**1. Introduction**

In this paper, we study Krasner hyperfields. These structures are a generalization of the concept of the field, where the addition is allowed to be a multivalued operation, i.e., *x* + *y* in general denotes a subset and not only an element. Apart from the applications for which they have been introduced in [1] by Krasner, recently, these structures have arisen naturally in several mathematical contexts. For instance, Viro in [2] used hyperfields in tropical geometry and Lee in [3] studied these structures in connection to the model theory of valued fields. Regarding hyperfields, it is certainly also worth mentioning the work of Connes and Consani in number theory [4,5].

Since hyperfields represent a generalization of the concept of a field, it is natural to ask which classical notions and theorems of the theory of fields can be generalized to the theory of hyperfields. M. Marshall in [6] started the investigation towards a theory of real hyperfields, generalizing the Artin-Schreier theory of real fields (for a general reference on the latter, we refer the reader to [7]). The work of Marshall provided the basis for the investigations made later in [8] and in [9]. In the sections below, real hyperfields will be studied further.

Historically, a subhyperfield *L* of a hyperfield *F* is required to be closed under the multivalued addition in the sense that *x* + *y* ⊆ *L* for all *x*, *y* ∈ *L*. However, Jun in (Definition 2.4) [10], felt the need for a less restrictive notion and started to talk about a multivalued operation, which can be "induced" by certain subsets. In Section 3, we take a model theoretical point of view (encoding the multivalued operation + via the ternary relation *z* ∈ *x* + *y*) to justify Jun's feeling and precisely define the notion of the multivalued operation induced by a subset (see also [9]). This leads us to the notion of relational subhyperfields. The interest for relational subhyperfields is motivated by the fact that they correspond to the submodels of hyperfields in a natural first-order language, which we describe in Section 3.

**Citation:** Kedzierski, D.E.; Linzi, A.; Stojałowska, H. Characteristic, C-Characteristic and Positive Cones in Hyperfields. *Mathematics* **2023**, *11*, 779. https://doi.org/10.3390/ math11030779

Academic Editor: Emeritus Mario Gionfriddo

Received: 27 December 2022 Revised: 20 January 2023 Accepted: 30 January 2023 Published: 3 February 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

The notion of this characteristic is fundamental in classical field theory and, when it is finite, it can only be a prime number. Moreover, the characteristic of a field is preserved by subfields. In this paper, we will demonstrate that a natural generalization of the notion of characteristic for hyperfields (see Definition 3 below), does not have to behave in the same way: we prove that the characteristic of a hyperfield does not have to be preserved by relational subhyperfields (Example 13), and that any integer greater than 1 can be realized as the characteristic of some hyperfield (Theorem 3).

In addition, we study an alternative notion of the characteristic for hyperfields, known as the C-characteristic, which in the case of fields coincides with the usual characteristic. Moreover, this quantity is not preserved by relational subhyperfields (Example 14), and we prove in Theorem 4 that any positive integer can be realized as the C-characteristic of some real hyperfield. In addition, we demonstrate how useful the interplay between these two notions of characteristic can be by providing a criterion (Theorem 6) for deciding whather certain hyperfields cannot be obtained via Krasner's quotient construction (see [11,12]). We apply this result to the finite real hyperfield of Example 4.

In Section 5, we define a strict partial order relation induced by a positive cone in a real hyperfield and study the associated directed graph in various interesting examples.

## **2. Preliminaries**

In this section, we provide an overview of the definitions and facts that are necessary for the rest of the paper, with several examples.

#### *2.1. Hyperfields*

Let *H* be a non-empty set and P(*H*) be its power-set. A *multivalued operation* + on *H* is a function which associates with every pair (*x*, *y*) ∈ *H* × *H* an element of P(*H*), denoted by *x* + *y*. A *hyperoperation* + on *H* is a multivalued operation, such that *x* + *y* = ∅ for all *x*, *y* ∈ *H*. If + is a multivalued operation on *H* = ∅, then for *x* ∈ *H* and *A*, *B* ⊆ *H*, we set

$$A + B := \bigcup\_{a \in A, b \in B} a + b,\tag{1}$$

*A* + *x* := *A* + {*x*} and *x* + *A* := {*x*} + *A*. If *A* or *B* is empty, then so is *A* + *B*.

A *hypergroup* can be defined as a non-empty set *H* with a multivalued operation +, which is *associative* (see Definition 1 (CH1) below) and *reproductive* (i.e., *x* + *H* = *H* + *x* = *H* for all *x* ∈ *H*). This notion was first considered by F. Marty in [13–15]. Let us mention [16] for an extended historical overview and [17,18] for a description of some applications.

If (*H*, +) is a hypergroup, then it follows that + is a hyperoperation. Indeed, suppose that *x* + *y* = ∅ for some *x*, *y* ∈ *H*. Then,

$$H = \mathbf{x} + H = \mathbf{x} + (y + H) = (\mathbf{x} + y) + H = \mathcal{Q} + H = \mathcal{Q}\_r$$

which is excluded (cf. (Theorem 12) in [16]).

The following special class of hypergroups will be of interest for us.

**Definition 1.** *A canonical hypergroup is a tuple* (*H*, +, 0)*, where H* = ∅*,* + *is a multivalued operation on H and* 0 *is an element of H such that the following axioms hold:*

$$\text{(CH1)}\; + \text{ is associative, i.e., } (\text{x} + \text{y}) + z = \text{x} + (\text{y} + z) \text{ for all } \text{x}, \text{y}, z \in \text{H}, \text{}$$


The axiom (CH4) is known as the *reversibility* axiom.

**Remark 1.** *Some authors (see e.g., (Definition 1.2) in [19]) define canonical hypergroups requiring explicitly that x* + 0 = {*x*} *for all x* ∈ *H. However, as already noted in (Section III, (b)) [12], this property follows from (CH3) and (CH4). Indeed, suppose that y* ∈ *x* + 0 *for some x*, *y* ∈ *H. Then,* 0 ∈ *y* − *x by (CH4). Presently, y* = *x follows from the uniqueness required in (CH3). For this reason, we call* 0 *the neutral element for* +*.*

**Remark 2.** *The multivalued operation of a canonical hypergroup* (*H*, +, 0) *is reproductive. To observe this fix, a* ∈ *H. For x* ∈ *H* + *a, there exists h* ∈ *H, such that x* ∈ *h* + *a* ⊆ *H, demonstrating that H* + *a* ⊆ *H. For the other inclusion, take x* ∈ *H, then*

$$\mathbf{x} \in \mathfrak{x} + \mathbf{0} \subseteq \mathfrak{x} + (a - a) = (\mathfrak{x} - a) + a\_r$$

*so there exists h* ∈ *x* − *a* ⊆ *H, such that x* ∈ *h* + *a* ⊆ *H* + *a. It follows, in particular, that* + *is a hyperoperation.*

The following structures have been considered by Krasner in [1,11].

**Definition 2.** *A hyperfield is a tuple* (*F*, +, ·, 0, 1)*, which satisfies the following axioms:*

*(HF1)* (*F*, +, 0) *is a canonical hypergroup; (HF2)* (*F* \ {0}, ·, 1) *is an abelian group and x* · 0 = 0 · *x* = 0 *for all x* ∈ *F; (HF3) the operation* · *is distributive with respect to* +*. That is, for all x*, *y*, *z* ∈ *F,*

$$
\mathbf{x} \cdot (\mathbf{y} + \mathbf{z}) = \mathbf{x} \cdot \mathbf{y} + \mathbf{x} \cdot \mathbf{z},
$$

*where for x* ∈ *F and A* ⊆ *F, we have set*

$$xA := \{xa \mid a \in A\}.$$

*We denote the multiplicative group of a hyperfield F by F*×*.*

**Remark 3.** *One can think about other kinds of hyperfields by modifying the axioms that the additive hypergroup should fulfill. The hyperfields for which the additive hypergroup is a canonical hypergroup (as above) are commonly known as Krasner hyperfields. As we mentioned in the introduction, we will consider only these kinds of structures and call them simply hyperfields, as indicated in the above definition.*

**Remark 4.** *The double distributivity law, i.e.,*

$$(a+b)(c+d) = a\cdot c + a\cdot d + b\cdot c + b\cdot d$$

*does not hold in general in hyperfields. However, the fact that the inclusion*

$$(a+b)(c+d) \subseteq ca+ad+bc+bd$$

*holds is not difficult to verify from the definitions and has been known for long time. For instance, it was stated without proof in [12,20]. A proof has been written in (Theorem 4B) of [2].*

*By induction, it is straightforward to show that*

$$(a\_1 + \dots + a\_n)(b\_1 + \dots + b\_m) \subseteq a\_1b\_1 + a\_1b\_2 + \dots + a\_1b\_m + a\_2b\_1 + \dots + a\_nb\_m$$

*for any natural numbers n*, *m.*

Examples of hyperfields can be obtained in the following way. Let *K* be a field and *G* a subgroup of *K*×. For *x* ∈ *K*×, we denote by [*x*]*<sup>G</sup>* the coset *xG* ∈ *K*×/*G*. Further, let [0]*<sup>G</sup>* denote the singleton containing only 0 ∈ *K*. Then, the *quotient hyperfield* of the *K* modulo *G* is the set *KG* := *K*×/*G* ∪ {[0]*G*} with the hyperoperation

$$[\mathfrak{x}]\_{\mathcal{G}} + [\mathfrak{y}]\_{\mathcal{G}} := \{ [\mathfrak{x} + \mathfrak{y}\mathfrak{g}]\_{\mathcal{G}} \mid \mathfrak{g} \in \mathcal{G} \} \quad (\mathfrak{x}, \mathfrak{y} \in \mathbb{K})$$

and the operation

$$[\mathfrak{x}]\_G[\mathfrak{y}]\_G := [\mathfrak{x}\mathfrak{y}]\_G \quad (\mathfrak{x}, \mathfrak{y} \in K).$$

This construction was demonstrated to always yield a hyperfield by Krasner himself in [11].

Not all hyperfields can be obtained in this way, i.e., there are hyperfields that are not quotient hyperfields. This has been demonstrated by Massouros in [12], who then improved his results in [21]. Afterwards, Baker and Jin in [22] have found the following theorem of Bergelson-Shapiro and Turnwald useful to prove that certain hyperfields cannot be obtained via Krasner's quotient construction.

**Theorem 1** (Theorem 1.3 in [23]; Theorem 1 in [24])**.** *If F is an infinite field and G is a subgroup of F*<sup>×</sup> *of the finite index, then G* − *G* = *F.*

The next observation gives a necessary condition for a hyperfield to be a field. This fact is an immediate corollary of a result already noted in [20] (p. 369). We wish to state it for later reference and we will take the opportunity to write a quick proof.

**Proposition 1** ([20])**.** *Let F be a hyperfield. If* 1 − 1 = {0}*, then F is a field.*

**Proof.** By distributivity (HR3), our assumption implies that *a* − *a* = {0} for all *a* ∈ *F*. Let *a*, *b* ∈ *F*<sup>×</sup> and take *x*, *y* ∈ *a* + *b*. Then

$$x - y \subseteq (a + b) - (a + b) = (a - a) + (b - b) = \{0\},$$

so *x* = *y* and *a* + *b* is always a singleton in *F*.

In the literature, one can find different interesting notions of the characteristic for hyperfields. Maybe the oldest is the one introduced by Mittas (a student of Krasner, see [20]). Later, Viro in [2] highlighted two other possible such notions. All these coincide with the usual characteristic of fields when the addition of the hyperfield under consideration is singlevalued (i.e., the hyperfield is in fact a field). However, they can be different in the general case. In this paper, we focus on the latter two notions, highlighted by Viro and which appear also in the work of P. Gładki [8] as well as in [4,5].

**Definition 3.** *Let F be a hyperfield. We set* 1 ×*<sup>F</sup>* 1 := {1} *and for n* ≥ 2

$$n \times\_F 1 := \underbrace{1 + \dots + 1}\_{n \text{ times}} \cdot 1$$

*If there is no risk of confusion, we simply write n* × 1 *in place of n* ×*<sup>F</sup>* 1*.*

