*4.2. On the C-Characteristic*

We now demonstrate that any positive integer can be realized as the C-characteristic of some real hyperfield.

**Theorem 4.** *For every natural number n* ∈ N*, there exists an infinite real hyperfield F, such that* C-char *F* = *n.*

**Proof.** We are going to demonstrate that for every positive integer *n* ∈ N, the quotient hyperfield *F* = Q&*n*+1' is real and has C-char *F* = *n*. First, observe that

$$n+1 = \underbrace{1+\ldots+1}\_{n+1 \text{ times}} \text{ so } [1]\_{\langle n+1\rangle} \in (n+1) \times [1]\_{\langle n+1\rangle}.$$

Hence, C-char *F* ≤ *n*. If *n* = 1, then C-char *F* = 1. Let *n* > 2. and suppose that C-char *F* < *n*, i.e.,

$$|1|\_{\langle n+1\rangle} \in k \times |1|\_{\langle n+1\rangle'} \text{ where } 2 \le k < n.$$

Then, there exist *xi* ∈ Z, *i* ∈ {1, ..., *k* + 1}, such that

$$(n+1)^{x\_{k+1}} = (n+1)^{x\_1} + \dots + (n+1)^{x\_k}.$$

Let *N* = min{*x*1, ..., *xk*<sup>+</sup>1} and denote *mi* := *xi* − *N*. Then,

$$(n+1)^{m\_{k+1}} = (n+1)^{m\_1} + \dots + (n+1)^{m\_k},$$

where *mi* ∈ N ∪ {0} and 2 ≤ *k* ≤ *n*. We obtain that

$$(n+1)^{m\_1} + \dots + (n+1)^{m\_k} - (n+1)^{m\_{k+1}} = 0.\tag{3}$$

Presently, since for any *m* ∈ N ∪ {0}, we have

$$(n+1)^m \equiv 1 \pmod{n},$$

the left hand side of the Equation (3) is congruent to

$$\underbrace{1 + \dots + 1}\_{(k \text{ times})} - 1 = k - 1$$

modulo *n*, while the right hand side is congruent to 0 modulo *n*. Hence,

$$k - 1 \equiv 0 \pmod{n},$$

which is a contradiction, since 2 ≤ *k* ≤ *n*. Hence, C-char *F* = *n* must hold.

Consider now the set of natural numbers

$$S := \{ (n+1)^p + 1 \mid p \in \mathbb{N} \}.$$

By definition, for *s*, *t* ∈ *S* we have that [*s*]&*n*+1' = [*t*]&*n*+1' if and only if there exists some *g* ∈ &*n* + 1' such that *s* = *gt*. Suppose that *g* = 1. Without loss of generality, we can assume that *g* = (*n* + 1)*<sup>m</sup>* for some *m* ∈ N (if not, we apply the following reasoning to the equality

*t* = *g*−1*s*). Since *s* ≡ 1 (mod *n* + 1) and *gt* ≡ 0 (mod *n* + 1), we obtain a contradiction. Hence, [*s*]&*n*+1' = [*t*]&*n*+1' if and only if *s* = *t*; thus,

$$\{ [(n+1)^p + 1]\_{\langle n+1 \rangle} \mid p \in \mathbb{N} \}$$

is an infinite subset of *F*, implying that *F* is infinite. Moreover, the set

$$P\_{\langle n+1 \rangle} := \{ [\mathfrak{x}]\_{\langle n+1 \rangle} \mid \mathfrak{x} > 0 \}$$

is a positive cone in *F* by Theorem 2.

Let us now demonstrate that a finite hyperfield *F* must satisfy C-char *F* < ∞.

**Proposition 8.** *Let F be a finite hyperfield of cardinality n* > 1*, which is not the field* F2*. Then,* C-char *F* ≤ 2*n* − 3*.*

**Proof.** Let *F* = F<sup>2</sup> be a finite hyperfield of cardinality *n* > 1. If 1 + 1 = {0}, then by Proposition 1, *F* is a field of characteristic 2, so *n* ≥ 4 and thus C-char *F* = 2 ≤ 2*n* − 3. If 1 ∈ 1 + 1, then C-char *F* = 1 ≤ 2*n* − 3. Otherwise, let *a* ∈ 1 + 1. Since *F*<sup>×</sup> is an abelian group of cardinality *n* − 1, by Remark 4, we have that

$$1 = a^{n-1} \in (1+1)^{n-1} \subseteq 2(n-1) \times\_F 1$$

and hence C-char *F* ≤ 2*n* − 3.

From Proposition 7 and Remark 5, the following result follows immediately.

**Proposition 9.** *Let K be a field and G a finite subgroup of K*×*. If n* ∈ N><sup>1</sup> *divides the cardinality of G, then* C-char *KG* ≤ *n.*

An ordered field has to be infinite. This is a consequence of the compatibility of the order relation induced by the positive cone, with the addition of the field (see Section 5, below). However, we have observed that there are real hyperfields, which are finite (cf. Example 1 and Example 4). The following result shows that we can construct finite real hyperfields with the C-characteristic 1 of any odd cardinality. Note that a finite real hyperfield has to have an odd number of elements by Proposition 6.

Let *p* be a prime number. In the proof of the next result, we will use the *p-adic valuation vp* on the field of rational numbers Q. Let us briefly recall how is that is defined (for more details on valuations, we refer to [29]). Let *vp*(0) := ∞, and for *<sup>a</sup> <sup>b</sup>* <sup>∈</sup> <sup>Q</sup>×, write

$$\frac{a}{b} := p^{v} \frac{a'}{b'},$$

where *ν* ∈ Z and *a* , *b* ∈ Z are not divisible by *p*. Define *vp*( *<sup>a</sup> <sup>b</sup>* ) := *ν*. Thus, *vp* is a map from Q to Z ∪ {∞}.

**Theorem 5.** *For every odd number n* ≥ 3*, there exists a finite real hyperfield F with* C-char *F* = 1*, such that* |*F*| = *n.*

**Proof.** Consider the field of rational numbers Q, a positive integer *k* ∈ N and the subgroup of Q×:

$$G\_k := \{ \mathfrak{x} \in \mathbb{Q} \mid v\_{\mathfrak{p}}(\mathfrak{x}) \equiv 0 \pmod{k} \text{ and } \mathfrak{x} > 0 \}$$

where *vp* is the *p*-adic valuation on Q, for some prime number *p*. We are going to show that the quotient hyperfield Q*Gk* is a finite, real hyperfield with C-characteristic 1 and cardinality <sup>2</sup>*<sup>k</sup>* <sup>+</sup> 1. First, observe that *Gk* <sup>⊆</sup> <sup>Q</sup>+; thus, <sup>Q</sup>*Gk* is real by Theorem 2. Moreover, the index (Q<sup>+</sup> : *Gk*) = *<sup>k</sup>*, so (Q<sup>×</sup> : *Gk*) = <sup>2</sup>*<sup>k</sup>* and <sup>|</sup>Q*Gk* <sup>|</sup> <sup>=</sup> <sup>2</sup>*<sup>k</sup>* <sup>+</sup> 1. Observe also that

$$1 = \frac{1}{p^k} + \frac{p^k - 1}{p^k} \in G\_k + G\_{k\prime}$$

hence [1]*Gk* ∈ 2 × [1]*Gk* , which means that C-char Q*Gk* = 1.

The following result provides a criterion for deciding whether certain hyperfields cannot be obtained via Krasner's quotient construction.

**Theorem 6.** *Every finite hyperfield F with* char *F* = ∞ *and* C-char *F* > 1 *is not a quotient hyperfield.*

**Proof.** Consider a finite hyperfield, which is a quotient hyperfield *KG*. Observe first that if char *KG* = ∞, then char *K* = ∞ by Lemma 2. Hence, *K* must be infinite. On the other hand, since *KG* is finite, *G* has a finite index in *K*×. From Theorem 1, we obtain that *G* − *G* = *K*, so in particular [1]*<sup>G</sup>* ∈ [1]*<sup>G</sup>* − [1]*G*. From the reversibility axiom, we obtain that [1]*<sup>G</sup>* ∈ [1]*<sup>G</sup>* + [1]*G*. This shows that C-char *KG* = 1.

In particular, the hyperfield that we have introduced in Example 4 cannot be obtained with Krasner's quotient construction.

## **5. The Strict Partial Order Induced by a Positive Cone**

We begin this section recalling the following definition.

**Definition 6.** *A* strict partial order *is a set S with an binary relation* <*, which is:*

*(O1) irreflexive (x* < *x for all x* ∈ *S);*

*(O2) asymmetric (x* < *y implies y* < *x for all x*, *y* ∈ *S);*

*(O3) transitive (x* < *y and y* < *z imply x* < *z for all x*, *y*, *z* ∈ *S).*

*A strict partial order* (*S*, <) *is called a* strict linear order *if for all x*, *y* ∈ *S one has x* < *y, y* < *x or x* = *y.*

In the theory of ordered fields, any positive cone *P* induces a strict linear order. This is defined as follows: *x* < *y* if and only if *y* − *x* ∈ *P*. In the hyperfield case, one can define the relation *x* < *y* as *y* − *x* ⊆ *P*. One then obtains a strict partial order. Indeed, *x* < *x* because 0 ∈/ *P* and if *x* < *y*, then *y* < *x* cannot hold since *P* ∩ −*P* = ∅. In order to show transitivity, take *x*, *y*, *z* ∈ *F*, such that *x* < *y* and *y* < *z*. We have to demonstrate that *x* < *z*. Since *y* − *x*, *z* − *y* ⊆ *P* and *P* + *P* ⊆ *P*, we obtain that

$$z - \mathfrak{x} \subseteq y - y + z - \mathfrak{x} = (y - \mathfrak{x}) + (z - y) \subseteq P.$$

Nevertheless, this strict partial order does not have to be linear, as the following example shows.

**Example 15.** *Consider the quotient hyperfield* Q(Q×)<sup>2</sup> *with its unique positive cone*

$$P := \{ [\mathfrak{x}]\_{(\mathbb{Q}^\times)^2} \mid \mathfrak{x} \in \mathbb{Q}^+ \}.$$

*Observe that since* 1 = 2 · 12 − 1 · 12 *and* 2 = 1 · 22 − 2 · 12*, we have that*

$$[1]\_{(\mathbb{Q}^\times)^2} \in [2]\_{(\mathbb{Q}^\times)^2} - [1]\_{(\mathbb{Q}^\times)^2} \quad \text{and} \quad [2]\_{(\mathbb{Q}^\times)^2} \in [1]\_{(\mathbb{Q}^\times)^2} - [2]\_{(\mathbb{Q}^\times)^2} \cdot$$

*Therefore, both* [2](Q×)<sup>2</sup> − [1](Q×)<sup>2</sup> *and* [1](Q×)<sup>2</sup> − [2](Q×)<sup>2</sup> *contain elements of P and thus* [1](Q×)<sup>2</sup> *and* [2](Q×)<sup>2</sup> *are incomparable with respect to the order relation associated to P, since P* ∩ −*P* = ∅*.*

In an ordered field *K*, the order relation < associated to a positive cone *P* is *compatible* with the addition of *K* in the sense that *a* < *b* implies that *a* + *c* < *b* + *c* for all *a*, *b*, *c* ∈ *K*. In the next example, we consider a real hyperfield *F*, such that *a* < *b* and *a* + *c* = *b* + *c* for some *a*, *b*, *c* ∈ *F*, where < is the order induced by a positive cone of *F*.

**Example 16.** *Consider the sign hyperfield* S *and its unique positive cone P* = {1}*. Observe that* 0 < 1*, but* 0 + 1 = {1} *and* 1 + 1 = {1}*.*

We now note that in the case of the sign hyperfield S, the strict partial order relation induced by its positive cone *P* = {1} is a strict linear order.

At this point, let us consider another example of the real hyperfield.

**Example 17** (Example 3.6 in [9])**.** *Consider the following cartesian product* {−1, 1} × Γ*, where* (Γ, +, 0, <) *is an ordered abelian group. Denote F* := {−1, 1} × Γ ∪ {0}*. Then, the tuple* (*F*, , ·, 0,(1, 0)) *is a hyperfield, with the hyperaddition defined as follows:*


*The result of the group multiplication* · *by 0 is defined to be 0, and for nonzero elements of F, we set:*

$$(s\_1\gamma\_1)\cdot(s\_2\gamma\_2) = (s\_1s\_2\gamma\_1+\gamma\_2).$$

*Moreover,* {(1, *γ*) | *γ* ∈ Γ} *is a positive cone in F.*

**Remark 9.** *The hyperfield F from the previous example is a quotient hyperfield. It is obtained as RE*+(*R*)*, where R is a real closed field and E*+(*R*) *is the group of totally positive units with respect to the natural valuation associated with the unique positive cone of R. For more details, we refer the reader to [9].*

One can observe that the positive cone of the real hyperfield that we have introduced in the above example also induces a strict linear order relation.

The property that the sign hyperfield and the real hyperfield of Example 17 have in common is that they are stringent hyperfields.

**Definition 7** ([30])**.** *A Krasner hyperfield F is said to be* stringent *if for all x*, *y* ∈ *F, we have that x* + *y is a singleton unless y* = −*x.*

In fact, we have the following general result.

**Proposition 10.** *Let F be a stringent real hyperfield with positive cone P. Then, the relation*

*a* < *b* ⇐⇒ *b* − *a* ⊆ *P* (*a*, *b* ∈ *F*)

*is a strict linear order relation on F.*

**Proof.** Take two distinct elements *a* = *b* of *F*. Then, *b* − *a* = {*c*} for some *c* ∈ *F*×. Therefore, either *c* ∈ *P*, in which case *a* < *b*, or *c* ∈ −*P*, in which case *b* < *a*. Hence, *a* and *b* are comparable and < is indeed a linear order.

To any strict partial order one can easily associate a directed graph. Let (*S*, <) be a strict partial order. The *(directed) graph associated to* < has *S* as its set of vertices, and an edge goes from a vertex *a* to a vertex *b* precisely when *a* < *b*. The reader should note that in the following illustrations, we do not draw the edges that can be deduced from the transitivity of <. For instance, for *S* = {*a*, *b*, *c*} with *a* < *b* < *c*, we draw the following graph

instead of

In the case of stringent real hyperfields, we obtain a linear order by Proposition 10. The directed graph obtained in this case can be found in Figure 1 below.

$$1\dots - \text{x} \longrightarrow \dots \longrightarrow -1 \longrightarrow \dots \longrightarrow \dots \longrightarrow 0 \longrightarrow \dots \longrightarrow 1 \longrightarrow \dots \longrightarrow \dots \longrightarrow \dots \longrightarrow \dots$$

**Figure 1.** The directed graph associated to a strict linear order.

In the case described in Example 15, the directed graph associated to the strict partial order induced by the positive cone is illustrated in Figure 2 below.

**Figure 2.** The directed graph associated to the strict partial order induced by the positive cone described in Example 15.

Indeed, in that case, one can demonstrate that if *a*, *b* ∈ *P* are two distinct elements, then they are not comparable.

In the following example, we consider a more complex situation.

**Example 18.** *Consider the field of rational functions over the real numbers* R(*X*)*. This field admits infinitely many positive cones. Below, we define a specific one, but our reasoning would apply to any of them. Every rational function h* ∈ R(*X*) *can be written uniquely in the following form:*

$$h = \mathfrak{x}^k \frac{f}{\mathfrak{z}'} \text{ where } f(0), \mathfrak{z}(0) \neq 0, k \in \mathbb{Z}.$$

*We consider the positive cone P* := {*h* ∈ R(*X*) | *<sup>f</sup>*(0) *<sup>g</sup>*(0) > 0}*. We set U* := {*h* ∈ R(*X*) | *k* = 0} *and* ∑(R(*X*)×)<sup>2</sup> := {*h* ∈ R(*X*) | *h* = *h*<sup>2</sup> <sup>1</sup> + ... + *<sup>h</sup>*<sup>2</sup> *<sup>n</sup> for some h*1, ... , *hn* ∈ R(*X*)×, *n* ∈ N}*. Consider the quotient hyperfield F* := R(*X*)*E*<sup>+</sup> *, where E*<sup>+</sup> := *U* ∩ ∑(R(*X*)×)2*. In particular, E*<sup>+</sup> ⊆ *P, since sums of non-zero squares are contained in any positive cones, and from Theorem 2 we obtain that F is real with the positive cone PE*<sup>+</sup> = {[*h*]*E*<sup>+</sup> | *h* ∈ *P*}*.*

*Take two elements* [*h*1], [*h*2] ∈ *F, such that*

$$h\_1 = \mathbf{x}^{k\_1} \frac{f\_1}{\mathfrak{g}\_1}, h\_2 = \mathbf{x}^{k\_2} \frac{f\_2}{\mathfrak{g}\_2} \text{ with } \frac{f\_1(0)}{\mathfrak{g}\_1(0)} > 0, \frac{f\_2(0)}{\mathfrak{g}\_2(0)} > 0.$$

*First, assume that k*<sup>1</sup> = *k*2*. Without a loss of generality, let k*<sup>1</sup> > *k*2*. We compute*

$$h\_2 - hh\_1 = \mathfrak{x}^{k\_2} \frac{f\_2 \mathfrak{z}\_1 \mathfrak{z} - \mathfrak{x}^{k\_1 - k\_2} f\_1 \mathfrak{z}\_2 f}{\mathfrak{z}\_1 \mathfrak{z}\_2 \mathfrak{z}}, \text{ where } h = \frac{f}{\mathfrak{z}} \in E^+.$$

*Then*

$$\frac{\left(f\_2\mathfrak{g}\_1\mathfrak{g} - \mathfrak{x}^{k\_1 - k\_2}f\_1\mathfrak{g}\_2f\right)(0)}{\left(\mathfrak{g}\_1\mathfrak{g}\_2\mathfrak{g}\right)(0)} = \frac{f\_2(0)}{\mathfrak{g}\_2(0)} > 0.$$

*Hence* [*h*2] − [*h*1] ⊆ *PE*<sup>+</sup> *, so* [*h*2] > [*h*1]*. Now assume that k* := *k*<sup>1</sup> = *k*2*. We have*

$$h\_2 - hh\_1 = \mathfrak{x}^k \frac{f\_2 \mathfrak{z}\_1 \mathfrak{z} - f\_1 \mathfrak{z}\_2 f}{\mathfrak{z}\_1 \mathfrak{z}\_2 \mathfrak{z}}, \text{ where } h = \frac{f}{\mathfrak{z}} \in E^+.$$

*Take h* = *<sup>f</sup> <sup>g</sup> , such that g*(0) = <sup>1</sup> *and f*(0) <sup>&</sup>lt; (*f*<sup>2</sup> *<sup>g</sup>*1)(0) (*f*<sup>1</sup> *<sup>g</sup>*2)(0)*. Then*

$$\frac{(f\_2g\_1g - f\_1g\_2f)(0)}{(g\_1g\_2g)(0)} > 0.$$

*Hence,* ([*h*2] − [*h*1]) ∩ *PE*<sup>+</sup> = ∅*. On the other hand, let g*(0) = 1 *and f*(0) > (*f*<sup>2</sup> *<sup>g</sup>*1)(0) (*f*<sup>1</sup> *<sup>g</sup>*2)(0)*. Then*

$$\frac{(f\_2g\_1g - f\_1g\_2f)(0)}{(g\_1g\_2g)(0)} < 0.$$

*Hence,* ([*h*2] − [*h*1]) ∩ −*PE*<sup>+</sup> = ∅*, so* ([*h*1] − [*h*2]) ∩ *PE*<sup>+</sup> = ∅*. This means that* [*h*1] *and* [*h*2] *are incomparable with respect to the partial order induced by PE*<sup>+</sup> *.*

*We illustrate in Figure 3 below the graph associated to the strict partial order induced by P on* R(*X*)*.*

**Figure 3.** The graph associated to the strict partial order induced by *P* on R(*X*).

*The nodes situated above the central node labelled by 0 correspond to elements of P. The nodes below correspond to elements of* −*P. Each level of this graph, which is above the central node labelled by 0, corresponds to an integer k. For instance, the level of the node labelled by 1 consists of all the nodes situated on the left and on the right of the node labelled by 1 and corresponds to the integer k* = 0*. The levels below this level correspond to positive integers k* > 0 *and the levels above to negative integers k* < 0*. Similarly, each level below the central node labelled by 0 correspond to an integer. There are no edges between any two nodes of the same level, as they are incomparable. If two nodes are in different levels, then they are connected in the upwards direction.*

#### **6. Further Research**

In this paper, we have studied the notion of the positive cone in hyperfields. We have investigated the (directed) graph associated to the strict partial order induced by a positive cone in a hyperfield in some examples. What we have observed suggests a particular structure of this graph (that of a linear order and that of a star, as in Example 15, or a combination of these two, as in Example 18).

Moreover, we have considered the characteristic and the C-characteristic of hyperfields, and we have demonstrated how these interact to produce interesting results in the theory of hyperfields. In particular, we have obtained Theorem 6, which gives a criterion for deciding whether a given finite hyperfield cannot be obtained via Krasner's quotient construction.

We have demonstrated that any positive integer larger than 1 can be realized as the characteristic of an infinite hyperfield (Theorem 3). We ask if it is possible to realize any characteristic with finite hyperfields as well. To try to answer this question, we have initially focused on the finite hyperfields of the form (F*p*)*<sup>G</sup>* and developed an algorithm to compute their characteristic. At this point, we can provide the data in Table 1 below.


**Table 1.** Finite hyperfields of non-prime characteristic ≤ 12.

The characteristic of a hyperfield of the form (F*p*)*<sup>G</sup>* depends on *p* and on the multiplicative subgroup *G* of F<sup>×</sup> *<sup>p</sup>* . Since *G* is a cyclic group, we conclude that char(F*p*)*<sup>G</sup>* depends on the prime number *p* and on the choice of a divisor *d* of *p* − 1. This means that the number *N* of possible hyperfields increases very fast with *p*. For example, in the case of characteristic 12, before finding the hyperfield (F62323)&28216', the algorithm would a priori have to generate and check the hyperfields corresponding to 6262 prime numbers, which gives a total of *N* = 449,569 possibilities. Nevertheless, we can use Proposition 7 to substantially reduce the number of cases to be considered.

For example, if we are looking for a hyperfield of characteristic 12, then we would assume that char(F*p*)*<sup>G</sup>* = 12, which by Proposition 7 cannot hold if |*G*| is divisible by some prime number < 12. Hence, we can restrict our attention to those prime numbers *p*, such that *p* − 1 is divisible at least by one prime number ≥ 13. Moreover, for these primes *p*, we can restrict our choice of the divisor of *p* − 1 to those *d*, which are not divisible by primes < 12. These restrictions reduce the number of hyperfields to be checked by the algorithm to 9871, which is approximately 2.19% of *N*.

#### **Remark 10.** *We know that* char(F*p*)*<sup>G</sup>* = 14 *for all prime numbers p* ≤ *160,000.*

An analogous problem can be posed for the realization of C-characteristics, as Theorem 5 provides only infinite hyperfields. For the hyperfields of the form (F*p*)*G*, our algorithm provided the data in Table 2 below.


**Table 2.** Finite hyperfields of non-prime C-characteristic ≤ 10.

**Remark 11.** *We know that* C-char(F*p*)*<sup>G</sup>* = 12 *for all prime numbers is p* ≤ 152897*.*

Our algorithm does not consider hyperfields of the form (F*p<sup>k</sup>* )*<sup>G</sup>* with *k* > 1.

Another natural question is if it is possible to somehow generalise Example 4 to construct finite real hyperfields of cardinality > 7 with C-characteristic > 1, thus providing further examples of non-quotient hyperfields.

**Author Contributions:** Conceptualization, D.E.K., A.L. and H.S.; methodology, D.E.K., A.L. and H.S.; investigation, D.E.K., A.L. and H.S.; writing—original draft preparation, D.E.K., A.L. and H.S.; writing—review and editing, D.E.K., A.L. and H.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors would like to express their gratitude to Franz-Viktor and Katarzyna Kuhlmann and to the anonymous referees for their suggestions and remarks which helped to improve the paper significantly. Many thanks also to Irina Cristea for encouraging us to write this manuscript.

**Conflicts of Interest:** The authors declare no conflict of interest.
