About the fundamental relations defined on the hypergroupoids associated with binary relations

European Journal of Combinatorics 32 (2011) 72–81

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European Journal of Combinatorics journal homepage: www.elsevier.com/locate/ejc

About the fundamental relations defined on the hypergroupoids associated with binary relations Irina Cristea a , Mirela Ştefănescu b , Carmen Angheluţă c a

DIEA, University of Udine, Via delle Scienze 206, 33100, Udine, Italy

b

Faculty of Mathematics and Informatics, ‘‘Ovidius’’ University, Bd. Mamaia 124, 900527 Constanţa, Romania

c

Faculty of Mathematics and Computer Science, University of Bucharest, str. Academiei 14, 010014 Bucharest, Romania

article

info

Article history: Received 22 April 2010 Accepted 13 July 2010 Available online 16 August 2010

abstract This paper deals with connections between hypergroupoids and n-ary relations. First we prove that the study of the reduced hypergroupoids associated with n-ary relations can be linked to the simpler case of binary relations. On the basis of the properties of some fundamental relations defined on a hypergroupoid, we determine necessary and sufficient conditions for two elements in a hypergroupoid associated with a binary relation to be operationally equivalent or inseparable; moreover we characterize the reduced hypergroupoids in some special cases. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction It may happen that on a non-empty set H, the composition of two elements of H does not give an element of H, as in classical algebra, but gives a non-empty subset of H. These kinds of operations are called hyperoperations or hyperproducts. A set H endowed with one or more hyperoperations is called an algebraic hyperstructure. The simplest hyperstructure is the hypergroupoid, that is a couple ⟨H , ◦⟩, where ◦ is a mapping from H × H to the power set P ∗ (H ), the set of all non-empty subsets of H. If this hyperproduct satisfies the associativity and the reproduction law, then ⟨H , ◦⟩ is a hypergroupoid. The first example of a hypergroup, which motivated the introduction of these new structures, was the following one [19]. Let (G, ·) be a group and H be a subgroup of G; then G/H = {xH | x ∈ G} with the hyperoperation defined by xH ◦ yH = {zH | z ∈ xH · yH } is a hypergroup. In the last few decades numerous connections between hypergroups and various domains of pure and applied mathematics (like binary relations, geometry, graphs and hypergraphs, fuzzy sets, rough

E-mail addresses: [email protected] (I. Cristea), [email protected] (M. Ştefănescu), [email protected] (C. Angheluţă). 0195-6698/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ejc.2010.07.013

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sets, topology, codes, cryptography, automata, probabilities) have been established and investigated (see [3,8,16,21]). The study of the correspondence between hypergroups and binary relations started with the paper of Chvalina [1], when he used ordered sets to construct semihypergroups and hypergroups. Later on, using binary or n-ary relations, several hyperoperations were obtained by Corsini [4], Rosenberg [25], and Cristea and Ştefănescu [12]; the new hyperstructures have represented the main object of the researches of Corsini [5,6], Corsini and Leoreanu [7], Cristea [9], Cristea and Ştefănescu [11], De Salvo and Lo Faro [13,14], Leoreanu-Fotea and Davvaz [18], Massouros and Tsitouras [20], Spartalis [26], Spartalis and Mamaloukas [28], Spartalis et al. [27]. Several algorithms have been created [10,20,28] in order to calculate the number of non-isomorphic hypergroups determined by a binary relation on a finite set, associated as in [4,25]. For more details on the n-ary relations, the reader is refereed to [22,24]. Using the fundamental relations on a hypergroupoid introduced by Jantosciak [17], the study of the hypergroups can be divided into two parts: the study of reduced hypergroups and the study of all hypergroups having the same reduced form [17]. In this note we continue the work started in [11] on the reduced hypergroups associated with binary relations. The paper is organized as follows. In Section 2, introductory concepts concerning hypergroups and relevant results on the connection between hypergroups and binary relations are briefly reviewed. Then we show that the hypergroupoid associated with an n-ary relation as in [25], or [4], or [12], coincides with the one associated with a certain induced binary relation. In Section 3, we investigate whether and when the fundamental relations introduced by Jantosciak [17] are (strong) regular equivalences. Further discussion on these equivalences in connection with reduced hypergroups is included in Section 4 and in Section 5, where we present some links with the Boolean matrices associated with binary relations. Finally we indicate some conclusions and research directions covered in the last section. 2. Hypergroups and binary relations For the sake of convenience and completeness of our study, we recall some basic definitions and properties. For a non-empty set H, we denote by P ∗ (H ) the set of all non-empty subsets of H. A non-empty set H, endowed with a mapping, called the hyperoperation, ◦ : H 2 −→ P ∗ (H ), is called a hypergroupoid. A hypergroupoid which verifies the following conditions: 1. (x ◦ y) ◦ z = x ◦ (y ◦ z ), for all x, y, z ∈ H, 2. x ◦ H = H = H ◦ x, for all x ∈ H (reproduction axiom), is called a hypergroup.  If A and B are non-empty subsets of H, then we write A ◦ B = a∈A a ◦ b. b∈B

A hypergroupoid ⟨H , ◦⟩ is called an Hv -semigroup if the weak associativity law is valid, i.e., (x ◦ y) ◦ z ∩ x ◦ (y ◦ z ) ̸= ∅, for all x, y, z ∈ H. An Hv -semigroup is called an Hv -group if the reproduction axiom is valid. For each pair (a, b) ∈ H 2 , we define a/b = { x | a ∈ x ◦ b} and b \ a = {y | a ∈ b ◦ y}. If A and B are non-empty subsets of H, then we define A/B = a∈A a/b. b∈B

A commutative hypergroupoid ⟨H , ◦⟩ is called a join space if, for any (a, b, c , d) ∈ H 4 , the following implication holds: a/b ∩ c /d ̸= ∅ H⇒ a ◦ d ∩ b ◦ c ̸= ∅ (‘‘transposition axiom’’). For more details on hypergroup theory, see [2]. Already, various hyperoperations have been defined using a binary relation ρ on a non-empty set H. We recall here those introduced by Rosenberg [25] and by Corsini [4]. Let ρ be a binary relation on a non-empty set H. The sets

D(ρ) = {x ∈ H | ∃y ∈ H : (x, y) ∈ ρ}, R(ρ) = {y ∈ H | ∃x ∈ H : (x, y) ∈ ρ} are called the domain and the range of the relation ρ , respectively.

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Moreover, for any x ∈ H, set Lρ (x) = {y ∈ H | (y, x) ∈ ρ},

Rρ (x) = {z ∈ H | (x, z ) ∈ ρ}. If it is clear which relation we are talking about, then we use the notation L(x) and R(x) instead of Lρ (x) and Rρ (x). I. Rosenberg defined the following hyperproduct: x ◦ y = R(x) ∪ R(y),

for any x, y ∈ H

and he gave necessary and sufficient conditions such that the associated hypergroupoid ⟨H , ◦⟩ is a hypergroup or a join space. P. Corsini introduced and studied the partial hypergroupoid ⟨H , ⊙⟩, considering, for any x, y ∈ H, x ⊙ y = R(x) ∩ L(y). In this paper we deal with the hyperoperation defined by x ⊗ρ y = L(x) ∪ R(y),

for any x, y ∈ H .

(1)

This hyperoperation has been considered for the first time by Cristea and Ştefănescu in a more general case, using an n-ary relation ρ ⊆ H n . Like for the case of a binary relation, for a given n-ary relation ρ on H and for any x ∈ H, we set Lρ (x) = {y ∈ H | ∃u1 , . . . , un−2 ∈ H : (y, x, u1 , . . . , un−2 ) ∈ ρ ∨ (u1 , . . . , un−2 , y, x) ∈ ρ

∨(u1 , . . . , uk , y, x, uk+1 , . . . , un−2 ) ∈ ρ, for any k ∈ {1, . . . , n − 3}} and Rρ (x) = {y ∈ H | ∃u1 , . . . , un−2 ∈ H : (x, y, u1 , . . . , un−2 ) ∈ ρ ∨ (u1 , . . . , un−2 , x, y) ∈ ρ

∨(u1 , . . . , uk , x, y, uk+1 , . . . , un−2 ) ∈ ρ, for any k ∈ {1, . . . , n − 3}}. If it is clear which relation we are talking about, then we use the notation L(x) and R(x) instead of Lρ (x) and Rρ (x). We notice that, if L(x) = R(x), for any x ∈ H, the hyperproduct ‘‘⊗ρ ’’ coincides with Rosenberg’s hyperproduct. It is obvious that (see [12]): 1. y ∈ L(x) if and only if x ∈ R(y), for any x, y ∈ H. 2. x∈H L(x) = H if and only if, for any x ∈ H , R(x) ̸= ∅. 3. x∈H R(x) = H if and only if, for any x ∈ H , L(x) ̸= ∅. Proposition 1 (See [12, Proposition 14]). ⟨H , ⊗ρ ⟩ is an Hv -group if and only if L(x) ̸= ∅ and R(x) ̸= ∅, for any x ∈ H. For a binary relation ρ on H, this property can be written in the following way. Proposition 2. ⟨H , ⊗ρ ⟩ is an Hv -group if and only if ρ is a binary relation with full domain and full range. Remark 1. If ρ1 and ρ2 are two binary relations defined on the same set H, then Lρ1 (x) = Lρ2 (x), for any x ∈ H, if and only if Rρ1 (x) = Rρ2 (x) if and only if ρ1 = ρ2 . But for n-ary relations, with n ≥ 3, this is not always true, as we see in the following example. Example 3. Set H = {1, 2, 3, 4, 5}; we consider the following two 4-ary relations on H:

ρ1 = {(1, 4, 4, 2), (2, 1, 4, 2), (2, 5, 3, 5), (3, 3, 3, 3), (3, 1, 4, 2)}, ρ2 = {(1, 4, 2, 1), (2, 5, 3, 3), (3, 5, 3, 1), (4, 4, 4, 4)}. For any x ∈ H, we obtain Lρ1 (x) = Lρ2 (x), which is equivalent to Rρ1 (x) = Rρ2 (x), but ρ1 ̸= ρ2 .

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Therefore, on the set of n-ary relations defined on a non-empty set H, with n ≥ 3, we introduce an equivalence:

ρ1 ∼ ρ2 ⇐⇒



Lρ1 (x) = Lρ2 (x) , Rρ1 (x) = Rρ2 (x)

for any x ∈ H .

(2)

Definition 4. With any n-ary relation ρ, n ≥ 3, defined on a non-empty set H we associate a binary relation ρ 0 in the following way: if (a1 , . . . , an ) ∈ ρ then (ai , ai+1 ) ∈ ρ 0 ,

for any i, 1 ≤ i ≤ n − 1.

(3)

0 0 It is obvious that, for any x ∈ H , Lρ (x) = Lρ (x), Rρ (x) = Rρ (x) and thus we obtain the following result.

Proposition 5. Let ρ be an n-ary relation on a set H and ρ 0 be its associated binary relation as in (3). Then the hypergroupoid associated with ρ 0 (in the sense of Rosenberg, or Corsini, or Cristea and Ştefănescu) coincides with the hypergroupoid associated (in the same sense) with any n-ary relation λ ∼ ρ . Therefore, the study of the hypergroupoid associated with an n-ary relation ρ , in the sense of Rosenberg, or Corsini, or Cristea and Ştefănescu, is reduced to the study of the hypergroupoid associated, in the same sense, with the corresponding binary relation ρ 0 . Definition 6 (See [23]). With any n-ary relation ρ on a set H , n ≥ 3, we may associate a binary relation ρ b on H as follows. For any (x, y) ∈ H 2 , we put (x, y) ∈ ρ b if there exist (x1 , x2 , . . . , xn ) ∈ ρ and natural numbers i, j such that 1 ≤ i < j ≤ n, x = xi , y = xj . Remark 2. It is obvious that, for any n-ary relation ρ on a set H, the corresponding binary relations

ρ 0 and ρ b verify the relation: ρ 0 ⊆ ρ b .

Example 7. On the set H = {1, 2, 3, 4, 5} we consider the following 5-ary relations: ∆ = {(x, x, x, x, x) | x ∈ H }, ρ = {(1, 3, 4, 5, 2), (1, 3, 3, 3, 3)}. We find that ∆0 = ∆b = {(x, x) | x ∈ H }, ρ 0 = {(1, 3), (3, 4), (4, 5), (5, 2), (3, 3)}, ρ b = {(1, 3), (3, 4), (4, 5), (5, 2), (3, 3), (1, 4), (1, 5), (1, 2), (3, 5), (3, 2), (4, 2)}; thus ρ 0 ( ρ b . Proposition 8 ([23, Theorem 3.1]). Let ρ be an n-ary preordering (or an n-ary equivalence) on a set H. Then ρ b is a preordering on H. In a similar way, one can prove the following result: Corollary 9. Let ρ be an n-ary preordering (or an n-ary equivalence) on a set H. Then ρ 0 is a preordering on H. 3. Fundamental relations on hypergroupoids It may happen that the hyperoperation ‘‘◦’’ does not discriminate between a pair of elements of H when two elements play interchangeable roles with respect to the hyperoperation. On a hypergroupoid ⟨H , ◦⟩, the following three equivalence relations, called the operational equivalence, the inseparability and the essential indistinguishability, respectively, may be defined (see [17]):

• x ∼o y ⇐⇒ x ◦ a = y ◦ a and a ◦ x = a ◦ y, for any a ∈ H; • x ∼i y ⇐⇒ for a, b ∈ H, and we have x ∈ a ◦ b ⇐⇒ y ∈ a ◦ b; • x ∼e y ⇐⇒ x ∼o y and x ∼i y. xi and  xe , respectively, denote the equivalence classes of x with respect to the For any x ∈ H, let  xo , relations ∼o , ∼i and ∼e . We say that a hypergroupoid ⟨H , ◦⟩ is reduced if, for any x ∈ H , xe = {x}.

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Definition 10 (See [2]). An equivalence relation R on a hypergroupoid ⟨H , ◦⟩ is called regular to the right if, for any (x, y) ∈ H 2 , we have xRy H⇒ ∀u ∈ x ◦ a, ∃v ∈ y ◦ a such that uRv and ∀v ′ ∈ y ◦ a, ∃u′ ∈ x ◦ a such that u′ Rv ′ , for any a ∈ H. We say that R is strongly regular to the right if, for any (x, y) ∈ H 2 , we have the implication xRy H⇒ ∀u ∈ x ◦ a, ∀v ∈ y ◦ a we have uRv,

for any a ∈ H .

Similarly, we define the (strong) regularity to the left. An equivalence relation R is (strongly) regular if it is (strongly) regular to the left and to the right. To start with, we investigate whether the previous fundamental relations are regular or strongly regular. Proposition 11. 1. The operational equivalence is regular, but in general not strongly regular. 2. The inseparability is not a regular relation. 3. The essential indistinguishability is regular, but in general not strongly regular. Proof. (i) We consider x, y ∈ H such that x ∼o y and u is arbitrary in H. Then x ◦ u = y ◦ u (respectively, u ◦ x = u ◦ y) and, for any a ∈ x ◦ u (respectively, a ∈ u ◦ x), there exists b = a ∈ y ◦ u (respectively, b = a ∈ u ◦ y) such that a ∼o b (since ∼o is reflexive). Therefore ∼o is regular. Now we give an example of a hypergroup H on which the relation ∼o is not strongly regular. Let H = {a, b, c } be the following hypergroup: H a b c

a a a a, b , c

b a a a, b , c

c a, b , c a, b , c c

We observe that a ∼o b and we do not have a ∼o c. We suppose that ∼o is strongly regular; that is, a ∼o b H⇒ ∀u ∈ H , ∀x ∈ a ◦ u, ∀y ∈ b ◦ u : x ∼o y. For u = c , x = a and y = c it follows that a ∼o c, which is a contradiction. (ii) Let H = {a, b, c } be the following hypergroupoid: H a b c

a a a a, b , c

b a a b, c

c a, b, c b, c b, c

We suppose that the relation ∼i is regular, so from b ∼i c it results that, for any u ∈ H and any x ∈ b ◦ u, there exists y ∈ c ◦ u such that x ∼i y. For u = b and x = a we obtain that there exists y ∈ {b, c } such that a ∼i b or a ∼i c, which is a contradiction. (iii) We prove that the relation ∼e is regular, that is, a ∼e b H⇒ ∀u ∈ H , ∀x ∈ a ◦ u, ∃y ∈ b ◦ u : x ∼e y. From a ∼e b we have a ∼o b, so, for any u ∈ H , a ◦ u = b ◦ u (respectively, u ◦ a = u ◦ b). Then, there exists y = x ∈ b ◦ u (respectively, y = x ∈ u ◦ b) such that x ∼e y (since the relation ∼e is reflexive). Now we give an example of a hypergroup H on which the relation ∼e is not strongly regular. Let H = {a, b, c } be the following hypergroup: H a b c

a a, b a, b a, b , c

b a, b a, b a, b , c

c a, b , c a, b , c a, b , c

We easily observe that a ∼e b and we suppose that ∼e is strongly regular; then, for any u ∈ H and any x ∈ a ◦ u, y ∈ b ◦ u, it results that x ∼e y. For u = c , x = a, y = c it follows that a ∼e c, which is a contradiction. 

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Proposition 12. Let ⟨H , ◦⟩ be a hypergroupoid. The relation ∼i is regular on H if and only if a ∼i b H⇒ a ∼o b, for a, b ∈ H. Proof. Let us suppose that the relation ∼i is regular on H. We consider a, b ∈ H such that a ∼i b. We have to prove that for all u ∈ H , a ◦ u = b ◦ u and u ◦ a = u ◦ b. For x ∈ a ◦ u, by the regularity of ∼i , there exists y ∈ b ◦ u such that x ∼i y; since x ∼i y it follows that x ∈ b ◦ u, so a ◦ u ⊂ b ◦ u. Similarly, b ◦ u ⊂ a ◦ u, so a ◦ u = b ◦ u, for any u ∈ H. In the same way we prove u ◦ a = u ◦ b, for any u ∈ H. Let’s suppose now that a, b ∈ H with a ∼i b, so a ∼o b, by hypothesis. We show that, for any u ∈ H and any x ∈ a ◦ u (respectively, x ∈ a ◦ u), there exists y ∈ b ◦ u (respectively, y ∈ u ◦ b) such that x ∼i y. We take u ∈ H and x ∈ a ◦ u (respectively, x ∈ u ◦ a); it results that there exists y = x ∈ b ◦ u = a ◦ u (respectively, y = x ∈ u ◦ b = u ◦ a) such that x ∼i y. So, ∼i is regular on H.  4. Reduced hypergroupoids In this section we determine necessary and sufficient conditions for the Hv -group ⟨H , ⊗ρ ⟩ to be reduced. Proposition 13. Let ρ be a binary relation on H with full domain and full range such that, for any x ∈ H , x ̸∈ L(x). For any x, y ∈ H, the following implications hold: (i) x ∼o y ⇐⇒ L(x) = L(y) and R(x) = R(y); (ii) x ∼i y ⇐⇒ L(x) = L(y) and R(x) = R(y); (iii) therefore x ∼e y ⇐⇒ L(x) = L(y) and R(x) = R(y). Proof. (i) Let x, y be arbitrary elements in H such that x ∼o y, that is, for any a ∈ H , L(x) ∪ R(a) = L(y) ∪ R(a) and L(a) ∪ R(x) = L(a) ∪ R(y). First we prove that x ̸∈ L(y) and y ̸∈ L(x). If, by absurdity, y ∈ L(x), then y ∈ L(y) ∪ R(a), for any a ∈ H and since y ̸∈ L(y) by hypothesis, it follows that y ∈ R(a), for any a ∈ H, and thus y ∈ R(y) ⇐⇒ y ∈ L(y), which is a contradiction. Similarly it results that x ̸∈ L(y). Let now u ∈ L(x) \ L(y); thus u ∈ R(a) for an a ∈ H and again, as mentioned above, we have a contradiction. Therefore L(x) ⊂ L(y) and similarly L(y) ⊂ L(x); thereby L(x) = L(y) and similarly R(x) = R(y). Besides, the sufficiency is evident and this completes the proof of the first assertion. (ii) Assume that x, y ∈ H such that x ∼i y, that is x ∈ L(a) ∪ R(b) iff y ∈ L(a) ∪ R(b), for a, b ∈ H. As in the previous part, we can prove that x ̸∈ L(y) and y ̸∈ L(x). Now let u ∈ L(x); then x ∈ L(x)∪R(u), and therefore y ∈ L(x)∪R(u); by the previous considerations, it follows that y ∈ R(u), that is u ∈ L(y) and thus L(x) ⊂ L(y). Similarly one can prove the other inclusion, obtaining L(x) = L(y), and similarly R(x) = R(y). Conversely, let x, y be in H such that L(x) = L(y) and R(x) = R(y). Since R(x) ̸= ∅, there exists a ∈ H \ {x} such that a ∈ R(x) = R(y); then x ∈ L(a) iff y ∈ L(a). Similarly we obtain that x ∈ R(b) iff y ∈ R(b). Now it is clear that x ∈ L(a) ∪ R(b) iff y ∈ L(a) ∪ R(b), which means that x ∼i y and the proof is complete.  Corollary 14. Let ρ be a binary relation on H with full domain and full range such that, for any x ∈ H , x ̸∈ L(x). Then the Hv -group ⟨H , ⊗ρ ⟩ is reduced if and only if there are no x ̸= y ∈ H such that L(x) = L(y) and R(x) = R(y). Proposition 15. Let ρ be a reflexive binary relation on H and let x ̸= y be two arbitrary elements in H. Then in the Hv -group ⟨H , ⊗ρ ⟩ we have x ∼o y if and only if one of the following assertions holds: (i) L(x) = L(y) and R(x) = R(y); (ii) L(u) = H, for any u ∈ L(x)1L(y), and R(v) = H, for any v ∈ R(x)1R(y), where, for any two sets A and B, we define A1B = (A \ B) ∪ (B \ A).

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Proof. Let x, y be in H such that x ∼o y, that is, for any a ∈ H,



L(x) ∪ R(a) = L(y) ∪ R(a) L(a) ∪ R(x) = L(a) ∪ R(y).

(4)

Since x ∈ L(x), it follows that x ∈ L(x) ∪ R(y) = L(y) ∪ R(y) and thus x ∈ L(y) or x ∈ R(y). We distinguish the following situations. (a) If x ̸∈ L(y), then x ∈ R(y) ⇐⇒ y ∈ L(x). It is clear that L(x) ̸= L(y) and R(x) ̸= R(y). Since L(y) ∪ R(a) = L(x) ∪ R(a) ∋ x, for any a ∈ H, it follows that x ∈ R(a) ⇐⇒ a ∈ L(x), for any a ∈ H; thus L(x) = H. Similarly, since y ̸∈ R(x), it follows that R(y) = H. Then the formula (4) becomes, for any a ∈ H,



L(y) ∪ R(a) = H L(a) ∪ R(x) = H .

(5)

Thereby, for any u ∈ H \ L(y), u ∈ R(a), for any a ∈ H, that is L(u) = H, and, for any v ∈ H \ R(x), we obtain v ∈ L(a), for any a ∈ H, that is R(v) = H. These two conditions are equivalent to the second assertion of the hypothesis. (b) If x ̸∈ R(y), then we obtain the same result as in the previous case. (c) If x ∈ L(y)∩ R(y), then L(x) = L(y) and R(x) = R(y) (that is the first assertion of the proposition), or L(x) ̸= L(y) or R(x) ̸= R(y). We suppose that L(x) ̸= L(y); thus there exists u ∈ L(x)1L(y); for instance we consider u ∈ L(x) \ L(y). Since L(x) ∪ R(a) = L(y) ∪ R(a), for any a ∈ H, it follows that u ∈ R(a), for any a ∈ H, and therefore L(u) = H. Similarly, if u ∈ L(y) \ L(x), then L(u) = H, for any u ∈ L(x)1L(y). By similar considerations, if R(x) ̸= R(y), then R(v) = H, for any v ∈ R(x)1R(y). Reciprocally, if L(x) = L(y) and R(x) = R(y), it is clear that x ∼o y. We suppose that L(x) ̸= L(y) and we prove that formula (4) holds. If x ̸∈ L(y) or x ̸∈ R(y), we have proved that (4) is equivalent to condition (ii) of the hypothesis. If x ∈ L(y) ∩ R(y), we show that, for any a ∈ H , L(x) ∪ R(a) = L(y) ∪ R(a). Set an arbitrary u ∈ L(x) ∪ R(a); if u ∈ R(a), then u ∈ L(y) ∪ R(a). If u ∈ L(x) and u ∈ L(y), it follows that L(x) ∪ R(a) ⊂ L(y) ∪ R(a). If u ∈ L(x) and u ̸∈ L(y), then u ∈ L(x)1L(y) and therefore L(u) = H, that is a ∈ L(u), for any a ∈ H, which is equivalent to u ∈ R(a), for any a ∈ H, and again we obtain that L(x) ∪ R(a) ⊂ L(y) ∪ R(a). Analogously we prove the other inclusion and the other relation L(a) ∪ R(x) = L(a) ∪ R(y), for any a ∈ H, which concludes the proof.  Proposition 16. Let ρ be a reflexive binary relation on H. In the Hv -group ⟨H , ⊗ρ ⟩ we obtain that x ∼i y if and only if one of the following holds: 1. 2. 3. 4. 5.

L(x) = R(y) = H. L(y) = R(x) = H. L(x) = L(y) = H. R(x) = R(y) = H. L(x) = L(y) ̸= H and R(x) = R(y) ̸= H.

Proof. Let x ̸= y be in H such that x ∼i y. Since x ∈ L(x), it follows that x ∈ L(x) ∪ R(a), for any a ∈ H; thus y ∈ L(x) ∪ R(a), for any a ∈ H. On the one hand, if y ̸∈ L(x), then y ∈ R(a), for any a ∈ H, and therefore L(y) = H. Moreover x ̸∈ R(y), but y ∈ L(a) ∪ R(y), for any a ∈ H. Since x ∼i y, it follows that x ∈ L(a), for any a ∈ H, that is R(x) = H. On the other hand, if there exists u ∈ H such that y ̸∈ R(u), we distinguish the following situations:

• u = x, so y ̸∈ R(x), which is equivalent to x ̸∈ L(y); we obtain (as in the case y ̸∈ L(x)) that L(x) = R(y) = H. • u ̸= x, so there exists u ∈ H such that u ̸∈ L(y), and thus L(y) ̸= H.

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We suppose that L(x) ̸= L(y); for instance, set v ∈ L(x) \ L(y). Then x ∈ R(v), y ̸∈ R(v), but x ∈ L(a) ∪ R(v), for any a ∈ H, that is R(y) = H and then L(a) ∪ R(b) ∋ x, for any a, b ∈ H. If L(x) ̸= H and R(x) ̸= H, this means that there exist a¯ and b¯ in H such that a¯ ̸∈ L(x) and b¯ ̸∈ R(x). Then x ̸∈ L(¯a) ∪ R(b¯ ), which is false. Summarizing, if L(x) ̸= L(y), then R(y) = L(x) = H or R(y) = R(x) = H. Now we suppose that L(x) = L(y) ̸= H. If there exists u ∈ R(x) \ R(y), then x ∈ L(u) ⊂ L(u) ∪ R(a), whenever a ∈ H, and since x ∼i y, it follows that y ∈ R(a), whenever a ∈ H; thus L(y) = H which is false. We obtain a similar result if R(y) \ R(x) ̸= ∅. In conclusion, if L(x) = L(y) ̸= H, then R(x) = R(y). Conversely, if one of the cases 1–4 holds, it follows that a ⊗ρ b ⊃ {x, y}, for any a, b ∈ H; therefore x ∼i y. If L(x) = L(y) ̸= H and R(x) = R(y) ̸= H, then x ∈ L(a), for any a ∈ H, if and only if y ∈ L(a) and x ∈ R(b), for b ∈ H, if and only if y ∈ R(b). Thus x ∈ L(a) ∪ R(b), for a, b ∈ H, if and only if y ∈ L(a) ∪ R(b), so again x ∼i y.  Now, using Propositions 15 and 16, we can look for necessary and sufficient conditions such that the Hv -group ⟨H , ⊗ρ ⟩, obtained from a reflexive binary relation defined on H, is a reduced hypergroupoid. In the previous case, when ρ is a binary relation on H, with full domain and full range, such that x ̸∈ L(x), for any x ∈ H, the conditions are very simple: ⟨H , ⊗ρ ⟩ is reduced if and only if there are no x ̸= y ∈ H such that L(x) = L(y) and R(x) = R(y). First we illustrate in the following example that these conditions are not sufficient in the second case when ρ is a reflexive binary relation. Example 17. Let H = {x, y, z , t , u} and let ρ be defined as follows: L(x) = L(y) = H , L(x) = {x, z }, L(t ) = {z , t }, L(u) = {z , u} and R(x) = {x, y, z }, R(y) = {x, y}, R(z ) = H , R(t ) = {x, y, t }, R(u) = {x, y, u}. We notice that R(x) ̸= R(y), but x ∼e y, since the conditions of Propositions 15(ii) and 16(3) are satisfied. Therefore ⟨H , ⊗ρ ⟩ is not a reduced hypergroupoid. Moreover, it should be noted that, if L(x) ̸= L(y) and R(x) ̸= R(y), for any x ̸= y, we may obtain a non-reduced hypergroupoid. Let us see the example below. Example 18. Let H = {x, y, z , t , u} and let ρ be defined as follows: L(x) = H , L(y) = {y, z , u}, L(z ) = {y, z }, L(t ) = H , L(u) = {x, y, z , u} and R(x) = {x, t , u}, R(y) = R(z ) = H , R(t ) = {x, t }, R(u) = {x, y, t , u}. We notice that x ∼o y ∼o t ∼o u and x ∼i y ∼i z ∼i t and thus ⟨H , ⊗ρ ⟩ is not a reduced hypergroupoid. Instead, if L(x) ̸= L(y) ̸= H ̸= L(x) or R(x) ̸= R(y) ̸= H ̸= R(x), for any x ̸= y ∈ H, then, by Propositions 15 and 16, we obtain that xˆ e = {x}, for any x ∈ H, that is ⟨H , ⊗ρ ⟩ is reduced. This is not a necessary condition, as we can note from the following example. Example 19. Let H = {x, y, z , t , u, v} and let ρ be defined as follows: L(x) = {x, y, z , t }, L(y) = {x, y, t }, L(z ) = H , L(t ) = {y, t }, L(u) = {u, t }, L(v) = {v, t } and R(x) = {x, y, z }, R(y) = {x, y, z , t }, R(z ) = {x, z }, R(t ) = H , R(u) = {z , u}, R(v) = {z , v}. We notice that x ∼o y, but x i y and z ∼i t, but z o t and moreover that ⟨H , ⊗ρ ⟩ is reduced. It is not simple to find necessary and sufficient conditions for two elements of the hypergroupoid

⟨H , ⊗ρ ⟩, when ρ is a reflexive relation on H, such that both conditions of Propositions 15 and 16 are satisfied. 5. Binary relations and Boolean matrices

Our ultimate aim is to characterize, in terms of matrices, the fundamental relations defined on a hypergroupoid, using the interpretation of binary relations as Boolean matrices. Let H be a finite set of cardinality n and ρ be a binary relation defined on H. To simplify the notation we set, without loss the generality, H = {1, 2, . . . , n}. We denote by M (ρ) = (aij ), with i, j ∈ {1, 2, . . . , n}, the Boolean matrix representing ρ , that is aij = 1, if (i, j) ∈ ρ , and aij = 0, if (i, j) ̸∈ ρ . We recall that in a Boolean algebra it holds that 0 + 0 = 0, 1 + 0 = 0 + 1 = 1 + 1 = 1 and 0 · 0 = 0 · 1 = 1 · 0 = 0, 1 · 1 = 1. First we characterize the matrix representing a binary relation such that its associated hypergroupoid is an Hv -group.

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Corollary 20. Let ρ be a binary relation on H and let M (ρ) = (aij ) be the Boolean matrix associated with it. ⟨H , ⊗ρ ⟩ is an Hv -group if and only if, for any i ∈ {1, 2, . . . , n}, there exist j1i , j2i ∈ {1, 2, . . . , n} such that aij1i = 1 and aj2i i = 1 (i.e. in any line and any column of the matrix M (ρ) there is at least one 1). Proof. Indeed, by Proposition 1, ⟨H , ⊗ρ ⟩ is an Hv -group if and only if, for any i ∈ {1, 2, . . . , n}, L(i) ̸= ∅ and R(i) ̸= ∅. It follows that there exist j1i , j2i ∈ {1, 2, . . . , n} such that (j1i , i), (i, j2i ) ∈ ρ , that is aij1i = 1 and aj2i i = 1.  As in [4], a Boolean matrix M (ρ) determined by a binary relation ρ is called a good matrix if ⟨H , ⊗ρ ⟩ is a hypergroupoid, that is, for any i ∈ {1, 2, . . . , n}, there exists j ∈ {1, 2, . . . , n} such that aij = 1 or aji = 1. Now we reformulate the results obtained in Section 4 concerning the reduced hypergroupoids. Proposition 21. Let ρ be a binary relation on H = {1, 2, . . . , n} such that M (ρ) = (aij ) is a good matrix with no 1 on the principal diagonal. Then, for i, j ∈ H, we have i ∼e j if and only if aki = akj and aik = ajk , for any k ∈ H. Proof. Since M (ρ) = (aij ) is a good matrix with no 1 on the principal diagonal, it follows that i ̸∈ L(i), for any i ∈ H and, according to Proposition 13, we have that i ∼e j if and only if L(i) = L(j) and R(i) = R(j), which is equivalent to aki = akj and aik = ajk , for any k ∈ H.  Therefore Corollary 14 becomes: Corollary 22. Let ρ be a binary relation on H = {1, 2, . . . , n} such that M (ρ) = (aij ) is a good matrix with no 1 on the principal diagonal. Then the hypergroupoid ⟨H , ⊗ρ ⟩ is reduced if and only if M (ρ) has not two identical lines i, j such that the columns i, j are identical and vice versa. Now we treat the case of a reflexive binary relation; thus the Boolean matrix M (ρ) has the principal diagonal formed only by 1’s. According to Propositions 15 and 16 we obtain: Proposition 23. For a reflexive binary relation ρ , in the Hv -group ⟨H , ⊗ρ ⟩ we find: (i) for i, j ∈ H , i ∼o j if and only if one of the following assertions holds: 1. the line and column i coincide with the line and column j, respectively; 2. alk = 1, for any l, k ∈ H such that (aki = 1 ∧ akj = 0) or (akj = 1 ∧ aki = 0), and similarly, akl = 1, for any l, k ∈ H such that (aik = 1 ∧ ajk = 0) or (ajk = 1 ∧ aik = 0); (ii) for i, j ∈ H , i ∼i j if and only if one of the following assertions holds: 1. aki = ajk = 1, for any k ∈ H; 2. akj = aik = 1, for any k ∈ H; 3. aki = akj = 1, for any k ∈ H; 4. aik = ajk = 1, for any k ∈ H; 5. aki = akj and ail = ajl , for any k, j ∈ H, and there exist k0 , l0 ∈ H such that ak0 i ̸= 1 and ail0 ̸= 1. Having the characterization with Boolean matrices of the reduced hypergroup associated with a binary relation as in (1), it is not difficult to create an informatics program (similar to those in [10,20,28]) which computes the number of the above reduced hypergroups. 6. Conclusions and future work In this paper we have provided new properties of the fundamental relations, called the operational equivalence, the inseparability and the essential indistinguishability, defined on a hypergroupoid [17]. We have continued the study of the reduced hypergroupoids associated with a binary relation started in [11]. Focusing on the above mentioned equivalences, necessary and sufficient conditions for the Hv -group associated with a binary relation in the sense of Cristea and Ştefănescu [12] to be reduced have been presented. It is well known that every binary relation ρ on a finite set H, with card H = n, may be represented by a Boolean matrix M (ρ) and, conversely, any Boolean matrix of order n may define a binary relation

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