(m,n)-Hyperideals in ordered semihypergroups

In this paper, first we introduce the notions of an (m,n)hyperideal and a generalized (m,n)-hyperideal in an ordered semihypergroup, and then, some properties of these hyperideals are studied. Thereafter, we characterize (m,n)-regularity, (m, 0)-regularity, and (0, n)-regularity of an ordered semihypergroup in terms of its (m,n)-hyperideals, (m, 0)-hyperideals and (0, n)-hyperideals, respectively. The relations mI, In,H m, and B m on an ordered semihypergroup are, then, introduced. We prove that B m ⊆ H m on an ordered semihypergroup and provide a condition under which equality holds in the above inclusion. We also show that the (m, 0)-regularity [(0, n)regularity] of an element induce the (m, 0)-regularity [(0, n)-regularity] of the whole H m-class containing that element as well as the fact that (m,n)regularity and (m,n)-right weakly regularity of an element induce the (m,n)regularity and (m,n)-right weakly regularity of the whole B m-class and H mclass containing that element, respectively. * Corresponding author


Introduction and preliminaries
By an ordered semigroup, we mean an algebraic structure (S, · ≤), which satisfies the following conditions: (1) S is a semigroup with respect to the multiplication "·"; (2) S is a partially ordered set by ≤; (3) if a and b are elements of S such that a ≤ b, then ac ≤ bc and ca ≤ cb for all c ∈ S. Many authors, especially Alimov [1], Clifford [2][3][4], Hion [13], Conrad [5], and Kehayopulu [15] studied such semigroups with some restrictions.
In 1934, Marty [21] introduced the concept of a hyperstructure and defined hypergroup. Later on several authors studied hyperstructure in various algebraic structures such as rings, semirings, semigroups, ordered semigroups, Γ-semigroups and Ternary semigroups, etc. The concept of a semihypergroup is a generalization of the concept of a semigroup and many classical notions such as of ideals, quasi-ideals and bi-ideals defined in semigroups and regular semigroups have been generalized to semihypergroups (see [8,9] for other related notions and results on semihypergroups). In [14], Heidari and Davvaz introduced the notion of an ordered semihypergroup as a generalization of the notion of an ordered semigroup. Davvaz et al. in [6,7,14,22,23,25,26] studied some properties of hyperideals and bihyperideals in ordered semihypergroups. Lajos [16] introduced the concept of (m, n)-ideals in semigroups (see also [17][18][19]). In [12], the authors defined the notion of an (m, n)-quasi-hyperideal in a semihypergroup and investigated several properties of these (m, n)-quasi-hyperideals.
A hyperoperation on a non-empty set H is a map • : H × H → P * (H) where P * (H) = P(H) \ {∅} (the set of all non-empty subsets of H). In such a case, H is called a hypergroupoid. Let H be a hypergroupoid and A, B be any non-empty subsets of H. Then We shall write, in whatever follows, A•x instead of A•{x} and x•A instead of {x} • A, for any x ∈ H. Also, for simplicity, throughout the paper, we shall write A n for A • A • · · · • A (n − copies of A) for any n ∈ Z + . Also the integers m, n will stand for positive integers throughout the paper until and unless otherwise specified. Moreover, the hypergroupoid H is called a semihypergroup if, for all x, y, z ∈ H, Let H be a non-empty set, the triplet (H, •, ≤) is called an ordered semihypergroup if (H, •) is a semihypergroup and (H, ≤) is a partially ordered set such that for all x, y, z ∈ H. Here, if A and B are non-empty subsets of H, then we say that A ≤ B if for every a ∈ A there exists b ∈ B such that a ≤ b.
Let H be an ordered semihypergroup. For a non-empty subset A of H, Lemma 1.1. [7] Let H be an ordered semihypergroup and A, B be any non-empty subsets of H. Then the following conditions hold: 2 (m, 0)-hyperideals, (0, n)-hyperideals and (m, n)-hyperideals in ordered semihypergroups In this section, the notions of (m, n)-hyperideals and generalized (m, n)hyperideals in ordered semihypergroups are introduced. Moreover, important some properties of these hyperideals are studied.
Definition 2.1. Let H be an ordered semihypergroup and m, n be the positive integers. Then a subsemihypergroup (respectively, non-empty subset) Note that in Definition 2.1, if m = 1 = n, then A is called a (generalized) bi-hyperideal of H. Moreover, a (generalized) bi-hyperideal of an ordered semihypergroup H is an (generalized) (m, n)-hyperideal of H for all positive integers m and n. It is clear that, for positive integers m and n, the notion of (generalized) (m, n)-hyperideal of H is a generalization of the notion of (generalized) bi-hyperideal of H. The following example shows that a generalized (m, n)-hyperideal of H need not be an (m, n)-hyperideal and generalized bi-hyperideal of H.
The covering relation ≺ and the figure of H are as follows: In Definition 2.3, if m = 1 = n, then A is called a right hyperideal (left hyperideal) of H. Clearly, each right hyperideal (respectively, left hyperideal) of H is an (m, 0)-hyperideal for each positive integer m (respectively, (0, n)-hyperideal for each positive integer n), that is, the notion of an (m, 0)-hyperideal ((0, n)-hyperideal) of H is a generalization of the notion of a right hyperideal (respectively, left hyperideal) of H. Conversely, an (m, 0)-hyperideal (respectively, (0, n)-hyperideal) of H need not be a right hyperideal (respectively, left hyperideal) of H. We illustrate it by the following example.
Theorem 2.10. [20] Let H be an ordered semihypergroup. Then the following conditions hold: Let H be an ordered semihypergroup and A be any non-empty subset of H.
Theorem 2.11. Let H be an ordered semihypergroup and A be a non-empty subset of H. Then for any positive integers m, n. .
Then, x ∈ z 1 • z 2 for some z 1 , z 2 ∈ m+n i=1 A i . Then, z 1 = A p , z 2 = A q for some 1 < p, q ≤ m + n. There are two cases arising. If p + q ≤ m + n, then Theorem 2.12. [20] Let H be an ordered semihypergroup and A be any non-empty subset of H. Then: Theorem 2.13. Let H be an ordered semihypergroup and A be a non-empty subset of H. Then for any positive integers m, n.

Proof. We have
Theorem 2.14. [20] Let H be an ordered semihypergroup and A be a nonempty subset of H. Then

(m, n)-regularity in ordered semihypergroups
In this section, we characterize (m, n)-regular, (m, 0)-regular and (0, n)regular ordered semihypergroup in terms of its (m, n)-hyperideals, (m, 0)hyperideals and (0, n)-hyperideals.  Proof. Let a, b ∈ A. Since H is an (m, n)-regular ordered semihypergroup, there exist x, y ∈ H such that a ≤ a m • x • a n , b ≤ b m • y • b n . Therefore,    Proof. The statement is trivially true for m = 0 = n. If m = 0 and n = 0 or m = 0 and n = 0, then the result follows by Lemma 3.4. So, let m = 0, n = 0, R be any (m, 0)-hyperideal and L be any (0, n)-hyperideal of H. Therefore Conversely, assume that L ∩ R = (R m • L n ] for each (m, 0)-hyperideal R and for each (0, n)-hyperideal L of H. Let a ∈ S. As Hence, H is (m, n)-regular.
Theorem 3.7. Let H be an ordered semihypergroup and m, n be positive integers (either m ≥ 2 or n ≥ 2). Then, the following are equivalent: Proof. (i) ⇒ (ii) Assume that each (m, n)-hyperideal of H is idempotent. Let A and B be any (m, n)-hyperideals of H. As A ∩ B is an (m, n)hyperideal of H, we have (ii) ⇒ (iii) and (iii) ⇒ (iv) are obvious.
(iv) ⇒ (v) Take any a ∈ A. Then, by (iv), we have (v) ⇒ (i) Take any (m, n)-hyperideal A of H. As H is (m, n)-regular and A is an (m, n)-hyperideal, Hence, each (m, n)-hyperideal of H is an idempotent.
The following example shows that the condition m ≥ 2 or n ≥ 2 in Theorem 3.7 is necessary.
Clearly, all the relations defined above are equivalence relations on H.    Proof. Let (a, b) ∈ B n m . Then, [a]  derstanding of different classes of ordered semihypergroups ((m, n)-regular, (m, 0)-regular, (0, n)-regular, (m, n)-right weakly regular) by considering the structural influence of the equivalence relations m I, I n , B n m , and H n m . In particular, if we take m = 1 = n, the equivalence relations m I, I n and H n m are reduced to the equivalence relations R, L and H in ordered semihypergroup, respectively, which mimic the definition of the usual Green's relations R, L and H in plain semihypergroups [11]. Also when we take m = 1 = n in Theorems 1.9, 1.11, 4.1, 3.6, and 4.2, and Lemmas 4.1, 4.2, 4.3, 4.3, 5.1, and 5.2, then we obtain all the results for bi-hyperideals in an ordered semihypergroup and some characterizations of regular ordered semihypergroups, which is the main application of the results presented in this paper.