works

In [3] we emphasize that DM-multirings and DP-multirings provide a new way to
look at the abstract theories of quadratic forms (for example, for special
groups we obtained an easy way to describe the axiom SG6 in the theory of
special groups).

In [8] is constructed a von Neumann hull functor from multiring category and
that, when restricted to semi-real rings, it commutes with real semigroup
functor. This allows us to obtain some quadratic forms properties of a semi-
real ring by looking to its von Neumann regular hull. In fact, the Witt ring
of a real semigroups is canonically isomorphic to the Witt ring of its Von
Neumann regular hull and this connection allows the full description of the
graded Witt ring of a real semigroup.

With these two observations in mind, it is natural consider possible
expansions of “Milnor’s triangle” of graded rings – where K-theory is
interpolating graded Witt ring and graded Cohomology ring – to the multiring
setting. In more details, below are some further roads to follow:

  1. 1.

Extension of the K-theory framework to more general multirings (for example,
to VN-multirings) with quadratic flavour.

  2. 2.

Compare graded K-theory with graded Witt ring for VN-real semigroups as in the
field case (Milnor [18]) and special groups (Dickmann-Miraglia [10]).

  3. 3.

The definition and analysis of the structure of Witt ring (and graded Witt
ring) of more general quadratic structures (not only obtained from special
groups and real semigroups): this subject have already appeared in section 4,
in connection with [15].

  4. 4.

Still in the hyperfield case, investigate the extension of the concept of
Galois group to hyperfields, comparing the Galois cohomology ring and analyse
the existence of some canonical arrow from K-theory to this cohomology ring,
in an attempt to recover the Milnor’s Conjecture available in the classic
algebraic quadratic forms context ([5], [6]).

## Acknowledgement

We want to thank the anonymous referee for her/his careful reading and
valuable suggestions.

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