Abstract
In every hypergroup, the equivalence classes modulo the fundamental relation are the union of hyperproducts of element pairs. Making use of this property, we introduce the notion of height of a -class and we analyze properties of hypergroups where the height of a -class coincides with its cardinality. As a consequence, we obtain a new characterization of 1-hypergroups. Moreover, we define a hierarchy of classes of hypergroups where at least one -class has height 1 or cardinality 1, and we enumerate the elements in each class when the size of the hypergroups is , apart from isomorphisms.
1. Introduction
The term algebraic hyperstructure designates a suitable generalization of a classical algebraic structure, like a group, a semigroup, or a ring. In classical algebraic structures, the composition of two elements is an element, while in an algebraic hyperstructure the composition of two elements is a set. In the last few decades, many scholars have been working in the field of algebraic hyperstructures, also called hypercompositional algebra. In fact, algebraic hyperstructures have found applications in many fields, including geometry, fuzzy/rough sets, automata, cryptography, artificial intelligence and probability [], relational algebras [], and sensor networks [].
Certain equivalence relations, called fundamental relations, introduce natural correspondences between algebraic hyperstructures and classical algebraic structures. These equivalence relations have the property of being the smallest strongly regular equivalence relations such that the corresponding quotients are classical algebraic structures [,,,,,,,]. For example, if is a hypergroup, then the fundamental relation is transitive [,,] and the quotient set is a group. Moreover, if : is the canonical projection, then the kernel is a subhypergroup, which is called the heart of . The heart of a hypergroup plays a very important role in hypergroup theory because it gives detailed information on the partition of H determined by the relation , since for all .
In this work, we focus on the fundamental relation in hypergroups, and we introduce a new classification of hypergroups in terms of the minimum number of hyperproducts of two elements whose union is the -class that contains these hyperproducts. Our main aim is to deepen the understanding of the properties of the fundamental relation in hypergroups and to enumerate the non-isomorphic hypergroups fulfilling certain conditions on the cardinality of the hearth. This task belongs to an established research field that deals with fundamental relations and enumerative problems in hypercompositional algebra [,,,,]. The plan of this article is the following: After introducing some basic definitions and notations to be used throughout this article, in Section 3, we define the notion of height of an equivalence class . We give examples of hypergroups with infinite size where the height of all -classes is finite. Denoting cardinality by , if is a hypergroup with a -class of finite size such that , then , for all and . Moreover, when is finite, we prove that if and only if is a 1-hypergroup. In Section 4, we use the notion of height of a -class to introduce new classes of hypergroups. We enumerate the elements in each class when the size of the hypergroups is not larger than 4, apart from isomorphisms. In particular, we prove that there are 4023 non-isomorphic hypergroups of size with a -class of size 1. Moreover, excluding the hypergroups with and , there exist 8154 non-isomorphic hypergroups of size with .
2. Basic Definitions and Results
Let H be a non-empty set and let be the set of all non-empty subsets of H. A hyperoperation ∘ on H is a map from to . For all , the set is called the hyperproduct of x and y. The hyperoperation ∘ is naturally extended to subsets as follows: If , then .
A semihypergroup is a non-empty set H endowed with an associative hyperproduct ∘, that is, for all . A semihypergroup is a hypergroup if for all we have ; this property is called reproducibility. A non-empty subset K of a semihypergroup is called a subsemihypergroup of if it is closed with respect to ∘ that is, for all . A non-empty subset K of a hypergroup is called a subhypergroup if , for all . If a subhypergroup is isomorphic to a group, then we say that it is a subgroup of .
Given a semihypergroup , the relation of H is the transitive closure of the relation , where is the diagonal relation in H and, for every integer , is defined as follows:
The relations and are among the so-called fundamental relations []. Their relevance in semihypergroup and hypergroup theory stems from the following facts []: If is a semihypergroup (resp., a hypergroup), the quotient set equipped with the operation for all and , is a semigroup (resp., a group). The canonical projection is a good homomorphism, that is, for all . If is a hypergroup, then is a group and the kernel of is the heart of . Moreover, if , then is called 1-hypergroup.
Let A be a non-empty subset of a semihypergroup . We say that A is a complete part of if, for every and ,
Clearly, the set H is a complete part, and the intersection of all the complete parts containing a non-empty set X is called the complete closure of X. If X is a complete part of then .
If is a semihypergroup and is the canonical projection, then, for every non-empty set , we have . Moreover, if is a hypergroup, then
A hypergroup is said to be complete if , for all . If is a complete hypergroup, then
for every and .
A subhypergroup K of a hypergroup is said to be conjugable if it satisfies the following property: for all , there exists such that . The interested reader can find all relevant definitions, many properties, and applications of fundamental relations, even in more abstract contexts, also in [,,,,,,,,,,].
For later reference, we collect in the following theorem some classic results of hypergroup theory from [,,].
Theorem 1.
Let be a hypergroup. Then,
- 1.
- The relation β is transitive, which is ;
- 2.
- , for all ;
- 3.
- a subhypergroup K of is conjugable if and only if it is a complete part of ;
- 4.
- the heart of is the smallest conjugable subhypergroup (or complete part) of , that is, is the intersection of all conjugable subhypergroups (or complete part) of .
3. Locally Finite Hypergroups
Let be a hypergroup and let ∼ be the following equivalence relation on the set : . Let be a transversal of the equivalence classes of the relation ∼. For every , there exists a non-empty set such that . In fact, by reproducibility of , if , then there exist such that . Clearly, we have and because is a complete part of H. Moreover, there exists such that and . Hence, there exists a non-empty set such that and for all . The other inclusion follows from the fact that is a complete part of .
In conclusion, each -class is the union of hyperproducts of pairs of elements that can be chosen within a transversal of ∼. This fact suggests the following definitions.
Definition 1.
Let be a hypergroup and let be a transversal of the equivalence classes of the relation ∼. For every , the class is called locally finite if there exists a finite set such that . If a class is not locally finite, we say that it is locally infinite.
Definition 2.
Let be a β-class of a hypergroup . If is locally finite, then the minimum positive integer m such that there is a non-empty set such that and is called height of , and we write . If is locally infinite, we write .
Definition 3.
A hypergroup is locally finite if all β-classes are locally finite. In particular, is called locally n-finiteif for every , and there is at least one element such that . Moreover, is strongly locally n-finiteif for every .
Clearly, is locally 1-finite if and only if is strongly locally 1-finite. Examples of hypergroups locally 1-finite are the complete hypergroups. Indeed, if is a complete hypergroup, then, for every , there exist such that and .
Example 1.
In the set , consider the hyperproducts defined by the following tables:
1 | 2 | 3 | 4 | 5 | 6 | |
1 | ||||||
2 | ||||||
3 | ||||||
4 | ||||||
5 | ||||||
6 |
1 | 2 | 3 | 4 | 5 | 6 | |
1 | ||||||
2 | ||||||
3 | ||||||
4 | ||||||
5 | ||||||
6 |
Then, and are hypergroups such that . In particular, is a locally 2-finite hypergroup since and , while is a strongly locally 2-finite hypergroup because .
Example 2.
Let be a hypergroup and the automorphism group of H. If , let be the subgroup of , generated by f. In , we define the following hyperproduct: For all , let
Firstly, we show that is a hypergroup. Then, we describe its β-classes and the heart. As a consequence, we obtain that is a locally n-finite hypergroup (resp., strongly locally n-finite hypergroup, complete hypergroup, or 1-hypergroup) if and only if is a locally n-finite hypergroup (resp., strongly locally n-finite hypergroup, complete hypergroup, or 1-hypergroup).
- 1.
- The hyperproduct ⋆ on is associative. In fact, if , then we obtain:Consequently, we have thatBy induction, if are elements in , then the hyperproduct is the set
- 2.
- The hyperproduct ⋆ is reproducible. Indeed, we have , for all elements . On the other hand, if , then, by reproducibility of , there exists such that . Now, if we consider , then because and . Hence, . Thus, we haveIn the same way, one shows that .
- 3.
- From 1 and 2, is a hypergroup. If and β are the fundamental relations in and respectively, we havefor all . Indeed, if , then there exist and such thatBy point 1, we have and . Hence, , and . Thus, . On the other hand, if , then and there exist and such that . Now, if in we consider , we obtainand we have . Hence, .
- 4.
- The set is a subhypergroup of . In fact, if , we haveHence, , for all . In the same way one proves that , so is a subhypergroup.
- 5.
- The subhypergroup is the heart of .Indeed, for all , there exists such that . If we consider , then , so is conjugable. From point 4 of Theorem 1, we obtain that . Finally, since and are -classes, we deduce that .
Proposition 1.
Let be a hypergroup. Then, , for all .
Proof.
Let . By reproducibility of H, there exists such that and so we have
Clearly, for every , there exists such that . Moreover, since is a complete part of H, we have . Hence, there exists a non-empty subset A of such that and so we obtain . □
By the previous proposition, we have the following results:
Corollary 1.
Let be a hypergroup. Then, , for every .
Corollary 2.
Let be a hypergroup. If there exists a β-class of size 1, then is a strongly locally 1-finite hypergroup.
Proposition 2.
Let be a hypergroup. If is such that is finite and , then for all and .
Proof.
Let . For all and , we have . If , then and so . Hence, . Now, let and by contradiction we suppose that . Let , by reproducibility of H, there exist such that for all . Since is a complete part of H and , we deduce that , for all . Therefore, and so , impossible. Thus, , for all and . In an analogous way, we have that , for every . □
An immediate consequence of the previous proposition is the following corollary:
Corollary 3.
Let be a hypergroup. If is finite and , then is a subgroup of .
In the preceding corollary, the finiteness of is a critical hypothesis. Indeed, in the next example, we show a hypergroup where and is not a group.
Example 3.
Let be the group of integers. In the set , we define the following hyperproduct:
Routine computations show that is a hypergroup and is a subhypergroup of . Hereafter, we firstly describe the core , then we compute .
To prove that , we will show that is the smallest conjugable subhypergroup of . For every element , we can consider the element . We obtain , hence is a conjugable subhypergroup. Now, let K be a conjugable subhypergroup of and let . Since K is conjugable, there exists such that , and so . By reproducibility of K, there exists such that . Clearly, because and . Since , there exists such that . Consequently, we have and . Hence, and since is conjugable. Obviously, .
Finally, we prove that . By Proposition 1, we have . If , then there exist n hyperproducts of elements in H such that
This result is impossible since .
Now, we give two examples of hypergroups whose heart is a group and . In particular, we have that , and , if .
Example 4.
Consider the group and a set such that . In the set , we define the following hyperproduct:
- ;
- ;
- .
Routine computations show that is a hypergroup. We have , and if . Clearly, we have and because is a singleton, for all and .
Example 5.
Let be a group of size and let . Moreover, let σ be a n-cycle of the symmetric group defined over . If is a set disjoint with G, in , we can define the following hyperproduct:
- ,
- ,
- if .
Then, is a hypergroup such that , and if . Moreover, we have and .
In the next theorem, we characterize the locally n-finite hypergroups such that .
Theorem 2.
Let be a hypergroup such that is finite. The following conditions are equivalent:
- 1.
- is an 1-hypergroup that is ;
- 2.
- .
Proof.
The implication is trivial, hence we prove that Let n a positive integer such that . By Corollary 3, the heart is a subgroup of . Let e be the identity of . If there exist such that , then we have and . Now, by contradiction, we suppose that , for all , and let x be an element of H. By reproducibility of , there exists such that . Clearly, we have , since is a complete part of . Therefore, there exists an integer k, with , and elements of such that . Hence, we obtain
and so because . Therefore, we obtain and so is a conjugable subhypergroup of . Consequently, we have , a contradiction. □
The following result is an immediate consequence of Theorem 2.
Corollary 4.
Let be a finite hypergroup, then is a group if and only if , for all .
Proof.
The implication ⇒ is obvious. On the other hand, if we suppose that for all , then we have and so , by Theorem 2. From Corollary 1 and the hypothesis, we have . Hence, is a group. □
4. Hypergroups with at Least One -Class of Height Equal to 1
From Theorem 2, the 1-hypergroups are characterized by the fact that . In this section, we use the notion of height of a -class to introduce new classes of hypergroups. We enumerate the elements in each class when the size of hypergroups is , apart from isomorphisms. We give the following definition:
Definition 4.
Let be a hypergroup. We say that
- a.
- is a -hypergroup if there exists such that ;
- b.
- is a locally 1-finite hypergroup if , for all ;
- c.
- is a 1-weak hypergroup if ;
- d.
- is a weakly locally 1-finite hypergroup if there exists such that ;
In the following, we denote by , , , and the classes of 1-hypergroups, -hypergroups, locally 1-finite hypergroups, 1-weak hypergroups, and weakly locally 1-finite hypergroups, respectively. By Definition 4 and Corollary 2, we have the inclusions . Actually, these inclusions are strict, as shown in the following example.
Example 6.
In this example, we show four hypergroups , , and such that
- 1.
- and ,
- 2.
- and ,
- 3.
- and ,
- 4.
- and .
They are the following:
- 1.
- with the hyperproduct
∘ 1 2 3 1 1 2 3 2 2 1 3 3 3 3 - 2.
- with the hyperproduct
∘ 1 2 3 4 1 1 2 3 4 - 3.
- with the hyperproduct
∘ 1 2 3 4 1 1 2 3 4 2 2 1 4 3 3 3 4 4 4 3 - 4.
- with the hyperproduct
∘ 1 2 3 4 5 1 2 3 4 5
4.1. (1, )-Hypergroups of Size 4
In [], Corsini introduced the class of 1-hypergroups and listed the 1-hypergroups of size , apart from isomorphisms. In this subsection, our interest is to study the hypergroups in class and, in particular, to determine their number, apart from isomorphisms. Since the class of 1-hypergroups is a subclass of , we recall the result proved by Corsini in [].
Theorem 3.
If is a 1-hypergroup with , then is a complete hypergroup. Moreover, is isomorphic to either a group or one of the hypergroups described in the following three hyperproduct tables:
∘ | 1 | 2 | 3 |
1 | 1 | ||
2 | 1 | 1 | |
3 | 1 | 1 |
∘ | 1 | 2 | 3 | 4 |
1 | 1 | |||
2 | 1 | 1 | 1 | |
3 | 1 | 1 | 1 | |
4 | 1 | 1 | 1 |
∘ | 1 | 2 | 3 | 4 |
1 | 1 | 4 | ||
2 | 4 | 4 | 1 | |
3 | 4 | 4 | 1 | |
4 | 4 | 1 | 1 |
Therefore, there exist eight 1-hypergroups of size .
Now, we study the hypergroups of size . Clearly, and can take the values 2 or 3 and so we distinguish the following cases:
- , and ;
- , and ;
- , and .
- In this case, we can suppose , and . Clearly, since , for reproducibility of H, we have the following partial hyperproduct table of :
∘ a b x a x b x x x x a,b To complete this table, the undetermined entries must correspond to the hyperproduct table of the subhypergroup . Apart from isomorphisms, there are eight hypergroups of size 2. Their hyperproduct tables were determined in [] and are reproduced here below:W1: ∘ a b W2: ∘ a b W3: ∘ a b W4: ∘ a b a a b a a b a a a,b a a,b a,b b b a b b a,b b b a,b b a b W5: ∘ a b W6: ∘ a b W7: ∘ a b W8: ∘ a b a a a,b a a a,b a a,b a,b a a,b a,b b a,b b b a,b a,b b a,b a b a,b a,b Hence, in this case, we have eight hypergroups. - Without loss of generality, we suppose that , , and . Since , we have the following partial hyperproduct table of :
∘ a b x y a x y b x y x x x y a,b y y y a,b x As in the previous case, the entries that are left empty must be determined so that is a hypergroup of order 2. Hence, also in this case, we have eight hypergroups. - Let and . We have the following partial hyperproduct table:
∘ a b c x a x b x c x x x x x a,b,c In this case, it is straightforward to see that we get as many hypergroups as there are of size three, apart from isomorphisms. In [], this number is found to be equal to 3999.
From Theorem 3 and the preceding arguments, we summarize the number of non-isomorphic -hypergroups with in Table 1.

Table 1.
The number of non-isomorphic -hypergroups with .
Result 1.
There are 4023 non-isomorphic -hypergroups of size .
4.2. Locally 1-Finite Hypergroups of Size 4
In this subsection, we focus on hypergroups with . By Definition 4, we have that , for all .
If , then and is isomorphic to one of the hypergroups listed in the previous subsection, for .
If , then , otherwise at least one -class has size 1 and . Hence, to determine the hypergroups in of size 3, we must assume that there exist such that . With the help of computer-assisted computations, we found that in this case there are exactly 3972 hypergroups, apart from isomorphisms.
If , then there are two possible cases, namely and . In the first case, the only information we can deduce about is that there are at least two elements such that . The number of hypergroups having that property is huge, and at present we are not able to enumerate them because the computational task exceeds our available resources. A detailed analysis of this case is challenging and may be the subject of further research. In the other case, if , then we have with . Moreover, is isomorphic to one of the hypergroups for listed beforehand, and there exist such that . On the basis of the information gathered from the preceding arguments, we are able to perform an exhaustive search of all possible hyperproduct tables with the help of a computer algebra system. In Table 2, we report the number of the hypergroups such that and , depending on the structure of , apart from isomorphisms.

Table 2.
The number of non-isomorphic hypergroups in with , according to the structure of .
Since the hypergroups corresponding to the cases in which the heart is one of the are quite a few, we list hereafter only those whose heart is isomorphic to . Apart from isomorphisms, we have the following hyperproduct tables:
∘ | a | b | x | y |
a | a | b | x | y |
b | b | a | y | x |
x | ||||
y |
∘ | a | b | x | y |
a | a | b | ||
b | b | a | ||
x | x | y | ||
y | y | x |
∘ | a | b | x | y |
a | a | b | ||
b | b | a | ||
x | ||||
y |
On the basis of the previous arguments, the number of hypergroups belonging to is summarized in Table 3, in relation to the size of H:

Table 3.
The number of non-isomorphic hypergroups in , depending on their size.
Finally, since there are 4023 hypergroups in class , see Result 1, we obtain the following result:
Result 2.
Excluding the hypergroups such that and , there are 8153 non-isomorphic locally 1-finite hypergroups of size .
4.3. 1-Weak Hypergroups of Size 4
In this subsection, we determine the hypergroups of size , apart from isomorphisms. We observe that, if , then there is at least one -class of height different from 1. Moreover, since , we have , for all . Hence, if , then and .
Lemma 1.
Let be a hypergroup in such that , then is isomorphic to the group .
Proof.
By hypotheses, we have and . By Proposition 1, if is the class different from , then . Moreover, for Proposition 2, we have , for all and . Now, if by contradiction we suppose that there exist such that , then and because , and . This fact is impossible since and . Hence, , for all , and so . □
Theorem 4.
Let be a hypergroup such that the heart is isomorphic to a torsion group. If ε is the identity of , then , for all .
Proof.
Let . By reproducibility of H, there exists such that . Clearly ; moreover, we have and so . Obviously, by induction, we obtain , for all . Finally, since is isomorphic to a torsion group, there exists such that , hence . In the same way, we have . □
By reproducibility, Lemma 1, and Theorem 4, the hypergroups in with have the following partial hyperproduct table, apart from isomorphisms:
∘ | a | b | x | y |
a | a | b | x | y |
b | b | a | y | x |
x | x | y | ||
y | y | x |
Since , we have . Now, we prove that . In fact, if we suppose that , then we have:
We obtain the same result also if we suppose that or or . Hence, in class , there is only one hypergroup of size 4, apart from isomorphisms. Its hyperproduct table is the following:
∘ | a | b | x | y |
a | a | b | x | y |
b | b | a | y | x |
x | x | y | ||
y | y | x |
We note that this hypergroup is a special case of the hypergroup described in Example 5. The group is and the cycle is a transposition.
Result 3.
Excluding the hypergroups such that and , in the class there are 8154 hypergroups of size , apart from isomorphisms.
We complete this section by showing a result concerning the weakly locally 1-finite hypergroups.
Theorem 5.
If , then .
Proof.
By hypothesis, there is a class different from such that and . Because of the inclusions , we have and , otherwise . Now, if we suppose that , by Proposition 1, we obtain and so . Consequently, with the help of Theorem 2, we have the contradiction . Hence, , and . □
Recall that the hypergroup shown in Example 6 belongs to but not to due to the previous theorem that the hypergroup has the smallest cardinality, among all hypergroups sharing that property.
5. Conclusions
In hypergroup theory, the relation is the smallest strongly regular equivalence relation whose corresponding quotient set is a group. If is a hypergroup and is the canonical projection, then the kernel is the hearth of . If the hearth is a singleton, then is a 1-hypergroup. We remark that the hearth is a -class and also a subhypergroup of . In particular, if , then we have . More generally, every -class is the union of hyperproducts of pairs of elements of H. In this work, we defined the height of a -class as the minimum number of such hyperproducts. This concept yields a new characterization of 1-hypergroups, see Theorem 2, and allows us to introduce new hypergroup classes, depending on the relationship between height and cardinality of the -classes; see Definition 4. These classes include 1-hypergroups as particular cases. Apart from isomorphisms, we were able to enumerate the elements of those classes when , with only one exception. In fact, the problem of enumerating the non-isomorphic hypergroups where , and remains open.
In conclusion, as a direction for further research, we point out that many hypergroups that arose in the analysis of the hypergroup classes introduced in the present work are join spaces or transposition hypergroups [,]. For example, the 10 hypergroups of size three in Table 1 are transposition hypergroups. Transposition hyperstructures are very important in hypercompositional algebra. Hence, it would be interesting to enumerate the join spaces or the transposition hypergroups belonging to the hypergroup classes introduced in Definition 4, at least for small cardinalities. Another question that is stimulated by the concept of height concerns the height of the -classes of the coset hypergroups, i.e., the hypergroups that are quotient of a non-commutative group with respect to a non-normal subgroup []. We leave these observations and suggestions as a possible subject for new works.
Author Contributions
Conceptualization, formal analysis, writing—original draft: M.D.S., D.F. (Domenico Freni), and G.L.F.; software, writing—review and editing: D.F. (Dario Fasino). All authors have read and agreed to the published version of the manuscript.
Funding
This research has been carried out in the framework of the departmental research projects “Topological, Categorical and Dynamical Methods in Algebra” and “Innovative Combinatorial Optimization in Networks”, Department of Mathematics, Computer Science and Physics (PRID 2017), University of Udine, Italy. The work of Giovanni Lo Faro has been partly supported by INdAM-GNSAGA, and the work of Dario Fasino has been partly supported by INdAM-GNCS.
Conflicts of Interest
The authors declare no conflict of interest.
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