EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 10, No. 5, 2017, 929-945 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Hypersemigroups and fuzzy hypersemigroups Niovi Kehayopulu University of Athens, Department of Mathematics, 15784 Panepistimiopolis, Greece Abstract. The aim is to show that the theory of hypersemigroups and the theory of fuzzy hyper- semigroups are parallel to each other, in the following sense: An hypersemigroup H is intra-regular, for example, if and only if A∩B ⊆ B∗A for every right ideal A and every left ideal B of H. And an hypersemigroup H is intra-regular if and only if f ∧g � g◦f for every fuzzy right ideal f and every fuzzy left ideal g of H. An hypersemigroup H is left quasi-regular if and only if A∩B ⊆ A ∗B for every ideal A and every nonempty subset B of H. And an hypersemigroup H is left quasi-regular if and only if f ∧ g � f ◦ g for every fuzzy ideal f and every fuzzy subset g of H. 2010 Mathematics Subject Classifications: AMS 20M99, 08A72 Key Words and Phrases: Hypersemigroup, right (left) ideal, bi-ideal, fuzzy right (left) ideal, fuzzy bi-ideal, regular, intra-regular, left (right) quasi-regular, semisimple 1. Introduction and prerequisites A semigroup (S, ·) is called regular (von Neumann regular) if for every a ∈ S there exists x ∈ S such that a = axa. This is equivalent to saying that a ∈ aSa for every a ∈ S or A ⊆ ASA for every A ⊆ S. A nonempty subset A of a groupoid (S, ·) is called a right (resp. left) ideal of S if AS ⊆ A (resp. SA ⊆ A). If A is both a right and a left ideal of S, then it is called an ideal of S. A nonempty subset B of a semigroup (S, ·) is called a bi-ideal of S if BSB ⊆ B. It is well known that a semigroup S is regular if and only if for every right ideal A and every left ideal B if S, we have A ∩ B = AB (Iséki [2]). A semigroup (S, ·) is called intra-regular [1] if for every a ∈ S there exist x, y ∈ S such that a = xa2y. This is equivalent to saying that a ∈ Sa2S for every a ∈ S or A ⊆ SA2S for every A ⊆ S. It is also well known that a semigroup S is intra-regular if and only if for any right ideal A and any left ideal B of S, we have A ∩ B ⊆ BA (Lajos and Szász [9]). For the concepts of left (right) quasi-regular and semisimple semigroups we refer to [11]: A semigroup S is called left (resp. right) quasi-regular if for every a ∈ S there exist x, y ∈ S such that a = xaya (resp. a = axay). We remark that a semigroup S is left (resp. right) quasi-regular if and only if a ∈ SaSa (resp. a ∈ aSaS) for every a ∈ S, equivalently if A ⊆ SASA (resp. A ⊆ ASAS) for every A ⊆ S. A semigroup S is called semisimple if for every a ∈ S there exist x, y, z ∈ S such that a = xayaz. We note that a semigroup S is Email address: nkehayop@math.uoa.gr (N. Kehayopulu) http://www.ejpam.com 929 c© 2017 EJPAM All rights reserved. N. Kehayopulu / Eur. J. Pure Appl. Math, 10 (5) (2017), 929-945 930 semisimple if and only if a ∈ SaSaS for every a ∈ S, equivalently if A ⊆ SASAS for every A ⊆ S. In the present paper we show that the theories of hypersemigroups and fuzzy hypersemigroups are parallel to each other, in the following sense: An hypersemigroup H is regular if and only if A ∩ B = A ∗ B, equivalently if A ∩ B ⊆ A ∗ B for every right ideal A and every left ideal B of H. On the other hand, an hypersemigroup H is regular if and only if f ∧ g = f ◦ g, equivalently if f ∧ g � f ◦ g for every fuzzy right ideal f and every fuzzy left ideal g of H. An hypersemigroup H is intra-regular if and only if for every right ideal A and every left ideal B of H we have A ∩ B ⊆ B ∗ A. An hypersemigroup H is intra-regular if and only if for every fuzzy right ideal f and every fuzzy left ideal g of H we have f ∧ g � g ◦ f. An hypersemigroup H is left quasi-regular if and only if A ∩ B ⊆ A ∗ B for every ideal A and every bi-ideal B of H, equivalently if A∩B ⊆ A ∗B for every ideal A and every left ideal B of H. An hypersemigroup H is left quasi-regular if and only if f ∧g � f ◦g for every fuzzy ideal f and every fuzzy bi-ideal g of H, equivalently if f ∧g � f ◦g for every fuzzy ideal f and every fuzzy left ideal g of H. An hypersemigroup H is left quasi-regular if and only if the left ideals of H are idempotent. And an hypersemigroup H is left quasi-regular if and only if the fuzzy left ideals of H are idempotent. An hypersemigroup H is semisimple if and only if the ideals of H are idempotent, equivalently if A∩B = A ∗B for all ideals A,B of H. An hypersemigroup H is semisimple if and only if the fuzzy ideals of H are idempotent, equivalently if for each fuzzy ideals f and g of H we have f ∧ g = f ◦ g. Our aim being to show that the theories of hypersemigroups and of fuzzy hypersemigroups are parallel to each other, we restrict ourselves to an hypersemigroup (without order) and further interesting information related to this parallelism will be given in a forthcoming paper. However, analogous results with the results given in this paper for ordered hypersemigroups also hold. For related results on ordered semigroups on which the present article is based we refer to [3, 7]. The left (right) quasi-regular fuzzy ordered semigroups under the name left (right) weakly regular and the semisimple ordered semigroups have been studied by Shabir and Khan in [12, 13], and by Kehayopulu in [4]. Hypersemigroups under the name semihypergroups have been first systematically studied in 2011 by Mahmood in his PhD thesis [10]. However a revision in the notation in this thesis is necessary. For the sake of completeness, let us mention some definitions and results already given in [5, 6]. An hypergroupoid is a nonempty set H with an hyperoperation ◦ : H ×H → P∗(H) | (a, b)→ a ◦ b on H and an operation ∗ : P∗(H) × P∗(H) → P∗(H) | (A,B) → A ∗ B on P∗(H) (induced by the operation of H) such that A ∗ B = ⋃ (a,b)∈A×B (a ◦ b) for every A,B ∈ P∗(H), P∗(H) being the set of nonempty subsets of H. As the operation “∗” depends on the hyperoperation “◦”, an hypergroupoid can be denoted by (H, ◦) (instead of (H, ◦, ∗)). If (H, ◦) is an hypergroupoid then, for every x, y ∈ H, we have {x} ∗ {y} = x ◦ y. If (H, ◦) is an hypergroupoid and A,B,C ∈ P∗(H), then A ⊆ B implies A ∗ C ⊆ B ∗ C and C ∗ A ⊆ C ∗ B. We also have H ∗H ⊆ H. If (H, ◦) is an hypergroupoid, x ∈ H and A,B ∈ P∗(H), then the following two properties, though clear, are essential to the investigation: (1) If x ∈ A ∗B, then x ∈ a ◦ b for some a ∈ A, b ∈ B. N. Kehayopulu / Eur. J. Pure Appl. Math, 10 (5) (2017), 929-945 931 (2) If a ∈ A and b ∈ B, then a ◦ b ⊆ A ∗B [5, 6]. Lemma 1.1 [5] If (H, ◦) is an hypergroupoid and Ai, B ∈ P∗(H), i ∈ I, then we have the following: (1) ( ⋃ i∈I Ai) ∗B = ⋃ i∈I (Ai ∗B). (2) B ∗ ( ⋃ i∈I Ai) = ⋃ i∈I (B ∗Ai). An hypergroupoid (H, ◦) is called hypersemigroup if (x ◦ y) ∗ {z} = {x} ∗ (y ◦ z) for every x, y, z ∈ H. For short, we can identify the singleton {x} by the element x and the {z} by z and define the hypersemigroup as (x ◦ y) ∗ z = x ∗ (y ◦ z). If (H, ◦) is an hypersemigroup, then the operation “∗” on P∗(H) is associative so, for any subsets A,B,C ∈ P∗(H) we can write (A ∗B) ∗C = A ∗ (B ∗C) = A ∗B ∗C and for any product A1 ∗A2 ∗ ..... ∗An of elements of P∗(H) we can put parentheses in any place beginning with some Ai and ending in some Aj (1 ≤ i, j ≤ n). Following the concepts of right and left ideals of groupoids, a nonempty subset A of an hypergroupoid (H, ◦) is called a right (resp. left) ideal of H if A ∗H ⊆ A (resp. H ∗A ⊆ A). It is called an ideal of H if it is both a right and left ideal of H. For a subset A of H, we denote by R(A) (resp. L(A)) the right (resp. left) ideal of H generated by A and by I(A) the ideal of H generated by A. If (H, ◦) is an hypersemigroup, then R(A) = A ∪ (A ∗H), L(A) = A ∪ (H ∗A) and I(A) = A∪(A∗H)∪(H ∗A)∪(H ∗A∗H). For A = {a} (a ∈ H), we write R(a), L(a), I(a) instead of L({a}), R({a}), I({a}). Following the concepts of bi-ideals of semigroups, a nonempty subset B of an hypersemigroup (H, ◦) is called a bi-ideal of H if B ∗H ∗B ⊆ B. Following the concepts of regular and intra-regular semigroups, an hypersemigroup (H, ◦) is called regular if for every a ∈ H there exists x ∈ H such that a ∈ (a ◦ x) ∗ {a}; it is called intra-regular if for every a ∈ H there exist x, y ∈ H such that a ∈ (x ◦ a) ∗ (a ◦ y). Lemma 1.2. Let (H, ◦) be an hypersemigroup. The following are equivalent: (1) H is regular. (2) a ∈ {a} ∗H ∗ {a} for every a ∈ H. (3) A ⊆ A ∗H ∗A for every A ∈ P∗(H). Indeed: If H is regular and a ∈ H, then there exists x ∈ H such that a ∈ (a ◦ x) ∗ {a} = {a} ∗ {x} ∗ {a} ⊆ {a} ∗H ∗ {a}. “Conversely”, if A ⊆ A ∗H ∗A for every A ∈ P∗(H) and a ∈ A, then {a} ⊆ ( {a} ∗H ) ∗ {a}, then there exists u ∈ {a} ∗H such that a ∈ u ◦ a and h ∈ H such that u ∈ a ◦ h. Then we have a ∈ u ◦ a = {u} ∗ {a} ⊆ (a ◦ h) ∗ {a}, where h ∈ H, so H is regular. In a similar way, an hypersemigroup H is intra-regular if and only if, for every a ∈ H, we have a ∈ H ∗ {a} ∗ {a} ∗H, equivalently if for any nonempty subset A of H we have A ⊆ H ∗A ∗A ∗H. Following Zadeh, any mapping f : H → [0, 1] of an hypergroupoid H into the closed interval [0, 1] of real numbers is called a fuzzy subset of H (or a fuzzy set in H) and the N. Kehayopulu / Eur. J. Pure Appl. Math, 10 (5) (2017), 929-945 932 mapping fA (the so called characteristic function of A) is the fuzzy subset of H defined as follows: fA : H → {0, 1} | x→ fA(x) = { 1 if x ∈ A 0 if x /∈ A. For an element a of H, we denote by Aa the subset of H ×H defined by Aa := {(y, z) ∈ H ×H | a ∈ y ◦ z}. For two fuzzy subsets f and g of H, we denote by f ◦ g the fuzzy subset of H defined as follows: f ◦ g : H → [0, 1] a→  ∨ (y,z)∈Aa min{f(y), g(z)} if Aa 6= ∅ 0 if Aa = ∅. As no confusion is possible, we denote the operation between fuzzy subsets of H and the hyperoperation on H by the same symbol. Denote by F (H) the set of all fuzzy subsets of H and by “�” the order relation on F (H) defined by f � g ⇐⇒ f(x) ≤ g(x) for every x ∈ H. For two fuzzy subsets f and g of an hypergroupoid H we denote by f ∧ g the fuzzy subset of H defined as follows: f ∧ g : H → [0, 1] | x→ (f ∧ g)(x) := min{f(x), g(x)}. One can easily see that the fuzzy subset f ∧ g is the infimum of the fuzzy subsets f and g, and this is why we write f ∧ g = inf{f, g}. If f is a fuzzy subset of H, then f ∧ f = f . Indeed, if x ∈ H, then (f ∧ f)(x) := min{f(x), f(x)} = f(x). The concepts of fuzzy right and fuzzy left ideal of a groupoid due to Kuroki [8] can be naturally transferred to hypergroupoids as follows: A fuzzy subset f of an hypergroupoid H is called a fuzzy right ideal of H if f(x ◦ y) ≥ f(x) for every x, y ∈ H, in the sense that if x, y ∈ H and u ∈ x ◦ y, then f(u) ≥ f(x). A fuzzy subset f of an hypergroupoid H is called a fuzzy left ideal of H if f(x ◦ y) ≥ f(y) for every x, y ∈ H, meaning that if x, y ∈ H and u ∈ x ◦ y, then f(u) ≥ f(y). If f is both a fuzzy right and a fuzzy left ideal of H, then it is called a fuzzy ideal of H. A fuzzy subset f of an hypersemigroup H is called a fuzzy bi-ideal of H if f ( (x ◦ y) ∗ {z} ) ≥ min{f(x), f(z)} for every x, y, z ∈ H, in the sense that if x, y, z ∈ H and u ∈ (x ◦ y) ∗ {z}, then f(u) ≥ min{f(x), f(z)}. Exactly as in groupoids–semigroups, the following hold: N. Kehayopulu / Eur. J. Pure Appl. Math, 10 (5) (2017), 929-945 933 (1) If H is an hypergroupoid, then A is a right (resp. left) ideal of H if and only if the characteristic function fA is a fuzzy right (resp. fuzzy left) ideal of H. (2) If H is an hypersemigroup, then A is a bi-ideal of H if and only if fA is a fuzzy bi-ideal of H. 2. Main results Theorem 2.1. Let (H, ◦) be an hypersemigroup. The following are equivalent: (1) H is regular. (2) A ∩B = A ∗B for every right ideal A and every left ideal B of H. (3) A ∩B ⊆ A ∗B for every right ideal A and every left ideal B of H. (4) R(A) ∩ L(A) ⊆ R(A) ∗ L(A) for every A ∈ P∗(H). (5) R(a) ∩ L(a) ⊆ R(a) ∗ L(a) for every a ∈ H. Proof. (1) =⇒ (2). Let A be a right ideal and B a left ideal of H. The set A ∩ B is a nonempty subset of H. Indeed: Take an element a ∈ A and an element b ∈ B (A,B 6= ∅). Then a ◦ b ⊆ A ∗H ⊆ A and a ◦ b ⊆ H ∗B ⊆ B, so a ◦ b ⊆ A∩B. Since a ◦ b 6= ∅, we have A ∩B 6= ∅. Since H is regular and A ∩B ∈ P∗(H), we have A ∩B ⊆ (A ∩B) ∗H ∗ (A ∩B) ⊆ A ∗H ∗B = (A ∗H) ∗B ⊆ A ∗B ⊆ (A ∗H) ∩ (H ∗B) ⊆ A ∩B. Thus we have A ∩B = A ∗B. The implications (2)⇒ (3)⇒ (4)⇒ (5) are obvious. (5) =⇒ (1). Let a ∈ H. Since R(a) is a right ideal of H and L(a) is a left ideal of H, by hypothesis, we have a ∈ R(a) ∩ L(a) ⊆ R(a) ∗ L(a) = ( {a} ∪ ( {a} ∗H )) ∗ ( {a} ∪ ( H ∗ {a} )) = (a ◦ a) ∪ ( {a} ∗H ∗ {a} ) ∪ ( {a} ∗H ∗H ∗ {a} ) (by Lemma 1.1) = (a ◦ a) ∪ ( {a} ∗H ∗ {a} ) . We have a ∈ a ◦ a, so a ∈ (a ◦ a) ∗ {a} or a ∈ {a} ∗H ∗ {a}. In each case, H is regular. � Proposition 2.2. Let (H, ◦) be an hypergroupoid, f a fuzzy right ideal and g a fuzzy left ideal of H. Then we have f ◦ g � f ∧ g. Proof. Let a ∈ H. Then (f ◦ g)(a) ≤ (f ∧ g)(a). In fact: If Aa = ∅, then (f ◦ g)(a) := 0. Since a ∈ H and f ∧ g is a fuzzy subset of H, we have (f ∧ g)(a) ≥ 0, thus we have (f ◦ g)(a) ≤ (f ∧ g)(a). Let now Aa 6= ∅. Then (f ◦ g)(a) := ∨ (x,y)∈Aa min{f(x), g(y)} (∗) N. Kehayopulu / Eur. J. Pure Appl. Math, 10 (5) (2017), 929-945 934 We have min{f(x), g(y)} ≤ (f ∧ g)(a) for every (x, y) ∈ Aa (∗∗) Indeed: Let (x, y) ∈ Aa. Then a ∈ x ◦ y. Since f is a fuzzy right ideal of H, we have f(x ◦ y) ≥ f(x), then we have f(a) ≥ f(x). Since g is a fuzzy left ideal of H, we have g(x ◦ y) ≥ g(y), then g(a) ≥ g(y), so (f ∧ g)(a) := min{f(a), g(a)} ≥ min{f(x), g(y)}, and condition (∗∗) is satisfied. By (∗∗), we have∨ (x,y)∈Aa min{f(x), g(y)} ≤ (f ∧ g)(a). Then, by (∗), (f ◦ g)(a) ≤ (f ∧ g)(a). � Proposition 2.3. Let (H, ◦) be a regular hypersemigroup, f a fuzzy right ideal of H and g a fuzzy subset of H. Then we have f ∧ g � f ◦ g. Proof. Let a ∈ H. Since H is regular, there exists x ∈ H such that a ∈ (a ◦x) ∗ {a}. Then a ∈ u ◦ a for some u ∈ a ◦ x. Since a ∈ u ◦ a, we have (u, a) ∈ Aa, then (f ◦ g)(a) := ∨ (y,z)∈Aa min{f(y), g(z)} ≥ min{f(u), g(a)}. Since f is a fuzzy right ideal of H, we have f(a ◦ x) ≥ f(a). Since u ∈ a ◦ x, we have f(u) ≥ f(a). Then we have (f ◦ g)(a) ≥ min{f(u), g(a)} ≥ min{f(a), g(a)} := (f ∧ g)(a), so f ∧ g � f ◦ g and the proof is complete. � Theorem 2.4. Let (H, ◦) be an hypersemigroup. The following are equivalent: (1) H is regular. (2) f ∧ g = f ◦ g for every fuzzy right ideal f and every fuzzy left ideal g of H. (3) f ∧ g � f ◦ g for every fuzzy right ideal f and every fuzzy left ideal g of H. Proof. (1) =⇒ (2). Let f be a fuzzy right ideal and g a fuzzy left ideal of H. By Proposition 2.2, we have f ◦ g � f ∧ g. By Proposition 2.3, we have f ∧ g � f ◦ g, and (2) is satisfied. The implication (2)⇒ (3) is obvious. (3) =⇒ (1). By Theorem 2.1, it is enough to prove that R(a) ∩ L(a) ⊆ R(a) ∗ L(a) for every a ∈ H. So, let a ∈ H and b ∈ R(a) ∩ L(a). Since R(a) is a right ideal of H, the characteristic function fR(a) is a fuzzy right ideal of H and, since L(a) is a left ideal of H, fL(a) is a fuzzy left ideal of H. By hypothesis, we have fR(a) ∧ fL(a) � fL(a) ◦ fR(a), then( fR(a) ∧ fL(a) ) (b) ≤ ( fR(a) ◦ fL(a) ) (b). N. Kehayopulu / Eur. J. Pure Appl. Math, 10 (5) (2017), 929-945 935 Thus min{fR(a)(b), fL(a)(b)} ≤ ( fR(a) ◦ fL(a) ) (b). Since b ∈ R(a) and b ∈ L(a), we have fR(a)(b) = fL(a)(b) = 1, and so 1 ≤ ( fR(a) ◦ fL(a) ) (b). If Ab = ∅, then ( fR(a) ◦ fL(a) ) (b) = 0 which is impossible. Thus we have Ab 6= ∅ and( fR(a) ◦ fL(a) ) (b) = ∨ (y,z)∈Ab min{fR(a)(y), fL(a)(z)}. Then there exists (y, z) ∈ Ab such that y ∈ R(a) and z ∈ L(a) (∗) Indeed: Suppose there is no (y, z) ∈ Ab such that y ∈ R(a) and z ∈ L(a). Then, for each (y, z) ∈ Ab, we have y /∈ R(a) or z /∈ L(a). Then, for each (y, z) ∈ Ab, we have fR(a)(y) = 0 or fL(a)(z) = 0. Then min{fR(a)(y), fR(a)(z)} = 0 for every (y, z) ∈ Ab. Then∨ (y,z)∈Ab min{fR(a)(y), fL(a)(z)} = 0, so ( fR(a) ◦ fL(a) ) (b) = 0 which is no possible. By (∗), we have b ∈ y ◦ z ⊆ R(a) ∗ L(a), and the proof is complete. � Theorem 2.5. (cf. also [10]) Let (H, ◦) be an hypersemigroup. The following are equiva- lent: (1) H is intra-regular. (2) A ∩B ⊆ B ∗A for every right ideal A and every left ideal B of H. (3) R(A) ∩ L(A) ⊆ L(A) ∗R(A) for every A ∈ P∗(H). (4) R(a) ∩ L(a) ⊆ L(a) ∗R(a) for every a ∈ H. Theorem 2.6. An hypersemigroup (H, ◦) is intra-regular if and only if, for every fuzzy right ideal f and every fuzzy left ideal g of H, we have f ∧ g � g ◦ f. Proof. =⇒. Let a ∈ H. Since H is intra-regular, there exist x, y ∈ H such that a ∈ (x ◦ a) ∗ (a ◦ y). Then a ∈ u ◦ v for some u ∈ x ◦ a, v ∈ a ◦ y. Since a ∈ u ◦ v, we have (u, v) ∈ Aa, then we have (g ◦ f)(a) := ∨ (h,k)∈Aa min{g(h), f(k)} ≥ min{g(u), f(v)}. Since g is a fuzzy left ideal of H, we have g(x ◦ a) ≥ g(a). Since u ∈ x ◦ a, we get g(u) ≥ g(a). Since f is a fuzzy right ideal of H, we have f(a ◦ y) ≥ f(a). Since v ∈ a ◦ y, we have f(v) ≥ f(a). Thus we have (g ◦ f)(a) ≥ min{g(u), f(v)} ≥ min{g(a), f(a)} = (f ∧ g)(a), N. Kehayopulu / Eur. J. Pure Appl. Math, 10 (5) (2017), 929-945 936 thus f ∧ g � g ◦ f . ⇐=. By Theorem 2.5, it is enough to prove that R(a) ∩ L(a) ⊆ L(a) ∗ R(a) for every a ∈ H. Let now a ∈ H and b ∈ R(a) ∩ L(a). As fR(a) is a fuzzy right ideal and fL(a) is a fuzzy left ideal of H, by hypothesis, we have( fR(a) ∧ fL(a) ) (b) ≤ ( fL(a) ◦ fR(a) ) (b). Thus min{fR(a)(b), fL(a)(b)} ≤ ( fL(a) ◦ fR(a) ) (b). Since b ∈ R(a) and b ∈ L(a), we have fR(a)(b) = fL(a)(b) = 1, and so 1 ≤ ( fL(a) ◦ fR(a) ) (b). If Ab = ∅, then ( fL(a) ◦ fR(a) ) (b) = 0 which is impossible. Then we have Ab 6= ∅ and then( fL(a) ◦ fR(a) ) (b) = ∨ (y,z)∈Ab min{fL(a)(y), fR(a)(z)}. Then there exists (y, z) ∈ Ab such that y ∈ L(a) and z ∈ R(a). Then we get b ∈ y ◦ z ⊆ L(a) ∗R(a). � The concept of left quasi-regular semigroups can be naturally transferred to hyper- semigroups in the definition below. Definition 2.7. An hypersemigroup (H, ◦) is called left quasi-regular if for every a ∈ H there exist x, y ∈ H such that a ∈ (x ◦ a) ∗ (y ◦ a). Proposition 2.8. Let (H, ◦) be an hypersemigroup. The following are equivalent: (1) H is left quasi-regular. (2) a ∈ H ∗ {a} ∗H ∗ {a} for every a ∈ H. (3) A ⊆ H ∗A ∗H ∗A for every A ∈ P∗(H). Theorem 2.9. Let (H, ◦) be an hypersemigroup. The following are equivalent: (1) H is left quasi-regular. (2) A ∩B ⊆ A ∗B for every ideal A and every nonempty subset B of H. (3) A ∩B ⊆ A ∗B for every ideal A and every bi-ideal B of H. (4) A ∩B ⊆ A ∗B for every ideal A and every left ideal B of H. (5) I(A) ∩ L(A) ⊆ I(A) ∗ L(A) for every A ∈ P∗(H). (6) I(a) ∩ L(a) ⊆ I(a) ∗ L(a) for every a ∈ H. N. Kehayopulu / Eur. J. Pure Appl. Math, 10 (5) (2017), 929-945 937 Proof. (1) =⇒ (2). Let A be an ideal, B a nonempty subset of H and a ∈ A ∩ B. Since a ∈ H and H is left quasi-regular, by Proposition 2.8 (1)⇒ (2), we have a ∈ H ∗ {a} ∗H ∗ {a} = ( H ∗ {a} ∗H ) ∗ {a} ⊆ (H ∗A ∗H) ∗B. We have H ∗A∗H = (H ∗A)∗H ⊆ A∗H since A is a left ideal of H and A∗H ⊆ A since A is a right ideal of H. Thus we have H ∗A ∗H ⊆ A. Then a ∈ (H ∗A ∗H) ∗B ⊆ A ∗B. The implications (2)⇒ (3) and (4)⇒ (5)⇒ (6) are obvious, and (3)⇒ (4) since the left ideals of H are bi-ideals of H as well. (6) =⇒ (1). Let a ∈ H. By hypothesis, we have a ∈ I(a) ∩ L(a) ⊆ I(a) ∗ L(a) = ( {a} ∪ ( H ∗ {a} ) ∪ ( {a} ∗H ) ∪ ( H ∗ {a} ∗H )) ∗ ( {a} ∪ ( H ∗ {a} )) = ( {a} ∗ {a} ) ∪ ( H ∗ {a} ∗ {a} ) ∪ ( {a} ∗H ∗ {a} ) ∪ ( H ∗ {a} ∗H ∗ {a} ) . If a ∈ {a} ∗ {a}, then a ∈ {a} ⊆ {a} ∗ {a} ⊆ ( {a} ∗ {a} ) ∗ ( {a} ∗ {a} ) ⊆ H ∗ {a} ∗H ∗ {a}. If a ∈ H ∗ {a} ∗ {a}, then a ∈ {a} ⊆ H ∗ {a} ∗ ( H ∗ {a} ∗ {a} ) ⊆ H ∗ {a} ∗ (H ∗H) ∗ {a} ⊆ H ∗ {a} ∗H ∗ {a}. If a ∈ {a} ∗H ∗ {a}, then {a} ⊆ {a} ∗H ∗ ( {a} ∗H ∗ {a} ) ⊆ (H ∗H) ∗ ( {a} ∗H ∗ {a} ) ⊆ H ∗ {a} ∗H ∗ {a}. In each case, a ∈ H ∗ {a} ∗H ∗ {a}, so H is left quasi-regular. � Theorem 2.10. An hypersemigroup (H, ◦) is left quasi-regular if and only if, for any left ideals A and B of H, we have A ∩B ⊆ A ∗B. Proof. =⇒. Let A, B be left ideals of H and a ∈ A ∩ B. Since H is left quasi-regular, there exist x, y ∈ H such that a ∈ (x ◦ a) ∗ (y ◦ a) = {x} ∗ {a} ∗ {y} ∗ {a} ⊆ (H ∗A) ∗ (H ∗B) ⊆ A ∗B. ⇐=. Let A ∈ P∗(H). Then A ⊆ H ∗A ∗H ∗A. In fact, by hypothesis, we have A ⊆ L(A) = L(A) ∩ L(A) ⊆ L(A) ∗ L(A) = ( A ∪ (H ∗A) ) ∗ ( A ∪ (H ∗A) ) N. Kehayopulu / Eur. J. Pure Appl. Math, 10 (5) (2017), 929-945 938 = (A ∗A) ∪ (H ∗A ∗A) ∪ (A ∗H ∗A) ∪ (H ∗A ∗H ∗A). Then we have A ∗A ⊆ (A ∗A ∗A) ∪ (A ∗H ∗A ∗A) ∪ (A ∗A ∗H ∗A) ∪ (A ∗H ∗A ∗H ∗A) ⊆ A ∗H ∗A, from which H ∗A ∗A ⊆ H ∗A ∗H ∗A. Thus we obtain A ⊆ (A ∗H ∗A) ∪ (H ∗A ∗H ∗A), then we get A ∗H ∗A ⊆ (A ∗H ∗A ∗H ∗A) ∪ (H ∗A ∗H ∗A ∗H ∗A) ⊆ H ∗A ∗H ∗A, so A ⊆ H ∗A ∗H ∗A, and H is left quasi-regular. � A subset A of an hypergroupoid (H, ◦) is called idempotent if A ∗A = A. Theorem 2.11. An hypersemigroup (H, ◦) is left quasi-regular if and only if the left ideals of H are idempotent. Proof. =⇒. If L is a left ideal of H then, by Theorem 2.10, we have L ⊆ L∗L ⊆ H ∗L ⊆ L, so L ∗ L = L. ⇐=. By Theorem 2.9, it is enough to prove that for every ideal A and every left ideal B of H, we have A ∩ B ⊆ A ∗ B. Let now A be an ideal and B a left ideal of H. The set A∩B is a nonempty subset of H and H ∗ (A∩B) ⊆ (H ∗A)∩ (H ∗B) ⊆ A∩B, so the set A∩B is a left ideal of H. By hypothesis, we have A∩B = (A∩B) ∗ (A∩B) ⊆ A ∗B. � Theorem 2.12. Let (H, ◦) be an hypersemigroup. The following are equivalent: (1) H is left quasi-regular. (2) f ∧ g � f ◦ g for every fuzzy ideal f and every fuzzy subset g of H. (3) f ∧ g � f ◦ g for every fuzzy ideal f and every fuzzy bi-ideal g of H. (4) f ∧ g � f ◦ g for every fuzzy ideal f and every fuzzy left ideal g of H. Proof. (1) =⇒ (2). Let f be a fuzzy ideal, g a fuzzy subset of H and a ∈ H. Since H is left quasi-regular, there exist x, y ∈ H such that a ∈ ( (x ◦ a) ∗ {y} ) ∗ {a}. Then a ∈ u ◦ a for some u ∈ (x ◦ a) ∗ {y}. In addition, u ∈ v ◦ y for some v ∈ x ◦ a. On the other hand, since (u, a) ∈ Aa, we have (f ◦ g)(a) := ∨ (h,k)∈Aa min{f(h), g(k)} ≥ min{f(u), g(a)}. N. Kehayopulu / Eur. J. Pure Appl. Math, 10 (5) (2017), 929-945 939 Since f is a fuzzy right ideal of H, we have f(v ◦ y) ≥ f(v) and since u ∈ v ◦ y, we have f(u) ≥ f(v). Since f is a fuzzy left ideal of H, we have f(x◦a) ≥ f(a) and since v ∈ x◦a, we have f(v) ≥ f(a). Thus we have f(u) ≥ f(a), and (f ◦ g)(a) ≥ min{f(a), g(a)} = (f ∧ g)(a), so f ∧ g � f ◦ g. The implication (2) ⇒ (3) is obvious and (3) ⇒ (4) since every fuzzy left ideal is a fuzzy bi-ideal of H. (4) =⇒ (1). By Theorem 2.9, it is enough to prove that I(a) ∩ L(a) ⊆ I(a) ∗ L(a) for every a ∈ H. Let now a ∈ H and b ∈ I(a) ∩ L(a). Since I(a) is an ideal of H, the characteristic function fI(a) is a fuzzy ideal of H and since L(a) is a left ideal of H, fL(a) is a fuzzy left ideal of H. By hypothesis, we have fI(a) ∧ fL(a) � fI(a) ◦ fL(a), and so( fI(a) ∧ fL(a) ) (b) ≤ ( fI(a) ◦ fL(a) ) (b), that is min{fI(a)(b), fL(a)(b)} ≤ ( fI(a) ◦ fL(a) ) (b). Since b ∈ I(a), we have fI(b)(b) = 1 and since b ∈ L(a), we have fL(b)(b) = 1, so min{fI(a)(b), fL(a)(b)} = 1, and so 1 ≤ ( fI(a)◦fL(a) ) (b). If Ab = ∅, then ( fI(a)◦fL(a) ) (b) = 0 which is impossible. Thus we have Ab 6= ∅. Then( fI(a) ◦ fL(a) ) (b) = ∨ (y,z)∈Ab min{fI(a)(y), fL(a)(z)}. Then there exists (y, z) ∈ Ab such that y ∈ I(a) and z ∈ L(a) (∗) Indeed: Suppose there is no (y, z) ∈ Ab such that y ∈ I(a) and z ∈ L(a). Then, for every (y, z) ∈ Ab we have y /∈ I(a) or z /∈ L(a). Then, for each (y, z) ∈ Ab we have fI(a)(y) = 0 or fL(a)(z) = 0, so for each (y, z) ∈ Ab, we have min{fI(a)(y), fL(a)(z)} = 0, then ( fI(a) ◦ fL(a) ) (b) = 0 which is no possible. By (∗), we have b ∈ y ◦ z ⊆ I(a) ∗ L(a), and the proof is complete. � Theorem 2.13. An hypersemigroup (H, ◦) is left quasi-regular if and only if for any fuzzy left ideals f and g of H, we have f ∧ g � f ◦ g. Proof. =⇒. Let f and g be fuzzy left ideals of H and a ∈ H. Then (f ∧g)(a) ≤ (f ◦g)(a). Indeed: By hypothesis, we have a ∈ (x◦a)∗ (y ◦a), so we have a ∈ u◦v for some u ∈ x◦a, v ∈ y ◦ a. Since (u, v) ∈ Aa, we have (f ◦ g)(a) = ∨ (h,k)∈Aa min{f(h), g(k)} ≥ min{f(u), g(v)}. Since f is a fuzzy left ideal of H, we have f(x ◦ a) ≥ f(a) and since u ∈ x ◦ a, we have f(u) ≥ f(a). Since g is a fuzzy left ideal of H, we have g(y ◦ a) ≥ g(a) and since v ∈ y ◦ a, we have g(v) ≥ g(a). Hence we obtain (f ◦ g)(a) ≥ min{f(a), g(a)} = (f ∧ g)(a). N. Kehayopulu / Eur. J. Pure Appl. Math, 10 (5) (2017), 929-945 940 ⇐=. By Theorem 2.11, it is enough to prove that the left ideals of H are idempotent. Let now A be a left ideal of H and a ∈ A. Since fA is a fuzzy left ideal of H, by hypothesis, we have fA = fA ∧ fA � fA ◦ fA. Then 1 = fA(a) ≤ fA ◦ fA)(a). If Aa = ∅, then (fA ◦ fA)(a) = 0 which is impossible. Thus we have Aa 6= ∅ and (fA ◦ fA)(a) := ∨ (h,k)∈Aa min{fA(h), fA(k)}. Then there exists (y, z) ∈ Aa such that y ∈ A and z ∈ A. Then we have a ∈ y ◦ z ⊆ A∗A, so A ⊆ A ∗A ⊆ H ∗A ⊆ A, thus A ∗A = A, and A is idempotent. � Theorem 2.14. An hypersemigroup (H, ◦) is left quasi-regular if and only if the fuzzy left ideals of H are idempotent. Proof. =⇒. Let f be a fuzzy left ideal of H and a ∈ H. Then (f ◦ f)(a) ≤ f(a). In fact, if Aa = ∅, then (f ◦ f)(a) = 0 ≤ f(a). Let Aa 6= ∅. Then (f ◦ f)(a) = ∨ (x,y)∈Aa min{f(x), f(y)}. On the other hand, min{f(x), f(y)} ≤ f(a) for every (x, y) ∈ Aa. Indeed: Let (x, y) ∈ Aa. Since f is a fuzzy left ideal of H we have f(x ◦ y) ≥ f(y), and since a ∈ x ◦ y, we have f(a) ≥ f(y) ≥ min{f(x), f(y)}. Thus we get (f ◦ f)(a) ≤ f(a). Moreover f � f ◦ f . Indeed: Let a ∈ H. Since H is left quasi-regular, there exist x, y ∈ H such that a ∈ (x◦a)∗(y◦a). Then a ∈ u◦v for some u ∈ x◦a, v ∈ y◦a. Since (u, v) ∈ Aa, we have (f ◦ f)(a) := ∨ (h,k)∈Aa min{f(h), f(k)} ≥ min{f(u), f(v)}. Since f is a fuzzy left ideal of H, we have f(x ◦ a) ≥ f(a) and since u ∈ x ◦ a, we get f(u) ≥ f(a). Again since f is a fuzzy left ideal of H, we have f(y ◦ a) ≥ f(a) and since v ∈ y ◦ a, we get f(v) ≥ f(a). Thus we have (f ◦ f)(a) ≥ min{f(a), f(a)} = f(a), so f � f ◦ f , and f is idempotent. ⇐=. By Theorem 2.11, it is enough to prove that the left ideals of H are idempotent. Let now A be a left ideal of H and a ∈ A. Since fA is a fuzzy left ideal of H, by hypothesis, we have fA = fA ◦ fA, thus 1 = fA(a) = (fA ◦ fA)(a). Then 1 ≤ (fA ◦ fA)(a) and for the rest of the proof we refer to the proof of the ⇐-part of the previous theorem. � The concept of right quasi-regular semigroups is naturally transferred to hypersemi- groups in the following definition. Definition 2.15. An hypersemigroup (H, ◦) is called right quasi-regular if for every a ∈ H there exist x, y ∈ H such that a ∈ (a ◦ x) ∗ (a ◦ y). Proposition 2.16. Let (H, ◦) be an hypersemigroup. The following are equivalent: N. Kehayopulu / Eur. J. Pure Appl. Math, 10 (5) (2017), 929-945 941 (1) H is right quasi-regular. (2) a ∈ {a} ∗H ∗ {a} ∗H for every a ∈ H. (3) A ⊆ A ∗H ∗A ∗H for every A ∈ P∗(H). The right analogues of Theorems 2.9–2.14 also hold and we have the following: Theorem 2.17. Let (H, ◦) be an hypersemigroup. The following are equivalent: (1) H is right quasi-regular. (2) A ∩B ⊆ A ∗B for every nonempty subset A and every ideal B of H. (3) A ∩B ⊆ A ∗B for every bi-ideal A and every ideal B of H. (4) A ∩B ⊆ A ∗B for every right ideal A and every ideal B of H. (5) R(A) ∩ I(A) ⊆ R(A) ∗ I(A) for every A ∈ P∗(H). (6) R(a) ∩ I(a) ⊆ R(a) ∗ I(a) for every a ∈ H. Let us prove the implication (1)⇒ (2): Let A be a nonempty subset of H, B an ideal of H and a ∈ A ∩B. Since H is right quasi-regular, we have a ∈ {a} ∗H ∗ {a} ∗H ⊆ A ∗ (H ∗B ∗H) ⊆ A ∗B. � Theorem 2.18. An hypersemigroup (H, ◦) is right quasi-regular if and only if, for any right ideals A and B of H, we have A ∩B ⊆ A ∗B. Theorem 2.19. An hypersemigroup (H, ◦) is right quasi-regular if and only if the right ideals of H are idempotent. Theorem 2.20. Let (H, ◦) be an hypersemigroup. The following are equivalent: (1) H is right quasi-regular. (2) f ∧ g � f ◦ g for every fuzzy subset f and every fuzzy ideal g of H. (3) f ∧ g � f ◦ g for every fuzzy bi-ideal f and every fuzzy ideal g of H. (4) f ∧ g � f ◦ g for every fuzzy right ideal f and every fuzzy ideal g of H. Theorem 2.21. An hypersemigroup (H, ◦) is right quasi-regular if and only if for any fuzzy right ideals f and g of H, we have f ∧ g � f ◦ g. A fuzzy subset f of an hypergroupoid is called idempotent if f ◦ f = f . Theorem 2.22. An hypersemigroup (H, ◦) is right quasi-regular if and only if the fuzzy right ideals of H are idempotent. N. Kehayopulu / Eur. J. Pure Appl. Math, 10 (5) (2017), 929-945 942 The concept of semisimple semigroups can be naturally transferred to hypersemigroups as follows: Definition 2.23. An hypersemigroup (H, ◦) is called semisimple if for every a ∈ H there exist x, y, z ∈ H such that a ∈ (x ◦ a) ∗ (y ◦ a) ∗ {z}. Proposition 2.24. Let (H, ◦) be an hypersemigroup. The following are equivalent: (1) H is semisimple. (2) a ∈ H ∗ {a} ∗H ∗ {a} ∗H for every a ∈ H. (3) A ⊆ H ∗A ∗H ∗A ∗H for every A ∈ P∗(H). Let us prove the implication (3)⇒ (1): Let a ∈ H. By (3), we have {a} ⊆ (( H ∗ {a} ) ∗ ( H ∗ {a} )) ∗H. Then a ∈ x ◦ z for some x ∈ ( H ∗ {a} ) ∗ ( H ∗ {a} ) , z ∈ H. Then x ∈ u ◦ v for some u, v ∈ H ∗ {a}, u ∈ x ◦ a for some x ∈ H and v ∈ y ◦ a for some y ∈ H. Then a ∈ x ◦ z, x ∈ u ◦ v, u ∈ x ◦ a, v ∈ y ◦ a, z ∈ H. Thus we have a ∈ x ◦ z = {x} ∗ {z} ⊆ (u ◦ v) ∗ {z} = {u} ∗ {v} ∗ {z} ⊆ (x ◦ a) ∗ (y ◦ a) ∗ {z}. Since x, y, z ∈ H and a ∈ (x ◦ a) ∗ (y ◦ a) ∗ {z}, H is semisimple and condition (1) is satisfied. � Theorem 2.25. Let (H, ◦) be an hypersemigroup. The following are equivalent: (1) H is semisimple. (2) The ideals of H are idempotent. (3) A ∩B = A ∗B for all ideals A,B of H. (4) I(A) = I(A) ∗ I(A) for every A ∈ P∗(H). (5) I(a) = I(a) ∗ I(a) for every a ∈ H. Proof. (1) =⇒ (2). Let A be an ideal of H. Since A ∈ P∗(H) and H is semisimple, by Proposition 2.24, we have A ⊆ (H ∗A) ∗H ∗ (A ∗H) ⊆ A ∗H ∗A = (A ∗H) ∗A ⊆ A ∗A ⊆ A ∗H ⊆ A, N. Kehayopulu / Eur. J. Pure Appl. Math, 10 (5) (2017), 929-945 943 so A ∗A = A, and A is idempotent. (2) =⇒ (3). Let A,B be ideals of H. Then A ∗B ⊆ A ∗H ⊆ A and A ∗B ⊆ H ∗B ⊆ B, so A ∗B ⊆ A∩B. On the other hand, A∩B is an ideal of H and, by hypothesis, we have A ∩B = (A ∩B) ∗ (A ∩B) ⊆ A ∗B. Thus we have A ∩B = A ∗B. The implications (3)⇒ (4) and (4)⇒ (5) are obvious. (5) =⇒ (1). Exactly as in the Lemma 2 in [3], we prove that I(a) = I(a) ∗ I(a) ∗ I(a) ∗ I(a) ∗ I(a) and that I(a) ∗ I(a) ∗ I(a) ∗ I(a) ∗ I(a) ⊆ H ∗ {a} ∗H ∗ {a} ∗H. Then we get a ∈ H ∗ {a} ∗H ∗ {a} ∗H, and H is semisimple. � Proposition 2.26. Let (H, ◦) be an hypersemigroup. Then we have the following: (1) If H is regular, then it is left and right quasi-regular. (2) If H is left (or right) quasi-regular, then it is semisimple. (3) If H is intra-regular, then it is semisimple. Proof. (1) Let H be regular and A ∈ P∗(H). Then we have A ⊆ A ∗H ∗A ⊆ A ∗H ∗ (A ∗H ∗A) ⊆ (H ∗H) ∗ (A ∗H ∗A) ⊆ H ∗A ∗H ∗A, so H is left quasi-regular. Similarly H is right quasi-regular. (2) Let H be left quasi-regular and A ∈ P∗(H). Then we have A ⊆ H ∗A ∗H ∗A ⊆ H ∗ (H ∗A ∗H ∗A) ∗ (H ∗A) = (H ∗H) ∗ (A ∗H ∗A) ∗ (H ∗A) ⊆ (H ∗H) ∗ (A ∗H ∗A) ∗ (H ∗H) ⊆ H ∗A ∗H ∗A ∗H, thus H is semisimple. If H is right quasi-regular, the proof is analogous. (3) Let H be intra-regular and A a nonempty subset of H. Then we have A ⊆ H ∗A ∗A ∗H ⊆ H ∗ (H ∗A ∗A ∗H) ∗A ∗H = (H ∗H) ∗A ∗ (A ∗H) ∗A ∗H ⊆ (H ∗H) ∗A ∗ (H ∗H) ∗A ∗H ⊆ H ∗A ∗H ∗A ∗H, and H is semisimple. � Theorem 2.27. Let H be an hypersemigroup. The following are equivalent: N. Kehayopulu / Eur. J. Pure Appl. Math, 10 (5) (2017), 929-945 944 (1) H is semisimple. (2) For every fuzzy ideals f and g of H, we have f ∧ g = f ◦ g. (3) For every fuzzy ideal f of H, we have f = f ◦ f . Proof. (1) =⇒ (2). Let f and g be fuzzy ideals of H. Since f is a fuzzy right ideal and g is a fuzzy left ideal of H, by Proposition 2.2, we have f ◦ g � f ∧ g. Let now a ∈ H. Then (f ∧ g)(a) ≤ (f ◦ g)(a). In fact: Since H is semisimple, there exist x, y, z ∈ H such that a ∈ (x ◦ a) ∗ (y ◦ a) ∗ {z}. Then there exist u ∈ x ◦ a and v ∈ (y ◦ a) ∗ {z} such that a ∈ u ◦ v. Since v ∈ (y ◦ a) ∗ {z}, there exists w ∈ y ◦ a such that v ∈ w ◦ z. Thus we have u ∈ x ◦ a, a ∈ u ◦ v, w ∈ y ◦ a and v ∈ w ◦ z. Since a ∈ u ◦ v, we have (u, v) ∈ Aa. Since (u, v) ∈ Aa, we have (f ◦ g)(a) := ∨ (h,k)∈Aa min{f(h), g(k)} ≥ min{f(u), g(v)}. Since f is a fuzzy left ideal of H, we have f(x ◦ a) ≥ f(a) and since u ∈ x ◦ a, we have f(u) ≥ f(a). Since g is a fuzzy right ideal of H, we have g(w ◦ z) ≥ g(w) and since v ∈ w ◦ z, we have g(v) ≥ g(w). Since g is a fuzzy left ideal of H, we have g(y ◦ a) ≥ g(a) and since w ∈ y ◦ a, we have g(w) ≥ g(a). Thus we get g(v) ≥ g(a). Hence we obtain (f ◦ g)(a) ≥ min{f(a), g(a)} = (f ∧ g)(a), so f ∧ g � f ◦ g. The implication (2)⇒ (3) is obvious. (3) =⇒ (1). Let a ∈ H. We prove that I(a) ⊆ I(a) ∗ I(a). Then, since I(a) is an ideal of H, we have I(a) = I(a) ∗ I(a) and, by Theorem 2.25, H is semisimple. Let now b ∈ I(a). Then b ∈ I(a) ∗ I(a). In fact: Since I(a) is an ideal of H, the characteristic function fI(a) is a fuzzy ideal of H. By hypothesis, we have fI(a) = fI(a) ◦ fI(a), then fI(a)(b) = ( fI(a)◦fI(a) ) (b). Since b ∈ I(a), we have fI(a)(b) = 1, then 1 = ( fI(a)◦fI(a) ) (b). If Ab = ∅, then ( fI(a) ◦ fI(a) ) (b) = 0 which is impossible. Thus we have Ab 6= ∅ and( fI(a) ◦ fI(a) ) (b) = ∨ (y,z)∈Ab min{fI(a)(y), fI(a)(z)}. Then there exists (y, z) ∈ Ab such that y ∈ I(a) and z ∈ I(a) (otherwise,( fI(a) ◦ fI(a) ) (b) = 0 which is impossible). Therefore, we have b ∈ y ◦ z ⊆ I(a) ∗ I(a), and then b ∈ I(a) ∗ I(a). � With my best thanks to the referee for reading the paper carefully (recently not very usual) and his prompt reply. REFERENCES 945 References [1] A H Clifford, G B Preston. The Algebraic Theory of Semigroups. Amer. Math. Soc., Math. Surveys 7, Providence, Rhode Island 1961. xv+224 pp. [2] K Iséki. A characterization of regular semi-group. Proc. Japan Acad. 32:676–677, 1956. [3] N Kehayopulu. On prime, weakly prime ideals in ordered semigroups. Semigroup Forum 44(3):341–346, 1992. [4] N Kehayopulu. Characterization of left quasi-regular and semisimple ordered semi- groups in terms of fuzzy sets. Int. J. Algebra 6(13–16):747–755, 2012. [5] N Kehayopulu. On hypersemigroups. Pure Math. Appl. (PU.M.A.) 25(2):151–156, 2015. [6] N Kehayopulu. On fuzzy prime and fuzzy semiprime ideals of ≤–hypergroupoids. J. Hyperstruct. 5(2):108–114, 2016. [7] N Kehayopulu, M Tsingelis. Regular ordered semigroups in terms of fuzzy subsets. Inform. Sci. 176(24):3675–3693, 2006. [8] N Kuroki. Fuzzy semiprime ideals in semigroups. Fuzzy Sets and Systems 8(1):71–79, 1982. [9] S Lajos, G Szász. On characterizations of certain classes of semigroups. Publ. Math. Debrecen 25(3–4):225–227, 1978. [10] T Mahmood. Some contributions to semihypergroups. PhD thesis, Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan 2011. [11] M Mitrović. Semilattices of Archimedean Semigroups. University of Nis, Faculty of Mechanical Engineering, Nis 2003. xiv+160 pp. [12] M Shabir, A Khan. Characterizations of ordered semigroups by the properties of their fuzzy ideals. Comput. Math. Appl. 59(1):539–549, 2010. [13] M Shabir, A Khan. On fuzzy ordered semigroups. Inform. Sci. 274:236–248, 2014.