Research article Special Issues

Semilattice strongly regular relations on ordered n-ary semihypergroups

  • In this paper, we introduce the concept of j-hyperfilters, for all positive integers 1jn and n2, on (ordered) n-ary semihypergroups and establish the relationships between j-hyperfilters and completely prime j-hyperideals of (ordered) n-ary semihypergroups. Moreover, we investigate the properties of the relation N, which is generated by the same principal hyperfilters, on (ordered) n-ary semihypergroups. As we have known from [21] that the relation N is the least semilattice congruence on semihypergroups, we illustrate by counterexample that the similar result is not necessarily true on n-ary semihypergroups where n3. However, we provide a sufficient condition that makes the previous conclusion true on n-ary semihypergroups and ordered n-ary semihypergroups where n3. Finally, we study the decomposition of prime hyperideals and completely prime hyperideals by means of their N-classes. As an application of the results, a related problem posed by Tang and Davvaz in [31] is solved.

    Citation: Jukkrit Daengsaen, Sorasak Leeratanavalee. Semilattice strongly regular relations on ordered n-ary semihypergroups[J]. AIMS Mathematics, 2022, 7(1): 478-498. doi: 10.3934/math.2022031

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  • In this paper, we introduce the concept of j-hyperfilters, for all positive integers 1jn and n2, on (ordered) n-ary semihypergroups and establish the relationships between j-hyperfilters and completely prime j-hyperideals of (ordered) n-ary semihypergroups. Moreover, we investigate the properties of the relation N, which is generated by the same principal hyperfilters, on (ordered) n-ary semihypergroups. As we have known from [21] that the relation N is the least semilattice congruence on semihypergroups, we illustrate by counterexample that the similar result is not necessarily true on n-ary semihypergroups where n3. However, we provide a sufficient condition that makes the previous conclusion true on n-ary semihypergroups and ordered n-ary semihypergroups where n3. Finally, we study the decomposition of prime hyperideals and completely prime hyperideals by means of their N-classes. As an application of the results, a related problem posed by Tang and Davvaz in [31] is solved.



    The investigation of partially ordered algebraic systems was initiated by Conrad[16,17]. Particularly, an ordered semigroup (also called a partially ordered semigroup or a po-semigroup) is a well-known example of the partially ordered algebraic systems, which is a semigroup together with a compatible partial order. The notion of ideals and filters on ordered semigroups play a significant role in studying the structure of ordered semigroups, especially the decompositions of ordered semigroups see [2,23,34], and there are many applications in various branches of mathematics and computer science. Many mathematicians have studied different aspects of ideals and filters on ordered semigroups. For examples, see [2,22,23,24,25,34].

    The theory of algebraic hyperstructures have been originated by Marty[28] in 1934 when he studied hypergroups, which are considered as a generalization of groups, by using hyperoperations. Later on, several kinds of hyperstructures were studied by many mathematicians, for instance, see [4,5,6,7,19,20,26]. In 2011, Davvaz and Heidari[18] applied the notion of ordered semigroups to hyperstructures and presented a new class of hyperstructures called ordered semihypergroups. Such hyperstructure can be considered as an extension and a generalization of ordered semigroups. The relationships between ordered semihypergroups and ordered semigroups were described by Davvaz et al.[11]. Some properties of hyperideals on ordered semihypergroups were investigated by Changphas and Davvaz[3]. Next, Tang et al.[32] introduced the concept of a (right, left) hyperfilter on ordered semihypergroups and discussed their related properties. Omidi and Davvaz[29] generalized the concept of the relation N on ordered semigroups into ordered semihypergroups and proved some remarkable results. Note that, in an ordered semihypergroup S, the relation N on S is defined by N:={(x,y)S×S:N(x)=N(y)} where N(x) is the hyperfilter generated by the element x of S. Kehayopulu[21] proved that the relation N is the least semilattice congruence on semihypergroups. Meanwhile, Kehayopulu illustrate by counterexample that N is not the least semilattice congruence on ordered semihypergroups in general. The author introduced the concept of complete semilattice congruence on ordered semihypergroups and proved that N is the smallest complete semilattice congruence. Recently, Tang and Davvaz[31] studied the relation N and Green's relations, especially the relations L,R and J, on ordered semihypergroups. They showed that every (respectively, right, left) hyperideal can be decomposable into its (R-, L-)J-classes and posed a problem: " Can every prime hyperideal be decomposable into its N-classes?"

    On the other hand, the investigation of algebraic hyperstructures have been widely expanded. In 2006, Davvaz and Vougiouklis[14] applied the notion of hypergroups to n-ary algebras and presented the notion of n-ary hypergroups for n2. Leoreanu-Fotea and Davvaz [27] studied some binary relations on n-ary hypergroups. Anvariyeh and Momeni [1] investigated the interesting properties of n-ary relations on n-ary hypergroups. In 2009, Davvaz et al.[13] introduced a new class of hyperstructure called n-ary semihypergroups. This new class can be considered as a natural extension of semigroups, semihypergroups, ternary semigroups, ternary semihypergroups and n-ary semigroups. The interesting results of n-ary semigroups and n-ary semihypergroups, the readers can be referred to [8,12,13,15,30]. Recently, Daengsaen and Leeratanavalee [9] applied the theory of n-ary algebras to ordered semihypergroups and introduced the notion of ordered n-ary semihypergroups for n2. Such new notion is a natural generalization of ordered semigroups, ordered semihypergroups, ordered ternary semigroups and ordered ternary semihypergroups. Also, they characterized several kinds of regularities of ordered n-ary semihypergroups. In this paper, we attempt to study the notion of hyperfilters on ordered n-ary semihypergroups. In particular, we introduce the concept of j-hyperfilters, for all positive integers 1jn and n2, of ordered n-ary semihypergroups and use such concept to construct the relation N for studying the properties of semilattice congruence on ordered n-ary semihypergroups where n2.

    The present paper is organized as follows. In Section 2, we recall some basic notions and elementary results of algebraic hyperstructures. In Section 3, we introduce the concept of j-hyperfilters, which is a generalization of left and right hyperfilters of ordered semihypergroups and discuss the connection between j-hyperfilters and completely prime j-hyperideals on ordered n-ary semihypergroups where n2. In Section 4, we investigate some properties of the relation N on n-ary semihypergroups. We illustrate by counterexample that N does not necessarily to be the least semilattice congruence on n-ary semihypergroup, where n3, in general. At the meantime, we provide a sufficient condition that makes the previous result true. In Section 5, we introduce the concept of a complete -semilattice (strongly) regular relation on ordered n-ary semihypergroups, where n3, and prove that N is the smallest complete -semilattice strongly regular relation. In the last section, we give an example to show that a prime hyperideal of ordered n-ary semihypergroups cannot be decomposable into its N-classes in general. However, we provide some special condition for prime hyperideals that lead to be decomposable into its N-classes. As an application of our results, the related problem posed by Tang and Davvaz is solved.

    In this section, we recall some elementary results of algebraic hyperstructures. For more detail, we refer to [8,9,12,13,14].

    Let S be a nonempty set and let P(S) denote the set of all nonempty subsets of S. A mapping f:S××SS appearsn2timesP(S) is called an n-ary hyperoperation. A structure (S,f) is called an n-ary hypergroupoid [14]. Throughout this paper, we use the symbol akj to denote a sequence of elements aj,aj+1,...,ak of S. For the case k<j, akj is the empty symbol. Then f(a1,a2,...,an)=f(an1) and

    f(a1,...,aj,bj+1,...,bk,ck+1,...,cn)=f(aj1,bkj+1,cnk+1).

    In case a1=...=aj=a and ck+1=...=cn=c, we write the second expression in the form f(aj,bkj+1,cnk). For the symbols of a sequence of subsets of S, we denote analogously. For X1,...,XnP(S), we define

    f(Xn1)=f(X1,...,Xn):={f(an1):ajXj,j=1,...,n}.

    If X1={a1}, then we write f({a1},Xn2) as f(a1,Xn2) and analogously in other cases. In case X1=...=Xj=Y and Xj+1=...=Xn=Z, we write f(Xn1) as f(Yj,Znj). The hyperoperation f is associative[13] if

    f(f(an1),a2n1n+1)=f(aj11,f(an+j1j),a2n1n+j)

    hold for all a2n11S and for all 1<jn. In this case, the n-ary hypergroupoid (S,f) is called an n-ary semihypergroup (also called an n-ary hypersemigroup)[13].

    An ordered n-ary semihypergroup (S,f,) (also called a partially ordered n-ary semihypergroup or a po-n-ary semihypergroup)[9] is an n-ary semihypergroup (S,f) and a partially ordered set (S,) such that a partial order is compatible with f. Indeed, for any a,bS,

    ab implies f(cj11,a,cnj+1)f(cj11,b,cnj+1)

    for all cn1S and for all 1jn. Note that, for any X,YP(S), XY means for every xX there exists yY such that xy. If H is an n-ary subsemihypergroup of an ordered n-ary semihypergroup (S,f,), i.e f(Hn)H, then (H,f,) is an ordered n-ary semihypergroup.

    Definition 2.1. [9] Let (S,f,) be an ordered n-ary semihypergroup. For any positive integer 1jn and n2, a nonempty subset I of S is called a j-hyperideal of S if it satisfies the following two conditions:

    (I1)I is a j-hyperideal of an n-ary semihypergroup (S,f), i.e., f(aj11,b,anj+1)I for all bI and aj11,anj+1S.

    (I2) If bI and cS such that cb then cI.

    If I is a j-hyperideal, for all 1jn, then I is said to be a hyperideal.

    Here, 1-hyperideal (n-hyperideal) is called a right hyperideal (left hyperideal), respectively. Throughout this paper, we let S stand for an ordered n-ary semihypergroup (S,f,). By applying Lemma 4 in [8], the following lemma is obtained.

    Lemma 2.2. [8] For any positive integer 1jn, let {Ik:kJ} be a collection of j-hyperideals (hyperideals) of S. Then kJIk is a j-hyperideal (hyperideal) of S and kJIk is also a j-hyperideal (hyperideal) of S if kJIk.

    A nonempty subset P of S is called prime[31] if, for every an1S, f(an1)P implies akP for some 1kn. Equivalently, if an1P then f(an1)P.

    P is called completely prime[32] if, for every an1S, f(an1)P implies akP for some 1kn. Equivalently, if an1P then f(an1)P=.

    A (j-)hyperideal I of S is called a prime (j-)hyperideal (completely prime (j-)hyperideal) of S if I is a prime (completely prime) subset of S. Clearly, every completely prime (j-)hyperideal of S is a prime (j-)hyperideal of S.

    Example 2.3. Let Z be the set of all integers. Define f:ZnP(Z) by

    f(xn1)={yZ : ymin{x1,...,xn}}

    for all xn1Z. Then (Z,f,) is an ordered n-ary semihypergroup where is a usual partial order on Z. Let N be the set of all natural numbers. Then (N,f,) is an ordered n-ary subsemihypergroup of Z.

    Example 2.4. Let S={w,x,y,z}. Define f:SnP(S) by f(xn1)=((...(x1x2)...)xn), for all xn1S, where is defined by the following table.

    By applying Example 2 in [31], (S,f) is an n-ary semihypergroup. Define a partial order on S as follows:

    ≤:={(w,w),(x,x),(y,y),(z,x),(z,y),(z,z)}.

    Such relation can be presented by the following diagram.

    Then (S,f,) is an ordered n-ary semihypergroup. It is easy to verify that the sets A1={x,z}, A2={y,z}, A3={w,x,z} and A4={x,y,z} are all proper 2-hyperideals of S. Then we obtain the results as follows:

    (1) A1 and A4 are not prime since there exists wSAk such that f(wn)Ak for all k=1,4. It follows that A1,A4 are also not completely prime.

    (2) A2 is prime since f(xn1)A2 for all xn1SA2. But, it is not completely prime since there exists wSA2 such that f(wn)A2.

    (3) A3 is completely prime since f(xn1)A3= for all xn1SA3. It follows that A3 is prime.

    In this section, we introduce the concept of j-hyperfilters on ordered n-ary semihypergroups, which generalizes the notion of right and left hyperfilters on ordered semihypergroups[32], and investigate their related properties.

    Definition 3.1. Let (S,f,) be an ordered n-ary semihypergroup. For any positive integer 1jn and n2, an n-ary subsemihypergroup K of S is said to be a j-hyperfilter of S if it satisfies the following two conditions:

    (F1)K is a j-hyperfilter of an n-ary semihypergroup (S,f), i.e., for all an1S, f(an1)K implies ajK.

    (F2) If for any aK and bS such that ab then bK.

    If K is a j-hyperfilter, for all 1jn, then K is called a hyperfilter.

    Here, 1-hyperfilter (n-hyperfilter) is said to be a right hyperfilter (left hyperfilter, respectively). For any aS and 1jn, we denote by Nj(a)(N(a)) the j-hyperfilter (hyperfilter) of S generated by a and called the principal j-hyperfilter (hyperfilter) generated by a. A j-hyperfilter (hyperfilter) K of S is said to be proper if KS. Clearly, Definition 3.1 coincides with Definition 3.1 given in [32] if n=2.

    Example 3.2. Let S={w,x,y,z}. Define f:SnP(S) by f(xn1)=((...(x1x2)...)xn), for all xn1S, where is defined by the following table

    and a partial order is defined as follows:

    ≤:={(w,w),(w,x),(x,x),(y,y),(y,z),(z,z)}.

    Such relation can be presented by the following diagram.

    By applying Example 3.3 in [33], (S,f,) is an ordered n-ary semihypergroup. Clearly, the sets {w,x} and S are all 1-hyperfilters of S.

    Proposition 3.3. Let (S,f,) be an ordered n-ary semihypergroup with n2. For any 1jn, let K1,K2 be two j-hyperfilters of S. Then K1K2 is a j-hyperfilter of S if K1K2.

    Proof. Let an1K1K2. Then an1Ki for all i=1,2. Since Ki is an n-ary subsemihypergroup, we obtain f(an1)K1K2. Thus K1K2 is an n-ary semihypergroup of S. Next, let bn1S and cf(bn1)(K1K2). Then cf(bn1)Ki for all i=1,2. Since Ki is a j-hyperfilter of S, we have bjKi for all i=1,2. So bjK1K2. Finally, let aK1K2 such that abS. Since Ki is a j-hyperfilter, we obtain bKi for all i=1,2. So bK1K2. Therefore K1K2 is a j-hyperfilter of S.

    By Proposition 3.3, we obtain the following result.

    Corollary 3.4. Let (S,f,) be an ordered n-ary semihypergroup with n2 and K1,K2 be two hyperfilters of S. K1K2 is a hyperfilter of S if K1K2.

    Proposition 3.3 and Corollary 3.4 show that the intersection of finitely many j-hyperfilters(hyperfilters) of S is always a j-hyperfilter(hyperfilter) if their intersection is nonempty. In 2015, Tang et al.[32] showed that the union of two hyperfilters of an ordered semihypergroup H (for a special case n=2) is not necessarily a hyperfilter of H (see Example 3.5 in [32]). Using the idea of Lemma 3.6 in [32], we obtain the results as follows.

    Proposition 3.5. Let (S,f,) be an ordered n-ary semihypergroup with n3. For any 1<j<n, let K1,K2 be two j-hyperfilters of S. Then K1K2 is a j-hyperfilter of S if and only if K1K2 or K2K1.

    Proof. Now, let j be a fixed positive integer with 1<j<n and n3. Let K1K2 be a j-hyperfilter of S. Assume that K1K2 and K2K1. There are a,bK1K2 such that aK1,aK2,bK1 and bK2. Since K1K2 is an n-ary subsemihypergroup, we have f(aj1,bnj+1)K1K2. If f(aj1,bnj+1)K1 then, since K1 is a j-hyperfilter of S, we obtain bK1. It is impossible. So f(aj1,bnj+1)K1=. This implies that f(aj1,bnj+1)K2. Since K2 is an n-ary subsemihypergroup, we obtain f(bj1,f(aj1,bnj+1),bnj)K2. Then

    f(bj1,f(aj1,bnj+1),bnj)={f(bj1,aj1,bn2j+1,f(bn))if 1<jn2,f(bj1,anj,f(a2jn1,b2n2j+1))if n2<j<n.

    Since K2 is a j-hyperfilter, we obtain aK2. It is impossible. Thus K1K2 or K2K1. Conversely, if K1K2 or K2K1 then K1K2 is a j-hyperfilter of S.

    Corollary 3.6. Let (S,f,) be an ordered n-ary semihypergroup with n2 and K1,K2 be two hyperfilters of S. Then K1K2 is a hyperfilter of S if and only if K1K2 or K2K1.

    Theorem 3.7. Let (S,f,) be an ordered n-ary semihypergroup with n2 and KP(S). For any 1jn, the following assertions are equivalent.

    (i)K is a j-hyperfilter of S.

    (ii)SK= or SK is a completely prime j-hyperideal of S.

    Proof. (i) (ii) Let K be a j-hyperfilter of S. Suppose that SK. We show that SK is a completely prime j-hyperideal of S. First of all, we show that K is a j-hyperideal of S. Let aSK and bj11,bnj+1S. If f(bj11,a,bnj+1)K then, since K is a j-hyperfilter, we get aK. It is impossible. So f(bj11,a,bnj+1)K=. It implies that f(bj11,a,bnj+1)SK. Next, let xS and ySK such that xy. If xSK then xK. By Definition 3.1 (F2), we have yK. It is impossible. Hence xSK. Thus SK is a j-hyperideal of S. Next, we show that SK is completely prime. Let an1S such that f(an1)(SK). Assume that aiSK for all i=1,...,n. Then aiK for all i=1,...,n. Since K is an n-ary subsemihypergroup, we obtain f(an1)K. It implies that f(an1)(SK)=, which is impossible. Hence aiSK for some i=1,...,n. Thus SK is a completely prime j-hyperideal of S.

    (ii) (i) If SK= then K=S is a j-hyperfilter of S. Now, suppose that SK is a completely prime j-hyperideal of S. We will show that K is a j-hyperfilter of S. First of all, let an1K. If f(an1)K then f(an1)(SK). Since SK is completely prime, we obtain aiSK for some i=1,...,n. It is impossible. Thus f(an1)K. Next, let bn1S and f(bn1)K. If bjK then bjSK. Since SK is a j-hyperideal, we obtain f(bn1)f(Sj1,bj,Snj)f(Sj1,(SK),Snj)SK. This is impossible. So bjK. Finally, let xK and yS such that xy. If yK then ySK. Since SK is a j-hyperideal, by Definition 2.1 (I2), we get xSK. It is impossible. So yK. Therefore K is a j-hyperfilter of S.

    Using the similar proof of the previous theorem, we obtain the result as follows.

    Corollary 3.8. Let (S,f,) be an ordered n-ary semihypergroup with n2 and KP(S). Then the following assertions are equivalent.

    (i)K is a hyperfilter of S.

    (ii)SK= or SK is a completely prime hyperideal of S.

    Example 3.9. From the ordered n-ary semihypergroup (S,f,) given in Example 3.2, we know that the set {w,x} is a proper 1-hyperfilter of S. By Theorem 3.7, we have {y,z}=S{w,x} is a completely prime 1-hyperideal of S. In fact, one can verify that a proper completely prime 1-hyperideal of S is {y,z} since f(xn1){y,z}= for all xn1S{y,z}.

    Lemma 3.10. Let (S,f,) be an ordered n-ary semihypergroup with n2 and a,bS. For any 1jn, the following assertions hold:

    (i) If bNj(a) then Nj(b)Nj(a).

    (ii) If ab then Nj(b)Nj(a).

    (iii) If ab then Nj(a)Nj(b) is a completely prime j-hyperideal of Nj(a) where Nj(a)Nj(b).

    Proof. (i) Let bNj(a). Since Nj(b) is the smallest j-hyperfilter of S containing b, we have Nj(b)Nj(a).

    (ii) Let ab. Since Nj(a) is a j-hyperfilter of S and aNj(a), by Definition 3.1 (F2), we obtain bNj(a). By (i), we have Nj(b)Nj(a).

    (iii) Let ab. Then Nj(b)Nj(a). By Theorem 3.7, we have Nj(a)Nj(b)= or Nj(a)Nj(b) is a completely prime j-hyperideal of Nj(a). Thus Nj(a)Nj(b) is a completely prime j-hyperideal of Nj(a) where Nj(a)Nj(b).

    Remark 3.11. Lemma 3.10 is still true if the principal j-hyperfilters Nj(a) and Nj(b) are replaced by the principal hyperfilters N(a) and N(b), respectively. Moreover, the results on this section also hold on n-ary semihypergroups (S,f).

    In this section, we introduce the concept of semilattice (strongly) regular relations on n-ary semihypergroups and investigate their related properties.

    Let (S,f) be an n-ary semihypergroup with n2 and ρ be a relation on S. For any two elements A,BP(S), we write

    AˉρB if and only if for any aA there exists bB such that aρb and for all bB there exists aA such that aρb,

    AˉˉρB if and only if aρb for all aA and bB, see [10].

    For any 1jn, an equivalence relation ρ is called j-regular relation (strongly j-regular relation) if, for every a,b,xn1S,

    aρbf(xj11,a,xnj+1)ˉρf(xj11,b,xnj+1)
    (aρbf(xj11,a,xnj+1)ˉˉρf(xj11,b,xnj+1)).

    Here, ρ is said to be a regular relation (strongly regular relation) if it is a j-regular relation (strongly j-regular relation) for all 1jn.

    Lemma 4.1. Let (S,f) be an n-ary semihypergroup with n2. For any An1,Bn1P(S), the following statements hold:

    (i) If ρ is regular and AkˉρBk, for all k=1,...,n, then f(An1)ˉρf(An1).

    (ii) If ρ is strongly regular and AkˉˉρBk, for all k=1,...,n, then f(An1)ˉˉρf(An1).

    Proof. The proof is straightforward.

    Definition 4.2. Let (S,f) be an n-ary semihypergroup with n2. A regular relation (strongly regular relation) ρ on S is said to be a semilattice regular relation (semilattice strongly regular relation) if

    xˉρf(xn) and f(xn1)ˉρf(xα(n)α(1))
    (xˉˉρf(xn) and f(xn1)ˉˉρf(xα(n)α(1)))

    for all x,xn1S and for all αSn where Sn is the set of all permutations on {1,2,...,n}.

    Note that any semilattice strongly regular relation, semilattice equivalence relation[29] and semilattice congruence[21] on semihypergroups (for case n=2) are the same.

    Example 4.3. Let S={1,2,3,4,5}. Define f:S×S×SP(S) by f(x1,x2,x3)=(x1x2)x3, for all x1,x2,x3S, where is defined by the following table.

    By applying Example 3.2 in [32], (S,f) is a ternary semihypergroup (an 3-ary semihypergroup). Let

    σ1:={(1,1),(2,2),(3,3),(4,4),(5,5),(1,2),(1,3),(2,1),(2,3),(3,1),(3,2)},σ2:={(1,1),(2,2),(3,3),(4,4),(5,5),(1,2),(1,3),(2,1),(2,3),(3,1),(3,2),(4,5),(5,4)}.

    It is easy to show that σ1,σ2 are equivalence relations on S. Furthermore, one can verify that σ2 is a semilattice strongly regular relation on S. On the other hand, σ1 is not a semilattice strongly regular relation on S because we have f(4,4,4)={4,5} but (4,5)σ1, so σ1 does not satisfy with the condition 4 ¯¯σ1f(4,4,4) in Definition 4.2.

    For any KP(S), we write δK:={(a,b)S×S:a,bK  or a,bK}. Clearly, δK is an equivalence relation on S.

    Theorem 4.4. Let (S,f) be an n-ary semihypergroup with n2 and K be a completely prime hyperideal of S. Then δK is a semilattice strongly regular relation on S.

    Proof. Let K be a completely prime hyperideal of S.

    (1) We show that δK is a strongly regular relation. Firstly, let j be a fixed positive integer with 1jn. Let a,b,xj11,xnj+1S such that aδKb. To show that f(xj11,a,xnj+1)¯¯δKf(xj11,b,xnj+1), let uf(xj11,a,xnj+1) and vf(xj11,b,xnj+1). Since aδKb, we have a,bK or a,bK.

    Case 1.1: a,bK. Since K is a j-hyperideal, we get uf(xj11,a,xnj+1)K and vf(xj11,b,xnj+1)K. So uδKv.

    Case 1.2: a,bK. Then we have 2 subcases to be considered as follows.

    Case 1.2.1: There exists xiK for some i=1,...,j1,j+1,...,n. Since K is an i-hyperideal, we obtain uf(xj11,a,xnj+1)K and vf(xj11,b,xnj+1)K. So uδKv.

    Case 1.2.2: xiK for all i=1,...,j1,j+1,...,n. Then f(xj11,a,xnj+1)K= and f(xj11,b,xnj+1)K=. In fact, if f(xj11,a,xnj+1)K then, since K is completely prime, we obtain xiK for some i=1,...,n (we write xi=a if i=j), which is a contradiction. Similarly, if f(xj11,b,xnj+1)K then we also get a contradiction. Thus f(xj11,a,xnj+1)K= and f(xj11,b,xnj+1)K=. Hence uf(xj11,a,xnj+1)SK and vf(xj11,b,xnj+1)SK. It follows that uδKv. From Case 1.1 and 1.2, we conclude that f(xj11,a,xnj+1)¯¯δKf(xj11,b,xnj+1) and so δK is a strongly j-regular relation. For arbitrary positive integer j, we obtain δK is a strongly regular relation.

    (2) We show that δK is semilattice. Let a,an1S and αSn. Firstly, we show that a¯¯δKf(an). Let uf(an). We consider two cases as follows.

    Case 2.1: aK. Since K is a hyperideal of S, we have uf(an)K. So aδKu.

    Case 2.2: aK. Then f(an)K=. In fact, if f(an)K then, since K is completely prime, we have aK. It is impossible. So f(an)K=. It follows uK. So aδKu. From Cases 2.1 and 2.2, we conclude that a¯¯δKf(an). Next, we show that f(an1)¯¯δKf(aα(n)α(1)). Let uf(an1) and vf(aα(n)α(1)). We consider two cases as follows.

    Case 1: f(an1)K. Since K is completely prime, we obtain aiK for some i=1,...,n. Since α is bijective, there is i{1,...,n} such that α(i)=i. Since K is an i-hyperideal and an i-hyperideal, we obtain f(aα(n)α(1))=f(aα(1),...,aα(i),....,aα(n))K and f(an1)=f(a1,...,ai,...,an)K. Thus u,vK and it follows that uδKv.

    Case 2: f(an1)K=. Then f(aα(n)α(1))K=. Indeed, if f(aα(n)α(1))K then, since K is completely prime, we have aα(i)K for some i{1,...,n}. We set i=α(i). Since K is an i-hyperideal, we obtain f(an1)=f(a1,...,ai,...,an)K. It is impossible. Thus f(aα(n)α(1))K=. Hence uf(an1)SK and vf(aα(n)α(1))SK and it follows that uδKv. From Cases 1 and 2, we conclude that f(an1)¯¯δKf(aα(n)α(1)) and the proof is complete.

    For any positive integer 1jn and n2, let Nj (N) denote the relation on an (ordered) n-ary semihypergroup S which is defined by

    Nj={(a,b)S×S:Nj(a)=Nj(b)}
    (N={(a,b)S×S:N(a)=N(b)}).

    Clearly, Nj and N are equivalence relations on S. For any aS, we denote by (a)Nj the Nj-class containing a and the set S/Nj:={(a)Nj:aS}. For the relation N, we define analogously.

    Theorem 4.5. Let (S,f) be an n-ary semihypergroup with n2. Then the following statements hold:

    (i)N is a semilattice strongly regular relation.

    (ii)N={δK:KCPI(S)},

    where CPI(S) is the set of all completely prime hyperideals of S.

    Proof. (i) First of all, we show that N is a strongly regular relation on S. Let a,b,xn1S such that aNb. Then N(a)=N(b). For any fixed positive integer j with 1jn, let uf(xj11,a,xnj+1) and vf(xj11,b,xnj+1). Since f(xj11,a,xnj+1)N(u) and N(u) is a hyperfilter, by Definition 3.1 (F1), we obtain a,xj11,xnj+1N(u). By Lemma 3.10, we obtain N(a)N(u). Since bN(b)=N(a), we get bN(u). Since b,xj11,xnj+1N(u) and N(u) is an n-ary subsemihypergroup, we obtain vf(xj11,b,xnj+1)N(u). By Lemma 3.10, we have N(v)N(u). Using the similar previous process, we also obtain that N(u)N(v). So N(u)=N(v) and it follows that uNv. Hence f(xj11,a,xnj+1)¯¯Nf(xj11,b,xnj+1). So N is a strongly regular relation on S. Next, we show that N is a semilattice strongly regular relation on S. Let a,an1S and αSn. To show that a¯¯Nf(an), let uf(an). Then f(an)N(u). By Definition 3.1 (F1), we obtain aN(u). By Lemma 3.10, we get N(a)N(u). Since N(a) is an n-ary subsemihypergroup and aN(a), we obtain uf(an)N(a). By Lemma 3.10, we have N(u)N(a). So N(a)=N(u) and it implies that a¯¯Nf(an). Finally, we show that f(an1)¯¯Nf(aα(n)α(1)). Let uf(an1) and vf(aα(n)α(1)). Since f(an1)N(u), by Definition 3.1 (F1), we obtain an1N(u). It follows that vf(an1)N(u) and so N(v)N(u). Conversely, since f(aα(n)α(1))N(v), by using the same process, we have N(u)N(v). So N(u)=N(v) and it follows that uNv. Hence f(an1)¯¯Nf(aα(n)α(1)). Thus N is a semilattice strongly regular relation on S.

    (ii) Firstly, we will show that N{δK:KCPI(S)}. Let (x,y)N. Then N(x)=N(y). Assume that (x,y)δK for some KCPI(S). Then there exist two cases:

    Case 1.1: xK and yK. Then ySKS. Since S(SK)=K is completely prime, by Corollary 3.8, we have SK is a hyperfilter of S. Since ySK, by Lemma 3.10, we obtain xN(x)=N(y)SK. It is impossible.

    Case 1.2: xK and yK. Using the similar proof as in Case 1.1, we also get a contradiction. From Cases 1.1 and 1.2, we conclude that (x,y)δK for all KCPI(S). Thus N{δK:KCPI(S)}. Conversely, let (x,y)δK for all KCPI(S). Assume that (x,y)N. Then there exist 2 cases, i.e. xN(y) or yN(x). Indeed, if xN(y) and yN(x) then, by Lemma 3.10, we get N(x)N(y) and N(y)N(x). Consequently, N(x)=N(y) and so (x,y)N. This is a contradiction. Thus xN(y) or yN(x).

    Case 2.1: xN(y). Then xSN(y). Since N(y) is a hyperfilter and SN(y), by Corollary 3.8, we have SN(y) is a completely prime hyperideal. Since xSN(y) and ySN(y), it follows that (x,y)δSN(y). This is a contradiction.

    Case 2.2: yN(x). Using the similar proof as in Case 2.1, we get a contradiction. Hence (x,y)N. Therefore {δK:KCPI(S)}N and the proof is complete.

    As we have known from [21] that the relation N is the smallest semilattice strongly regular relation on semihypergroups. However, we illustrate by the following example that the similar result in n-ary semihypergroups, for n3, is not necessary true.

    Example 4.6. Let S={1,1,i,i}. Then (S,) is a commutative semigroup under the usual multiplication over complex numbers. Define f:S×S×SP(S) by f(x,y,z)={xyz}. Then (S,f) is a ternary semihypergroup. Clearly, CPI(S)={S}. In fact, suppose that A is an arbitrary hyperideal of S.

    If 1A then S=f(S,1,1)f(S,S,1)A.

    If 1A then S=f(S,1,1)f(S,S,1)A.

    If iA then S=f(S,i,i)f(S,S,i)A.

    If iA then S=f(S,i,i)f(S,S,i)A.

    Consequently, S=A. By Theorem 4.5(ii), we have

    N=δS={(1,1),(1,1),(1,i),(1,i),(1,1),(1,1),(1,i),(1,i),

    (i,1),(i,1),(i,i),(i,i),(i,1),(i,1),(i,i),(i,i)}.

    Next, let

    ρ={(1,1),(1,1),(i,i),(i,i),(1,1),(1,1),(i,i),(i,i)}.

    It is not difficult to show that ρ is a strongly regular relation on S. Furthermore, we have

    1 ˉˉρ f(1,1,1)={1}, 1 ˉˉρ f(1,1,1)={1},
    i ˉˉρ f(i,i,i)={i} and i ˉˉρ f(i,i,i)={i}.

    Since (S,) is commutative, we get f(x31)=f(xα(3)α(1)) and so f(x31) ˉˉρf(xα(3)α(1)) for all αS3. By Definition 4.2, we conclude that ρ is a semilattice strongly regular relation on S and ρN.

    By the previous example, we observe that the relation N is not necessary the smallest semilattice strongly regular relation on n-ary semihypergroups, for n3, in general. To provide a sufficient condition that makes the above conclusion true, we need to introduce the following definition.

    Definition 4.7. Let (S,f) be an n-ary semihypergroup with n3. A semilattice regular relation (semilattice strongly regular relation) ρ on S is called a -semilattice regular relation (-semilattice strongly regular relation) if for every x,yS,

    x ˉρ f(x,yn1) implies x ˉρ f(xn1,y)
    (x ˉˉρ f(x,yn1) implies x ˉˉρ f(xn1,y)).

    Example 4.8. Consider a ternary semihypergroup (S,f) given in Example 4.6. Obviously, i ˉˉρ f(i,1,1)={i}. Since f(i,i,1)={1} and (i,1)ρ, we conclude that ρ is not a -semilattice strongly regular relation on S.

    However, the relation N is always a -semilattice strongly regular relation on n-ary semihypergroups where n3.

    Proposition 4.9. Let (S,f) be an n-ary semihypergroup with n3. Then the following assertions hold:

    (i) If K is a completely prime hyperideal of S, then δK is a -semilattice strongly regular relation on S.

    (ii) The relation N is a -semilattice strongly regular relation on S.

    Proof. From Theorems 4.4 and 4.5, we obtain that δK and N are semilattice strongly regular relations on S.

    (i) Let x,yS be such that x¯¯δKf(x,yn1). To show that x¯¯δKf(xn1,y), let af(xn1,y). We consider two cases as follows:

    Case 1: xK. Since x¯¯δKf(x,yn1), by the definition of δK, we have f(x,yn1)K=. It follows that f(x,yn1)(SK). Since K is a completely prime hyperideal of S, by Corollary 3.8, we have SK is a hyperfilter of S. By (F1) in Definition 3.1, we have x,ySK. Since SK is an n-ary subsemihypergroup of S, we obtain af(xn1,y)SK. By the definition of δK, we have xδKa. Thus x¯¯δKf(xn1,y).

    Case 2: xK. Since K is a hyperideal of S, by (I1) in Definition 2.1, we obtain af(xn1,y)K. Consequently, xδKa and so x¯¯δKf(xn1,y). From Cases 1 and 2, we conclude that δK is a -semilattice strongly regular relation on S.

    (ii) Let x,yS with x¯¯Nf(x,yn1). Then N(x)=N(b) for all bf(x,yn1). To show that x¯¯Nf(xn1,y), let cf(xn1,y). Since bf(x,yn1)N(x), by (F1) in Definition 3.1, we have x,yN(x). Since N(x) is an n-ary semihypergroup, we have cf(xn1,y)N(x). By Lemma 3.10, we get N(c)N(x). Conversely, since f(xn1,y)N(c), by (F1) in Definition 3.1, we have xN(c). So N(x)N(c). Consequently, N(x)=N(c) and it follows that xNc for all cf(xn1,y). Therefore x¯¯Nf(xn1,y) and the proof is complete.

    Theorem 4.10. Let (S,f) be an n-ary semihypergroup with n3. Let ρ be a -semilattice strongly regular relation on S. Then there exists a collection C of proper completely prime hyperideals of S such that

    ρ=KCδK.

    Proof. Let xS. Setting

    Fx:={yS:xˉˉρf(xn1,y)}.

    Since ρ is a -semilattice strongly regular relation, we obtain xˉˉρf(xn). So xFx. Firstly, we show that Fx is a hyperfilter of S.

    (1) We show that Fx is an n-ary subsemihypergroup of S. Let yn1Fx. Then f(yn1)Fx. Indeed, let uf(yn1). We will show that uFx, i.e., xˉˉρf(xn1,u). Let vf(xn1,u). Since ykFx for all k=1,...,n, we have xˉˉρf(xn1,yk) for all k=1,...,n. Since ρ is a -semilattice strongly regular relation on S, by associativity and Lemma 4.1, we obtain

    xˉˉρf(xn)ˉˉρf(f(xn1,y1),...,f(xn1,yn1),f(xn1,yn))ˉˉρf(f(xn),...,f(xn)n1 times,f(yn1))ˉˉρf(x,...,x,n1 timesf(yn1)).

    Hence xˉˉρf(xn1,f(yn1)). Since vf(xn1,u)f(xn1,f(yn1)), we have xρv. So xˉˉρf(xn1,u) and it implies that uFx. Thus f(yn1)Fx.

    (2) We show that Fx satisfies (F1) in Definition 3.1. Let yn1S such that f(yn1)Fx. Then there exists vf(yn1) and vFx. Then xˉˉρf(xn1,v). First of all, we show that xˉˉρf(xn1,f(yn1)). Let af(xn1,f(yn1)). Then af(xn1,d) for some df(yn1). Since f(yn1)ˉˉρf(yn1) and v,df(yn1), we have vρd. Since ρ is a strongly regular relation, we obtain f(xn1,v)ˉˉρf(xn1,d). Since xˉˉρf(xn1,v), we obtain xˉˉρf(xn1,d). Since af(xn1,d), we have xρa. Thus xˉˉρf(xn1,f(yn1)). Next, we show that ykFx for all k=1,...,n. For any 1kn, we have

    xˉˉρf(xn1,f(yn1)), since ρ is semilattice,ˉˉρf(f(xn1,yk),yk11,ynk+1),since ρ is semilattice and yiˉˉρf([yi]n) for all i=1,...,k1,k+1,...,n,ˉˉρf(f(xn1,yk),f([y1]n),...,f([yk1]n),f([yk+1]n),...,f([yn]n),since ρ is semilattice,ˉˉρf(...f(f(...f(f(xn1,f(yn1)),[y1]n1),...)f appears n+1 terms,[yk1]n1),[yk+1]n1),...,[yn]n1),since xˉˉρf(xn1,f(yn1)),ˉˉρf(...f(f(...f(x,[y1]n1),...)f appears n1 terms,[yk1]n1),[yk+1]n1),...,[yn]n1),since xˉˉρf(xn),ˉˉρf(...f(f(...f(f(xn),[y1]n1),...)f appears n terms,[yk1]n1),[yk+1]n1),...,[yn]n1),since ρ is semilattice,ˉˉρf(x,f(x,yk11,ynk+1),...,f(x,yk11,ynk+1)f appears n1 terms).

    Since ρ is a strongly regular relation, we obtain

    f(x,[yk]n1)ˉˉρf(f(x,f(x,yk11,ynk+1),...,f(x,yk11,ynk+1)f appears n1 terms),[yk]n1),since ρ is semilattice,ˉˉρf(f(xn1,x),[f(yn1)]n1)=f(xn1,f(x,[f(yn1)]n1)), since xˉˉρf(xn),ˉˉρf([f(xn)]n1,f(x,[f(yn1)]n1))=f([f(x,xn1)]n1,f(x,[f(yn1)]n1)), since ρ is semilattice,ˉˉρf(f(xn1,x),[f(xn1,f(yn1))]n1), since f(xn1,f(yn1)) ˉˉρ x,ˉˉρf(f(xn1,x),xn1)ˉˉρx.

    Since ρ is a -semilattice strongly regular relation on S, we have xˉˉρf(xn1,yk). So ykFx for all k=1,2,...,n. This follows that Fx satisfies the condition (F1) in Definition 3.1. By (1) and (2), we conclude that Fx is a hyperfilter of S. By Corollary 3.8, we have SFx= or SFx is a completely prime hyperideal of S. If SFx=S then, since FxS, we have Fx=. It is impossible. Thus SFx is a proper completely prime hyperideal of S. Define

    C:={SFx:xS and SFx is a proper completely prime hyperideal of S}.

    Next, we show that ρ=xSδSFx. Let (a,b)ρ and xS. We consider the following two cases.

    Case 1: aSFx. Then aFx. We obtain xˉˉρf(xn1,a). Since (a,b)ρ, we obtain f(xn1,a)ˉˉρf(xn1,b). Then xˉˉρf(xn1,b) and so bFx. It follows that bSFx. Thus (a,b)δSFx.

    Case 2: aSFx. If bSFx then, using the similar proof as in Case 1, we have aSFx, which is a contradiction. So bSFx. It follows that (a,b)δSFx. Thus ρxSδSFx. Conversely, let (a,b)xSδSFx. Then (a,b)δSFa. Since aFa, we have aSFa. Since (a,b)δSFa, we have bSFa. So bFa. By the definition of Fa, we have aˉˉρf(an1,b). On the other hand, by using the similar process, we have bˉˉρf(bn1,a). Then

    aˉˉρf(an1,b)ˉˉρf(an1,f(bn1,a))ˉˉρf(bn1,f(an)), since ρ is semilattice,ˉˉρf(bn1,a), since aˉˉρf(an),ˉˉρb.

    It implies that (a,b)ρ. Therefore xSδSFxρ and the proof is complete.

    Corollary 4.11. Let (S,f) be an n-ary semihypergroup with n3. Then the relation N is the least -semilattice strongly regular relation on S.

    Proof. Let ρ be a -semilattice strongly regular relation on S. By Theorem 4.10, there exists a collection C of proper completely prime hyperideals of S such that ρ=KCδK. By Theorem 4.5, we have N=KCPI(S)δK. Since CCPI(S), we obtain that

    N=KCPI(S)δKKCδK=ρ.

    As we have known from [21] that the relation N is not the smallest semilattice congruence on ordered semihypergroups in general. The following example show that the result is also true in case of the -semilattice strongly regular relation on ordered n-ary semihypergroups with n3.

    Example 5.1. Let S={a,b,c,d}. Define f:S×S×SP(S) by f(x1,x2,x3)=(x1x2)x3, for all x1,x2,x3S, where is defined by the following table.

    By applying Example 1 in [31], (S,f) is a ternary semihypergroup. Define a partial order on S by

    ≤:={(a,a),(b,b),(c,a),(c,c),(d,a),(d,d)}.

    Such relation can be presented by the following diagram.

    Then (S,f,) is an ordered ternary semihypergroup. Moreover, we have N(b)={b} and N(a)=S=N(c)=N(d). It follows that

    N:={(a,a),(a,c),(a,d),(b,b),(c,a),(c,c),(c,d),(d,a),(d,c),(d,d)}.

    It is not difficult to show that

    ρ={(a,a),(a,d),(b,b),(c,c),(d,a),(d,d)}

    is a -semilattice strongly regular relation on S and ρN.

    Next, we introduce the concept of complete -semilattice (strongly) regular relations on ordered n-ary semihypergroups, where n3, that generalizes Definition 4.1 in [21].

    Definition 5.2. Let S be an ordered n-ary semihypergroup with n3. A -semilattice regular relation (-semilattice strongly regular relation) ρ on S is said to be complete if for every x,yS,

    xy implies xˉρf(xn1,y)
    (xy implies xˉˉρf(xn1,y)).

    Note that Theorem 4.4, Theorem 4.5 and Proposition 4.9 are also true for ordered n-ary semihypergroups where n3.

    Proposition 5.3. Let (S,f,) be an ordered n-ary semihypergroup with n3. Then the relation N is a complete -semilattice strongly regular relation on S.

    Proof. Firstly, let x,yS such that xy. We show that x¯¯Nf(xn1,y). Let af(xn1,y). Since xy, by Lemma 3.10, we have yN(y)N(x). Then x,yN(x). Since N(x) is an n-ary subsemihypergroup, we obtain af(xn1,y)N(x). Hence N(a)N(x). Conversely, since f(xn1,y)N(a), by (F1), we obtain xN(a). By Lemma 3.10, we get N(x)N(a). So N(x)=N(a). It follows that xNa. Thus x¯¯Nf(xn1,y).

    Theorem 5.4. Let (S,f,) be an ordered n-ary semihypergroup with n3. Let ρ be a complete -semilattice strongly regular relation on S. Then there exists a collection C of proper completely prime hyperideals of S such that

    ρ=KCδK.

    Proof. Let xS. Define

    Fx:={yS:xˉˉρf(xn1,y)}.

    From Theorem 4.10, it remains to show that the set Fx satisfies (F2) in Definition 3.1. Let yFx and zS such that yz. Since yFx, we have xˉˉρf(xn1,y). Since yz and ρ is a complete -semilattice strongly regular relation on S, we have yˉˉρf(yn1,z). Since ρ is a -semilattice strongly regular relation on S, we have yˉˉρf(yn). Next, we consider

    f(xn1,z)=f(xn2,x,z), since xˉˉρf(xn1,y),ˉˉρf(xn2,f(xn1,y),z), since yˉˉρf(yn),ˉˉρf(xn2,f(xn1,f(yn)),z)ˉˉρf(xn2,f(f(xn1,y),yn1),z), since xˉˉρf(xn1,y),ˉˉρf(xn2,f(x,yn1),z)=f(xn1,f(yn1,z)), since yˉˉρf(yn1,z),ˉˉρf(xn1,y)ˉˉρx.

    Consequently, zFx and the proof is complete.

    Corollary 5.5. Let (S,f,) be an ordered n-ary semihypergroup with n3. Then N is the smallest complete -semilattice strongly regular relation on S.

    Proof. The proof is similar to Corollary 4.11.

    Example 5.6. Consider the ternary semihypergroup (S,f) given in Example 5.1. Define a partial order on S by

    ≤:={(a,a),(a,b),(a,c),(b,b),(c,c),(d,b),(d,c),(d,d)}.

    Such relation can be presented by the following diagram.

    By applying Example 1 in [31], (S,f,) is an ordered ternary semihypergroup. There are 4 complete -semilattice strongly regular relations as follows:

    σ1={(a,a),(a,d),(b,b),(c,c),(d,a),(d,d)},σ2={(a,a),(a,b),(a,d),(b,a),(b,b),(b,d),(c,c),(d,a),(d,b),(d,d)},σ3={(a,a),(a,c),(a,d),(b,b),(c,a),(c,c),(c,d),(d,a),(d,c),(d,d)},σ4={(a,a),(a,b),(a,c),(a,d),(b,a),(b,b),(b,c),(b,d),(c,a),(c,b),(c,c),(c,d),(d,a),(d,b),(d,c),(d,d)}.

    Furthermore, we have N(b)={b}, N(c)={c} and N(a)=N(d)=S. It follows that N:={(a,a),(a,d),(b,b),(c,c),(d,a),(d,d)}=σ1. Obviously, N is the smallest complete -semilattice strongly regular relation on S.

    In this section, we show that every completely prime hyperideal on an ordered n-ary semihypergroups can be decomposable into its N-classes. But the result is not true in the case of prime hyperideals, see Example 6.6. This leads to answer the open problem given in [31]. However, we will provide a sufficient condition that makes the above conclusion true for case of prime hyperideals. Note that Theorems 4.4 and 4.5 are also true for every ordered n-ary semihypergroup with n2.

    Theorem 6.1. Let (S,f,) be an ordered n-ary semihypergroup with n2. Let K be a completely prime hyperideal of S. Then

    K={(a)N:aK}.

    Proof. Firstly, we show that {(a)N:aK}K. Let b(a)N for some aK. By Theorem 4.5, we have (a,b)N={δA:ACPI(S)}. So (a,b)δK. Since aK, we get bK. Thus {(a)N:aK}K. As we have known that a(a)N for all aK, so K{(a)N:aK}. Therefore K={(a)N:aK}.

    To solve the problem raised by Tang and Davvaz in [31], we firstly prove the following lemmas.

    Lemma 6.2. Let (S,f,) be an ordered n-ary semihypergroup with n2. Let K be a nonempty subset of S satisfying the following property:

    ()eitherf(xn1)Korf(xn1)SKforallxn1S.

    For any 1jn, if a nonempty subset SK is a prime j-hyperideal of S then K is a j-hyperfilter of S.

    Proof. The proof is similar to the proof of the converse of Theorem 3.7 by applying the property ().

    Remark 6.3. For case n=2, we obtain that the notion of prime hyperideal on S satisfying the property () coincides with the notion of prime hyperideal on ordered semihypergroups introduced by Kehayopulu (see Definition 2.5 in [21]).

    The following results, Lemma 6.4 and Theorem 6.5, generalize Proposition 2.5 and Theorem 2.10 in [21].

    Lemma 6.4. Let (S,f,) be an ordered n-ary semihypergroup with n2. Then the following statements hold:

    (i) For any KP(S) satisfying the property (), K is a hyperfilter of S if and only if either SK= or SK is a prime hyperideal of S.

    (ii)N={δK:KPH(S)} where PH(S) is the set of all prime hyperideals of S satisfying the property ().

    Proof. As we have known that every completely prime hyperideal of S is always a prime hyperideal of S. By Corollary 3.8 and Lemma 6.2, the proof of (i) is complete. The proof of (ii) is similar to the proof of Theorem 4.5(ii) by applying (i).

    Using the similar proof of Theorem 6.1, by using Lemma 6.4(ii), we obtain the result as follows.

    Theorem 6.5. Let (S,f) be an ordered n-ary semihypergroup with n2. Let K be a prime hyperideal of S satisfying the property (). Then K={(a)N:aK}.

    The following example shows that a prime hyperideal of S that does not satisfy the property () cannot be decomposable into its N-classes.

    Example 6.6. From the ordered n-ary semihypergroup (S,f,) given in Example 2.4, we obtain that the sets A2={y,z} and A3={w,x,z} are prime 2-hyperideals of S. Furthermore, A2,A3 are also prime hyperideals of S. Clearly, A3 satisfies the property () but A2 does not satisfy the property () since there exists wS such that f(wn)A2 and f(wn)SA2. On the other hand, we can verify that N(w)=N(x)=N(z)=S and N(y)={y}. Hence

    N={(w,w),(w,x),(w,z),(x,w),(x,x),(x,z),(y,y),(z,w),(z,x),(z,z)}.

    Then (w)N={w,x,z}=(x)N=(z)N and (y)N={y}. Clearly,

    A2={y,z}S=(y)N(z)N

    and

    A3={w,x,z}=(w)N(x)N(z)N.

    Remark 6.7. If we set n=2 then the problem given in [31] is answered by Theorem 6.5 and Example 6.6.

    In this paper, we introduced the concept of j-hyperfilters of ordered n-ary semihypergroups, where a positive integer 1jn and n2, and discussed their related properties. Such notion can be considered as a generalization of left and right hyperfilters of ordered semihypergroups. The interesting properties of j-hyperfilters by means of completely prime j-hyperideals were established. Furthermore, we defined and investigated the semilattice strongly regular relation N on ordered n-ary semihypergroups by the hyperfilters. In particular, we illustrated by counterexample that the relation N is not necessary the least semilattice strongly regular relation on ordered n-ary semihypergroups where n3. Meanwhile, we introduced the concept of complete -semilattice strongly regular relations on ordered n-ary semihypergroups, where n3, and proved that N is the smallest complete -semilattice strongly regular relation. Finally, we provided some necessary conditions to show that every prime hyperideal can be decomposable into its N-classes. This result answers the related problem posed by Tang and Davvaz in [31].

    The authors are highly grateful to the referees for their valuable comments and suggestions for improving the article. This research was supported by Chiang Mai University, Chiang Mai 50200, Thailand.

    The authors declare no conflict of interest.



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