Fundamental relations in simple and 0-simple semihypergroups of small size

We consider the fundamental relations β and γ in simple and 0-simple semihypergroups, especially in connection with certain minimal cardinality questions. In particular, we enumerate and exhibit all simple and 0-simple semihypergroups having order 3 where β is not transitive, apart of isomorphisms. Moreover, we show that the least order for which there exists a strongly simple semihypergroup where β is not transitive is 4. Finally, we prove that γ is transitive in all simple semihypergroups, and determine necessary and sufﬁcient conditions for a 0-simple semihypergroup to have γ transitive. The latter results obviously hold also for simple and 0-simple semigroups.

U on the right the fundamental relation β is transitive if |H | ≤ 8, and exhibits a counterexample having order 9; furthermore, in all semihypergroups of type U on the right the fundamental relation γ is always transitive. The above mentioned results allowed to solve various minimal cardinality problems on finite semihypergroups of type U on the right.
The present paper deepens the knowledge of the fundamental relations β and γ in the classes of simple and 0-simple semihypergroups, and addresses certain minimal cardinality problems in connection with them. Simple and 0-simple semihypergroups have been recently introduced in [18] as a generalization of analogous well known and widely studied structures in semigroup theory [17]. We remark that the class of simple semihypergroups extends that of finite semihypergroups of type U on the right. Motivated by the results in [13], it is natural to consider whether or not the relations β and γ are transitive in the wider class of simple semihypergroups. Actually, we prove hereafter that γ is transitive in all simple semihypergroups, while the least order of a simple semihypergroup having β non transitive is 3. Analogous properties are also found in the class of 0-simple semihypergroups.
The outline of this paper is as follows: After introducing some fundamental definitions and results, in Sect. 2 we recall definitions and elementary properties of simple and 0-simple semihypergroups, and we define strongly simple semihypergroups, a subclass of simple semihypergroups which plays an important role in what follows. Furthermore, we give a list of examples and some special constructions of simple and strongly simple semihypergroups. Some of these examples will be exploited in successive sections. In Sect. 3 we determine all simple semihypergroups having order 3, apart of isomorphisms, where β is not transitive. Remarkably, all these semihypergroups are not strongly simple, since they are either left-or right-reproducible.
This fact motivates the study carried out in Sect. 4 where we prove the existence of strongly simple semihypergroups having order 4 where β is not transitive. The proof is constructive and exploits a special one-element extension of the semihypergroups found in the preceding section. In Sect. 5 we consider 0-simple semihypergroups, as their analysis cannot be carried out as in the simple case, and we determine all 0-simple semihypergroups having order 3 where β is not transitive. In Sect. 6 we consider the relation γ in both simple and 0-simple semihypergroups. In particular we prove that γ is transitive in all simple semihypergroups, thus extending the analogous result concerning semihypergroups of type U on the right [13]. Moreover, we show a necessary and sufficient condition for a 0-simple semihypergroup to have γ transitive. In the last section we draw some conclusions and state an open problem. Remark 1.1 Throughout the paper, we will often show hyperproduct tables of semihypergroups. These tables are usually obtained after long arguments that are aimed at proving the existence of semihypergroups having certain properties. We inform the reader that, after these tables are obtained, we always check their associativity either by hand or by means of computer routines, as those described in [7]. Hence, the corresponding semihypergroups are correctly defined, even if this is not always explicitly stated in what follows.

Basic definitions and results
Throughout this paper we use just a few basic concepts and definitions that belongs to common terminology in hyperstructure theory. A semihypergroup is a set endowed with an associative hyperproduct. A semihypergroup H is left-reproducible (respectively, right-reproducible) if H x = H (respectively, x H = H ) for all x ∈ H . A hypergroup is a semihypergroup that is both left-and right-reproducible, hence x H = H x = H, for all x ∈ H .
A non-empty subset S of a semihypergroup H is called a subsemihypergroup of H if it is closed with respect to multiplication, that is, if x y ⊂ S for all x, y ∈ S. If the subsemihypergroup S is a semigroup, we say that S is a subsemigroup of H .
The relations β, β * , γ and γ * are called fundamental relations on H [21]. Their relevance in semihypergroup theory stems from the following facts [11,19]: If H is a semihypergroup (resp., a hypergroup), then the relation β * is the smallest strongly regular equivalence on H and the quotient H/β * is a semigroup (resp., a group). Moreover, the relation γ * is the smallest strongly regular equivalence such that the quotient H/γ * is a commutative semigroup (resp., a commutative group). The interested reader can find all relevant definitions, many properties and applications of fundamental relations, even in more abstract contexts, also in [1,4,9,12,16,21].

Simple and 0-simple semihypergroups
In this section we recall the definition of simple and 0-simple semihypergroups; furthermore, we show some examples of such structures that we shall use in the subsequent section. The concept of simple semigroup is well known and widely studied in the framework of semigroup theory [17]. Recently, this concept has been extended to semihypergroup theory [18].
If H is a semihypergroup, an element 0 ∈ H such that 0x = 0 (resp., x0 = 0) for all x ∈ H is called left zero scalar element (resp., right zero scalar element) of H . If 0 is both left and right zero scalar element, then 0 is called zero scalar.
We remark that the definitions of simple and 0-simple semihypergroup as given by Definition 2.1 and Definition 2.2 are equivalent to the ones usually introduced by means of the ideals of the semihypergroup, see e.g., [17,18].
Our first examples are immediate: 1. All left-or right-reproducible semihypergroups are simple. Indeed, we have As a consequence, not only all hypergroups are simple semihypergroups, but also all finite semihypergroups of type U on the right are simple, because they are left-reproducible [8,9,13]. 2. The following example mirrors a construction of the so-called K H -semihypergroups given in [5]: If H is a simple semihypergroup and A = {A x } x∈H is a family of nonempty and pairwise disjoint sets, then the set K = ∪ x∈H A x is a simple semihypergroup with respect to the hyperproduct a • b = ∪ z∈xy A z , for a ∈ A x and b ∈ A y . 3. If (H, •) and (K , •) are two simple semihypergroups, then the cartesian product H × K is a simple semihypergroup with respect to the hyperoperation ⊗ naturally defined as We remark that, in this case, if H is only left-reproducible (but not right-reproducible) and K is only rightreproducible (but not left-reproducible), then H × K is a simple semihypergroup which is neither left-nor right-reproducible. 4. A more significant example of simple semihypergroup which is neither left-nor right-reproducible is described by the following hyperproduct table: In fact, it is apparent that this semihypergroup cannot be obtained as a direct product of smaller semihypergroups.
The semihypergroup (1) has a remarkable property: All its subsemihypergroups are simple. This fact suggests the following definition, which will be developed in subsequent sections: As an immediate consequence, we obtain the following claim, which will be useful in subsequent arguments: Corollary 2.5 Let H be a simple semihypergroup that is not right-reproducible (resp., left-reproducible), and let K ⊂ H be a proper subsemihypergroup. If a ∈ K is a right (resp., left) zero scalar element of K , then Proof If (H − K )a ⊂ K then, using the identity a 2 = {a} we obtain As a consequence, Ha = {a} and a is a right zero scalar element of H . Hence, by Proposition 2.4, H is right-reproducible, which is a contradiction.
In what follows, we consider two more examples of simple semihypergroups, which deserve some attention. The first example is an extreme generalization of Rees construction [17]. In the second example we provide a list of all simple semihypergroups having order 2, apart of isomorphisms. Some of these semihypergroups will be exploited in the forthcoming section to construct simple semihypergroups having order 3, whose fundamental relation β is not transitive.

Example 2.6
Let H be a hypergroup, let A = {A i } i∈I and B = {B j } j∈J be two families of nonempty and pairwise disjoint sets, and let ϕ : J × I → P * (H ). Introduce the notations A = ∪ i∈I A i , B = ∪ j∈J B j , E = A × H × B and ϕ(h, k) = P hk for all (h, k) ∈ J × I . On the set E we can define the following hyperproduct: This hyperproduct is associative. Indeed, for any triple (a , z, b ) ∈ A r × H × B s , in the hypergroup H we have (x P jk y)P hr z = x P jk (y P hr z) and for the hyperproduct in E it holds and the hyperproduct is associative. Furthermore, for any fixed pair (s, r ) ∈ J × I and for any elements b ∈ B s and a ∈ A r , from equation H P si x P jr H = H we obtain that there exists a pair (w, w) ∈ H × H such that y ∈ w P si x P jr w. As a consequence, (a , The preceding example gives us a very general technique to construct simple semihypergroups; various known constructions can be considered as particular cases, where all the sets in the families A and B are singletons, so that we can identify A and B with the index sets I and J, respectively. In that case, we shall say that E is a (H, I, J, ϕ)-semihypergroup.
1. If H is a group and the map ϕ is single-valued, then the (H, I, J, ϕ)-semihypergroup is a semigroup known as Rees matrix semigroup, see [17, §3.3]. 2. If H is a regular hypergroup (see e.g., [18] for the definition of regular hypergroup) and the map ϕ is single-valued, then the (H, I, J, ϕ)-semihypergroup is the Rees matrix semihypergroup introduced in Indeed, in the paper [10] it is shown that for any nonempty subset A of a hypergroup H, the hyperproduct Example 2.7 It is not difficult to prove (by an exhaustive procedure) that, apart of isomorphisms, all simple semihypergroups of order 2 are the following: 1. The eight hypergroups of order 2, see [6,20]; 2. The two semigroups x y x x y y x y x y x x x y y y 3. The following semihypergroups: x y x x x, y y x y x y x x x, y y x x, y x y x x x y x, y y x y x x x y x, y x, y (4) Observe that the four semihypergroups (4) are not (H, I, J, ϕ)-semihypergroups. Indeed, all (H, I, J, ϕ)semihypergroups of order 2 fall into the following cases: 1. |H | = |J | = 1 and |I | = 2; 2. |H | = |I | = 1 and |J | = 2; 3. |H | = 2 and |I | = |J | = 1.
In the first two cases the resulting (H, I, J, ϕ)-semihypergroup is isomorphic to one of the two semigroups in (3), while in the last case it is a hypergroup.

The relation β in simple semihypergroups
The recent paper [13] has solved the problem of the existence of semihypergroups of type U on the right where the fundamental relation β is not transitive. Indeed, in that paper the author exhibits a semihypergroup of type U on the right having cardinality 9 where β is not transitive, and moreover, proves that in all semihypergroups of type U on the right having cardinality less than 9 β is always transitive. Analogous problems arise for the class of strongly simple semihypergroups. In fact, in this section we will prove that there exist exactly 10 simple semihypergroups having order 3, apart of isomorphisms, where the relation β is not transitive. Their subsemihypergroups are simple, hence they fulfil the first condition of Definition 2.3; by the way, they are not strongly simple because they are either left-or right-reproducible. Now, since in all semihypergroups having order 2 the relation β is transitive, it is interesting to determine the minimal cardinality for which there exist a strongly simple semihypergroup where the relation β is not transitive. This problem will be solved by the forthcoming Theorem 4.4.

Remark 3.1
If H is a semihypergroup such that β is not transitive, then there exists a triple (a, b, c) of distinct elements of H such that aβb, aβc but (b, c) / ∈ β. Obviously, in this case, for any integer k ≥ 1 and for any Furthermore, it holds H H = H . Indeed, if this is not the case, the set H H is contained into one of the sets {a, b}, {a, c} or {b, c}; as a consequence, we would obtain the contradiction by way of π 1 ), π 2 ), π 3 ) and π 4 ) we obtain the contradiction bβc. Analogously, we prove that c / ∈ aa.
The forthcoming results are preliminary to the construction of simple semihypergroups having order 3 and β not transitive.  ∈ ab ∪ ba ∪ bb. Let us suppose by absurd that c ∈ ab. Firstly, we prove that ba = ca = {a}. Indeed, by π 1 ) and π 3 ), one has ba ∪ bb ∪ ca ∪ cb ⊂ {a, c}. Moreover, c ∈ ba. Actually, if c ∈ ba then by π 2 ) we obtain the inclusion Hence, by π 2 ) it follows that ac ⊂ {a, c} and moreover H = (Ha)H = a H ⊂ {a, c}, which is absurd. Hence we conclude that c ∈ ab. By reversing the roles of a and b we obtain c ∈ ba. Finally we prove that c ∈ bb. Indeed, if c ∈ bb, by 1) and π 1 ) we have aa = ab = ba = {a} and moreover ac ⊂ a(bb) = (ab)b = ab = {a}, whence ac = {a}. By exploiting the hyperproduct bba we also get ca = {a}. This fact would imply that a is a zero element in H but this is impossible since H is simple. Hence c ∈ P P. Hence The second part of the claim is shown analogously by interchanging the role of b and c. Thus a ∈ (ab)a and b ∈ (bb)a. As a consequence, we get Pa P = P(a P) = Pa = aa ∪ ba = P and P ⊂ (ab)a ∪ (bb)a ⊂ Pb P ⊂ P. Hence, also in this case we obtain Pa P = Pb P = P.
In what follows, we will describe the isomorphism classes of the simple semihypergroups having order 3 where the relation β is not transitive. By Remark 3.
Indeed, by π 3 ), one has c ∈ ca otherwise Q P = H and bβc. Moreover, since b ∈ Q Q, we have b ∈ ca and thus ca = {a}. Furthermore, ac = {a}, otherwise a is a zero element in Q and Q is not simple. Hence, by point 2 of Lemma 3.2, we obtain ac = {a, c}. Finally, by π 2 ), we get ba = {a}. Now, from Example 2.7 we know that, apart of isomorphisms, there exist only two simple semihypergroups S i = {x, y}, for i = 1, 2, such that x x = yx = {x} and x y = {x, y}. Their multiplicative tables are the following: x y x x x, y y x y S 2 : x y x x x, y y x x, y Consequently, from conditions (5) we obtain that the two simple subsemihypergroups P = {a, b} and Q = {a, c} must be isomorphic to S 1 or S 2 . Hence, the only possible cases in the present step are contained in the following partial tables: a a a, b a, c  b a a a, b a, c  b a b  c a a, c a a a, b a, c  b a a, b c a c a a a, b a, c  b a a, b  c a a, c Now, we observe that the hyperproducts π 2 ) and π 3 Finally, from (9) we get the following implications:  a a a, b a, c  b a a, b a, c  c a a, b a, The partial table T 3 leads to the two simple semihypergroups a a a, b a, c  b a a, b a, c  c a b c a a a, b a, c  b a a, b a, c  c a a, b c (11) Furthermore, the partial table T 2 can be completed as a a a, b a, c  b a b  c  c a a, b a, c a a a, b a, c  b a b a, c  c a a,  to {c} = {a, c}. 6. As a consequence, the partial table T 1 can be completed only by the following simple semihypergroups: a a a, b a, c  b a a a, b a, c  b a b a, c  c a a, b c (13) So far, we have obtained five simple semihypergroups having order 3 where the relation β is not transitive. These semihypergroups are pairwise non isomorphic. Furthermore, they are not strongly simple, because they are right-reproducible.

Second step: ab = {a}
We begin with the following remark: If (H, •) is a semihypergroup then H is also a semihypergroup with respect to the hyperproduct • defined as x • y = y • x for all x, y ∈ H . In what follows, the semihypergroup (H, •) will be called the transposed of (H, •) and will be denoted simply by H T . Clearly, the use of that term is motivated by the fact that, in the finite case, the multiplicative table of H T is obtained by transposing the multiplicative table of H . Obviously, if H and K are two isomorphic semihypergroups then also H T and K T are isomorphic. Furthermore, we note that in general H is not isomorphic to H T . For example, it is easy to check that for i = 1, . . . , 5 the transposed semihypergroups H T i are not isomorphic respectively to the semihypergroups H i obtained in Eqs. (10), (11) and (13). Finally, we observe that the relation β is transitive in H if and only if it is transitive in H T . Moreover, H is simple (resp., left-reproducible) if and only if H T is simple (resp., right-reproducible).
In the following we will show that, apart of isomorphisms, the semihypergroups H T i for i = 1, . . . , 5 are the only simple semihypergroups having order 3 where the relation β is not transitive and aa = ab = {a}. Indeed, if K is a semihypergroup verifying these conditions, then the simple subsemihypergroup P = {a, b} is isomorphic to one of the two semihypergrups S 3 = S T 1 : x y x x x y x, y y In particular, in K one has ba = {a, b}. This fact implies that in K T one has ab = {a, b}. As a consequence, there exists i ∈ {1, . . . , 5} such that In conclusion, we can state the following result:

Strongly simple extensions
The ten semihypergroups in Theorem 3.5 fulfil the first condition in Definition 2.3 but not the second one. In fact, for i = 1, . . . , 5 the semihypergroup H i is right-reproducible, while H T i is left-reproducible. This fact lead us to the problem of determining the least order for a strongly simple semihypergroup whose relation β is not transitive. In this section we will prove that this number is 4, see Theorem 4.4. The proof consists in the construction of suitable extensions having order 4 of the semihypergroups H i . The reader can find various extension techniques exploited in semigroup and hypergroup theory in [14,15].

Lemma 4.1 Let H be a semihypergroup, and let K be a subsemihypergroup of H such that H
1. We prove the first claim by proceeding in three steps: I) The claim follows immediately from Corollary 2.5, since we have H − K = {d}. II) If b ∈ da then one has c ∈ bc ⊂ (da)c and b ∈ da ⊂ d(ac). We arrive at the contradiction bβc. The case c ∈ da is treated analogously. III) Let us suppose by absurd that da = {d}. This leads to db = dc = {d} and consequently Consequently b ∈ db. Analogously, from ac = {a, c} one also get c ∈ dc. Finally, by Lemma 4.1(2), one has d ∈ db ∩ dc, and the claim follows. 3 The cases in table (17) are all those possible due to Lemma 4.2. The cases listed in the tables (18) and (19) take into account the following properties, that must be considered when H extends H 2 or H 4 : • If cb = {b} then a ∈ dc ⇒ a ∈ db. Indeed, it holds a ∈ ad ⊂ (dc)b = d(cb) = db.
By direct verification, the hyperproduct in H as defined by all cases listed in (17), (18) and (19) is associative. Furthermore, the resulting semihypergroups H and their transposed semihypergroups H T are strongly simple and the relation β is not transitive. Hence, we obtained the following result:

The relation β in 0-simple semihypergroups
In semigroup theory, a relevant family is that of 0-simple semigroups [17]. This concept has been extended in [18] to semihypergroups, see Definition 2.2. Obviously, a 0-simple semihypergroup is not simple. Hence, the analysis carried out in Sect. 4 (in particular, concerning the transitivity of β) cannot be extended immediately to 0-simple semihypergroups.
In this section we present all 0-simple semihypergroups having order 3 where β is not transitive, see Theorem 5.6. Owing to Remark 3.1, hereafter we can restrict ourselves to the case where H = {a, b, c} has a zero  scalar element, aβb, aβc and (b, c) / ∈ β. H = {a, b, c} is a 0-simple semihypergroup where aβb, aβc and (b, c) / ∈ β, then the zero element of H is a.

Lemma 5.1 If
Proof Proceeding by absurd, suppose that the zero element of H is b. As already shown in Remark 3.1, we have aa = {a}. Furthermore, by the equalities π 2 ) and π 3  Firstly, note that |cb| = 1. Indeed, if |cb| = 1 then |cc| = |c(bb)| = |(cb)b| = 1, since the second column of the hyperproduct table is made of singletons. As a consequence, all hyperproducts reduce to singletons, H is a semigroup and β is transitive, a contradiction.
Owing to the hypotheses placed on β, the remaining cases are cb = {a, b} and cb = {a, c}. If cb = {a, c} then cc = (bb)c = b(bc) = b{a, c} = {a, c}, whence b / ∈ H H, impossible. Hence, the only possible case is bc = {a, b}, which leads to a truly associative hyperproduct, and the proof is complete.
Note that the semihypergroups defined by the last 4 rows are the transposed semihypergroups of those defined by the first 4 rows. All preceding results in this Section, in particular, Proposition 5.3, Proposition 5.5 and the table (21), can be summarized in the following result: The relations γ and γ * were introduced in [11,12] in the context of hypergroups, in order to characterize the derived hypergroup by means of the notion of strongly regular equivalence. In particular, in [12] a geometric interpretation of γ and γ * is found, showing their relationships with the concepts of geometric space and polygonal. Subsequently, various authors exploited the concept of geometric space and extended the relation γ to other hyperstructures, see e.g., [1,4]. In what follows, we summarize the relationship between semihypergroups and geometric spaces [4,11,12].
Obviously, the relations γ, γ * are the relations ∼, ≈ defined on H arising from the blocks of P σ (H ). We recall from [12,Thm. 3.4] that if H is a hypergroup then the geometric space (H, P σ (H )) is strongly transitive, thus ∼, that is γ, is transitive. Recently in [13] it was proved that γ is transitive even if H is a finite semi-hypergroup of type U on the right. In what follow, we will show that γ is transitive is all simple semihypergroups, see Theorem 6.4. We remark that this fact is not necessarily true if H is only 0-simple. Indeed, for example, the 0-simple semihypergroup shown in Proposition 5.3 is commutative, thus β = γ, whence γ is not transitive. In the next theorem we characterize all 0-simple semihypergroups whose associated geometric space (H, P σ (H )) is strongly transitive. Proof 1. ⇒ 2. By Theorem 6.1 we have that γ is transitive. Hence, for all B ∈ B 0 , for all y ∈ B and for all x ∈ γ * (0) we have yγ x. As a consequence, there exists a block B ∈ P σ (H ) such that {y, x} ⊂ B . Since B ∩ B = ∅ and the geometric space (H, P σ (H )) is strongly transitive, there exists a block C ∈ P σ (H ) such that B ∪ {x} ⊂ C. Obviously C ∈ B 0 as 0 ∈ B ⊂ C.
2. ⇒ 1. Let B 1 = B(z 1 , . . . , z m ) and B 2 = B(x 1 , . . . , x n ) be two blocks of P σ (H ) such that B 1 ∩ B 2 = ∅ and moreover let x ∈ B 2 . Our goal is to prove that there exists a block C ∈ P σ (H ) such that B 1 ∪ {x} ⊂ C. We distinguish the two cases I) 0 ∈ B 1 ∪ B 2 and II) 0 ∈ B 1 ∪ B 2 . I) By (22) we have B 1 ∪ B 2 ⊂ γ * (0) because 0 ∈ B 1 ∪ B 2 and B 1 ∩ B 2 = ∅. Hence, in particular we have B 1 ∈ B 0 . Finally, by hypothesis, there exists a block C Hence there exist two pairs (y, y ) and (c, c ) of elements of H such that z m ∈ yx y and x ∈ cbc . Then, using the foregoing properties of the blocks in P σ (H ) we obtain From the previous theorem we obtain a sufficient condition for the transitivity of γ in a 0-simple semihypergroup:

Corollary 6.3
If H is a 0-simple semihypergroup fulfilling one of the hypotheses in Theorem 6.2 then γ is transitive.
Proof The claim follows immediately from Theorem 6.1 and the fact that γ is the relation ∼ defined on the geometric space (H, P σ (H )).
We conclude this section by proving that the relation γ is transitive in all simple semihypergroups. In order to attain that result, we make use of a special construction which is well known in the framework of semigroup theory, see e.g., [17]. If H is a semihypergroup then we denote by H 0 the semihypergroup built by adding a zero scalar element 0 / ∈ H to H ; the hyperproduct in H 0 extends naturally the one defined in H . We will refer to H 0 as the natural 0-extension of H . The following propositions are immediate: With the help of the natural 0-extension of a semihypergroup, the proof of our last theorem becomes almost immediate; the reader could note that, in order to prove a relevant property of simple semihypergroups, we rely on a related property on 0-simple semihypergroups. Actually, that procedure can help to extend the proof to different objects, and has been used also elsewhere, see e.g., [17,18], in related contexts.

Theorem 6.4 If H is a simple semihypergroup then the relation γ is transitive.
Proof Since H is simple, then its natural 0-extension H 0 is 0-simple. Moreover, γ * H 0 (0) = {0}. As a consequence, H 0 fulfills the second condition of Theorem 6.2, whence γ H 0 is transitive thanks to Corollary 6.3, and the claim follows.
As recalled by the first example in Sect. 2, the class of simple semihypergroups includes, as special subclasses, that of hypergroups and that of semihypergroups of type U on the right. Transitivity of γ in such subclasses has been shown in [11] and [13], respectively. Hence, the foregoing theorem provides an extension of the above mentioned results. Moreover, it is worth noting that all results in this section obviously hold also for simple and 0-simple semigroups, as defined e.g., in [17].

Conclusions and open problems
The class of semihypergroups is huge, even if one considers semihypergroups having rather small order. Within this class there are instances having rather unexpected properties, and certain implications that hold true e.g., for semigroups or hypergroups do not hold on semihypergroups. We believe that the manifold examples shown in this paper may improve knowledge of semihypergroups and help further investigations in this subject.
For example, the 34 semihypergroups described in Remark 4.3, see (17), (18) and (19), and their respective transposed semihypergroups, are strongly simple semihypergroups having a subsemihypergroup K = {a, b, c} where β is not transitive. Moreover, they also have two subsemihypergroups, K 1 = {a, b, d} and K 2 = {a, c, d}, where β is transitive. This fact opens the problem to prove whether or not there exists a strongly simple semihypergroup all whose subsemihypergroups K with |K | ≥ 3 have a relation β which is not transitive.