**Remark 5.**


*(iii) For any hyperfield F we have that* C-char *F* ≤ char *F. Indeed, if* 0 ∈ *n* ×*<sup>F</sup>* 1*, then*

$$(n+1)\times\_F 1 = n\times\_F 1 + 1 = \bigcup\_{a \in n\times\_F 1} (a+1) \supseteq 0 + 1 = \{1\}\_F$$

*so* C-char *F* ≤ *n.*

Let us describe some examples of finite hyperfields that will be of interest for us.

**Example 1.** *The sign hyperfield* S *is the set* {−1, 0, 1} *with the hyperoperation and operation defined by the following tables:*


*As it was noted in, e.g., [25], (Page 22 (b)) this hyperfield is isomorphic to the quotient hyperfield of an ordered field (e.g., the field* R *of real numbers) over the multiplicative subgroup given by its positive cone (in the example of real numbers that is the multiplicative subgroup* R><sup>0</sup> *of positive real numbers). This hyperfield has the C-characteristic 1 and characteristic* ∞*.*

**Example 2.** *This example is the hyperfield generated by the algorithm presented in [26] and called HF*521*. It has five elements* {0, 1, −1, *a*, −*a*} *and its multiplicative group is isomorphic to* Z4*. The table for the hyperoperation is as follows:*


*As it was noted in [26], this hyperfield is isomorphic to the quotient hyperfield of the finite field with 29 elements* F<sup>29</sup> *over the multiplicative subgroup of* F<sup>×</sup> <sup>29</sup> *generated by 7. This hyperfield has C-characteristic 1 and characteristic 4.*

**Example 3.** *This example is the hyperfield generated by the algorithm presented in [26] and called HF*56*. It has five elements,* {0, 1, −1, *a*, −*a*}*. The table for the hyperoperation is as follows:*


*The multiplicative group is isomorphic to* Z<sup>2</sup> × Z2*.*

*Note that this hyperfield is not a quotient hyperfield. Indeed, since its multiplicative group is not cyclic, it cannot be a quotient of a finite field. Moreover,* 1 − 1 *does not coincide with the whole hyperfield and thus Theorem 1 ensures that it cannot be a quotient of an infinite field. We observe that this hyperfield has the C-characteristic 2 and characteristic 3.*

**Example 4.** *Consider the set F* = {0, 1, −1, *a*, −*a*, *a*2, −*a*2} *and its subset P* := {1, *a*, *a*2}*. We define on F the following hyperaddition:*


*The multiplicative group is isomorphic to* Z6*. One can demonstrate that F is a hyperfield with straightforward direct computations. In Section 2.2, below, we will study some properties of the subset P. Moreover, we will prove that this cannot be obtained with Krasner's quotient construction. Note that F has the C-characteristic 2 and its characteristic is* ∞*.*

#### *2.2. Real Hyperfields*

The Artin-Schreier theory of ordered fields, which led Artin to his solution of Hilbert's 17th problem (see [7] for details), was generalised to hyperfields in [6]. Let us recall some basic facts and definitions.

**Definition 4.** *Let F be a hyperfield. A subset P* ⊆ *F is called a* positive cone *in F if*

*(P1) P* + *P* ⊆ *P*,*; (P2) P* · *P* ⊆ *P*,*; (P3) P* ∩ −*P* = ∅,*; (P4) P* ∪ −*P* = *F*×.*.*

*A hyperfield F is called real if it admits a positive cone.*

Note that 1 ∈ *P* for every positive cone *P* in a hyperfield *F*. Indeed, from the axioms, either 1 or −1 belongs to *P*, but not both. If −1 ∈ *P*, then 1 = (−1) · (−1) ∈ *P* again

by the axioms. Hence, −1 cannot be in *P*, implying our assertion. This implies that the characteristic of a real hyperfield must be ∞, since *P* + *P* ⊆ *P* and 0 /∈ *P*.

**Example 5.** *The sign hyperfield* S *introduced in Example 1 above is real with the positive cone* {1}*. This is clearly the unique possible positive cone of* S*.*

**Example 6.** *The hyperfield that we have introduced in Example 4 is real with the positive cone P* := {1, *a*, *a*2}*. Again, this can be observed by straightforward computations.*

**Example 7.** *The hyperfield which we have introduced in Example 2 is not real. Indeed, if P would be a positive cone, then since* 1 ∈ *P we must have* 1 + 1 ⊆ *P. On the other hand,* 1 + 1 = {1, *a*, −*a*} *and by the axioms only one among a and* −*a can belong to P.*

*A similar reasoning yields that the hyperfield that we have introduced in Example 3 is not real.*

Let us now briefly recall some results which have been proved in [9]. As in that paper, we will denote by X (*F*|*G*) the set of all positive cones in *F*, which contain some subset *G* of *F* and by X (*F*) the set of all positive cones in *F*.

**Theorem 2** ([9])**.** *Let P be a positive cone of a field K and assume that a multiplicative subgroup G of K is contained in P. Consider the quotient hyperfield KG* = {[*x*]*<sup>G</sup>* | *x* ∈ *K*}*.*


**Example 8.** *Consider the quotient hyperfield <sup>F</sup>* := Q(Q×)<sup>2</sup> *, where* (Q×)<sup>2</sup> *is the set of nonzero squares in* Q*. The set* Q<sup>+</sup> := ∑(Q×)<sup>2</sup> *of the sums of nonzero squares in* Q *is the unique positive cone of* Q*. Hence, by assertion* (*iii*) *of Theorem 2, F is real and has a unique positive cone <sup>P</sup>*(Q×)<sup>2</sup> = {[*x*](Q×)<sup>2</sup> | *<sup>x</sup>* ∈ Q+}*.*
