**3. On Relational Subhyperfields**

From the point of view of model theory (for a general reference, see, e.g., [27]), standard operations are usually encoded via binary function symbols. The same is not possible for multivalued operations, since function symbols are classically interpreted in a structure as functions with values in the universe of the structure and not in its power-set. Nevertheless, as it was observed in [3], we can use the ternary relation *z* ∈ *x* + *y* to encode a multivalued operation +. Thus, a hyperfield is naturally a structure on the first-order language having two constant symbols 0, 1 for the neutral elements, a binary function symbol for the multiplication and a ternary relation symbol to encode the hyperoperation. Considering hyperfields as structures on this language, the general model theoretical notion of the *submodel* leads to the following definition (we provide more details in Remark 7, below).

**Definition 5.** *Let F be a hyperfield. A subset L* ⊆ *F is a relational subhyperfield of F if* 0 ∈ *L, L* \ {0} *is a (multiplicative) subgroup of F* \ {0} *and with the* induced *multivalued operation, which is defined as*

$$\mathfrak{x} +\_L \mathfrak{y} := (\mathfrak{x} +\_F \mathfrak{y}) \cap L \quad (\mathfrak{x}, \mathfrak{y} \in L),$$

*we have that* (*L*, 0, 1, ·, +*L*) *is a hyperfield.*

**Remark 6.** *Note that a priori, the multivalued operation induced by a subset L of a hyperfield F might not be a hyperoperation, as it may admit empty values, i.e.,* (*x* +*<sup>F</sup> y*) ∩ *L* = ∅ *may hold for some x*, *y* ∈ *L. If the latter is the case, then* (*L*, 0, 1, ·, +*L*) *is certainly not a hyperfield; in particular, L would not be a relational subhyperfield of F, by definition.*

**Remark 7.** *Presently, we will motivate the study of the notion introduced in Definition 5 above. A first-order language* L *consists of relation, function and constant symbols. A structure on* L

*is (informally) a universe (i.e., a non-empty set) where any (well-formed) expression over* L *is interpreted (for a formal definition, see, e.g., (Section 1.5) in [27]). A first-order theory* T *over* L *is a list of axioms, i.e., expressions, which can be true or false when interpreted in a certain structure. A structure in which all the axioms of* T *are true is called a* model *of* T*. For example, the additive group of integers or the cyclic group of order* 5 *are models of the theory of groups, as is any other group. The field of rational numbers or the field of complex numbers are models of the theory of fields, as is any other field. Complete graphs or star graphs are models of the theory of graphs, as is any other graph.*

*Given a structure S on* L *and a non-empty subset A of S, it is possible to restrict to A the interpretations in S of the symbols of* L*. In this way, A itself becomes a structure on* L*, and A is called a substructure of S (cf. (Section 2.3) in [27]). One of the main differences between an n-ary (n* ∈ N*) relation symbol, interpreted in S as a relation R* ⊆ *S<sup>n</sup> and an n-ary function symbol, interpreted in S as a function f* : *S<sup>n</sup>* → *S, is that, when restricted to A, the latter has to satisfy the requirement f*(*a*¯) ∈ *A for all a*¯ ∈ *An; because an n-ary function symbol must, by definition, be interpreted on A as a function f* : *A<sup>n</sup>* → *A. On the other hand, the restricted relation on A is just defined to be R* ∩ *An, and there are no further requirements to be satisfied.*

*With the notation introduced above, let us stress that under the assumption that S is a model of* T*, it does not follow in general that A is a model of* T *too. For example, we may restrict the operations of the field of real numbers to the set of integers* Z*, but we do not obtain a field. If the substructure A happens to be itself a model of* T*, then it is called a* submodel *of S. For example, the field of rational numbers is a submodel of the field of real numbers. If the axioms of* T *are all (equivalent to) universal axioms (i.e., they can be written using only the* ∀ *quantifier), then substructures are automatically submodels (this is a consequence of, e.g., (Theorem 3.3.3) in [27]).*

*Presently, let* T*hf be the theory given by the axioms of hyperfields (see Definitions 2 and 1) written, encoding the symbol* + *with the ternary relation z* ∈ *x* + *y. Then, a hyperfield F is a model of* T*hf and L is a relational subhyperfield of F if and only if L is a submodel of F.*

*In mathematics, it is customary to call* sub*objects the submodels of a model of the theory of (those) objects. For example, subgroups are submodels of a group; subfields are submodels of a field; subgraphs are submodels of a graph. However,* subhyperfields *are historically defined as subsets L of a hyperfield* (*F*, 0, 1, ·, +)*, such that* 0, 1 ∈ *L, x*−<sup>1</sup> ∈ *L for all x* ∈ *L* \ {0}*, xy* ∈ *L and x* + *y* ⊆ *L for all x*, *y* ∈ *L (cf. [12,20,21]). This definition can be traced back to the definition of the subhypergroup already present in, e.g., Definition 2 and the subsequent remark in [28].*

*While it is clear that any subhyperfield L of a hyperfield F is a relational subhyperfield of F with* +*<sup>L</sup>* = +*, there are examples of relational subhyperfields L , which do not satisfy the condition x* + *y* ⊆ *L for all x*, *y* ∈ *L (see Examples 9 and 10, below). Thus, in this setting and perhaps for historical reasons, the use of the prefix "sub" seems to not match the common practice. Nevertheless, our point of view is based on the choice of encoding hyperoperations with relations; thus, we chose the name* relational *subhyperfield to distinguish our notion from the traditional one.*

**Example 9.** *Consider the hyperfield F* := *HF*<sup>521</sup> *of Example 2 and its subset L* := {−1, 0, 1}*. Equip L with the multivalued operation* +*L, as in Definition 5. Then, L is the sign hyperfield (cf. Example 1); in particular, it is a relational subhyperfield of F. Note that* 1 ∈ *L but* 1 +*<sup>F</sup>* 1 = {1, *a*, −*a*} ⊆ *L.*

**Example 10.** *Consider the hyperfield F* := *HF*<sup>56</sup> *of Example 3 and its subset L* := {−1, 0, 1}*. Equip L with the multivalued operation* +*L, as in Definition 5. Then, L is the the finite field with* 3 *elements* F3*; in particular, it is a relational subhyperfield of F. Note that* 1 ∈ *L but* 1 +*<sup>F</sup>* 1 = {−1, *a*, −*a*} ⊆ *L.*

One might think that the subset {−1, 0, 1} is a relational subhyperfield of any hyperfield. The next examples demonstrate that this is not the case.

**Example 11.** *Let F be a field (considered as a hyperfield) with* char *F* > 3*. Then, L* := {−1, 0, 1} *is not a relational subhyperfield of F, since* −1, 0, 1 /∈ 1 + 1 *and so* (1 + 1) ∩ *L* = ∅*.*

**Example 12.** *Consider the hyperfield F from Example 4 and its subset L* := {−1, 0, 1}*. Then,* 1 +*<sup>L</sup>* 1 = ∅ *and thus L is not a relational subhyperfield of F.*

The following easy observation will be useful later.

**Lemma 1.** *Let F be a hyperfield and L be a relational subhyperfield of F. For all n* ∈ N*, we have that*

$$n \times\_L 1 \subseteq n \times\_F 1.$$

**Proof.** Let us show this by induction on *n*. The base step is clear. For the induction step, given *n* > 1, we compute

$$\begin{aligned} n \times\_L 1 &= \bigcup\_{\mathbf{x} \in I\_{n-1}(L)} \mathbf{x} +\_L 1 = \bigcup\_{\mathbf{x} \in I\_{n-1}(L)} (\mathbf{x} +\_F 1) \cap L \\ &\subseteq \bigcup\_{\mathbf{x} \in I\_{n-1}(L)} (\mathbf{x} +\_F 1) \subseteq \bigcup\_{\mathbf{x} \in I\_{n-1}(F)} (\mathbf{x} +\_F 1) = n \times\_F 1, \end{aligned}$$

where we have used the induction hypothesis (*n* − 1) ×*<sup>L</sup>* 1 ⊆ (*n* − 1) ×*<sup>F</sup>* 1.

#### *3.1. Characteristic of Relational Subhyperfields*

It is not difficult to observe that if *K* is a field, considered as a hyperfield, then all relational subhyperfields of *K* are (traditional) subhyperfields of *K* and they coincide with the subfields of *K*. In field theory, the characteristic of a subfield coincides with the characteristic of the upper field. Nevertheless, in the multivalued setting, the same might not hold.

**Example 13.** *Consider the hyperfield F* = *HF*<sup>521</sup> *from Example 2. We have observed that* char *F* = 4 *and that the sign hyperfield* S *is a relational subhyperfield of F (cf. Example 9). Since* S *is real, we have that* char S = ∞*. Thus, the strict inequality* char *F* < char S *holds.*

On the basis of Lemma 1, we can demonstrate that the characteristic of a hyperfield is not greater than the characteristic of any of its relational subhyperfields.

**Proposition 2.** *Let F be a hyperfield and L be a relational subhyperfield of F. Then,* char *F* ≤ char *L.*

**Proof.** Directly from Lemma 1, we can argue that if char *L* < ∞, then also char *F* < ∞ and char *F* ≤ char *L*. Otherwise, char *L* = ∞ is automatically not smaller than char *F*.

As we have observed in Example 13, the strict inequality might occur. In that example, we considered the hyperfield *HF*521, which has characteristic 4. We now prove that that is the minimal characteristic that a hyperfield can have in order to produce such a situation.

**Proposition 3.** *Let F be a hyperfield and L be a relational subhyperfield of F. If* char *F* ∈ {2, 3}*, then* char *L* = char *F.*

**Proof.** As we will observe, this follows from the fact that, since *L* is a relational subhyperfield of *F*, we have that 0, −1 ∈ *L*.

Assume first that char *F* = 2. By assumption, we have 0 ∈ 1 +*<sup>F</sup>* 1 and since 0 ∈ *L*, we also have 0 ∈ (1 +*<sup>F</sup>* 1) ∩ *L* = 1 +*<sup>L</sup>* 1, showing that char *L* = 2 = char *F* in this case.

Now assume that char *F* = 3. This means that 0 ∈ 1 +*<sup>F</sup>* 1 +*<sup>F</sup>* 1 and hence −1 ∈ 1 +*<sup>F</sup>* 1. Since −1 ∈ *L*, we obtain that −1 ∈ (1 +*<sup>F</sup>* 1) ∩ *L* = 1 +*<sup>L</sup>* 1 and 0 ∈ 1 +*<sup>L</sup>* 1 +*<sup>L</sup>* 1 follows, since 0 ∈ *L*. Hence, char *L* ≤ 3 = char *F*. By Proposition 2, we obtain the equality.

#### *3.2. C-Characteristic of Relational Subhyperfields*

A result analogous to Proposition 2 for the C-characteristic follows similarly as above from Lemma 1.

**Proposition 4.** *Let F be a hyperfield and L be a relational subhyperfield of F. Then,* C-char *F* ≤ C-char *L.*

**Proof.** By Lemma 1, we have that if C-char *L* < ∞, then also C-char *F* < ∞ and C-char *F* ≤ C-char *L*. Otherwise, C-char *L* = ∞ is automatically not smaller than C-char *F*.

Also for C-characteristics, the strict inequality might hold, as the following example shows.

**Example 14.** *Consider the hyperfield F* = *HF*<sup>56</sup> *from Example 3. We have observed that* C-char *F* = 2 *and that the finite field* F<sup>3</sup> *of cardinality* 3 *is a relational subhyperfield of F (cf. Example 10). We have that*

C-char F<sup>3</sup> = 3 > 2 = C-char *F*.

Let *F* be a hyperfield and *L* be a relational subhyperfield of *F*. Again, similarly as we observed for the characteristic, since 1 ∈ *L*, we have that if C-char *F* = 1, then C-char *L* = 1 as well. Let us state this result.

**Proposition 5.** *Let F be a hyperfield with* C-char *F* = 1 *and L be a relational subhyperfield of F. Then,* C-char *L* = C-char *F.*

Thus, in Example 14, the C-characteristic of *F* has the minimal value which can produce the strict inequality.

#### **4. Realizing Characteristics and C-Characteristics**

In this section, we deal with the problem of realizing a given positive integer as the characteristic or the C-characteristic of some hyperfield.
