EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 13, No. 2, 2020, 346-350 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global On regular hypersemigroups Niovi Kehayopulu Abstract. It is shown that an hypersemigroup (S, ◦) is regular if and only if the set of all quasi- ideals of S with the operation “∗” is a von Neumann regular semigroup. It is both regular and intra-regular if and only if the set of all quasi-ideals of S with the operation “∗” is a band. 2020 Mathematics Subject Classifications: 20M99, 06F05 Key Words and Phrases: Hypersemigroup, regular, intra-regular, right (left) ideal, quasi-ideal, band It has been shown in Semigroup Forum [2] that an le-semigroup (S, ·,≤) is regular if and only if the set Q of all quasi-ideal elements of S with the multiplication “·” of S is a von Neumann regular semigroup. Moreover, it has been proved that if S is both regular and intra-regular, then (Q, ·) is a band. “Conversely”, if the quasi-ideal elements of S are idempotent, then S is both regular and intra-regular. As a consequence, an le-semigroup S is both regular and intra-regular if and only if (Q, ·) is a band. As an example to the paper in Turkish J. Math. [7], we examine the above results on lattice ordered semigroups in case of an hypersemigroup. An hypersemigroup (S, ◦) is called regular if for every a ∈ S there exists x ∈ S such that a ∈ (a ◦ x) ∗ {a}; that is, for every a ∈ S there exists y ∈ a ◦ x such that a ∈ y ◦ a. It is called intra-regular if for every a ∈ S there exist x, y ∈ S such that a ∈ (x ◦ a) ∗ (a ◦ y); that is, for every a ∈ S there exist x, y ∈ S, u ∈ x ◦ a and v ∈ a ◦ y such that a ∈ u ◦ v. A subset A of an hypersemigroup (S, ◦) is called idempotent if A ∗ A = A. For notations and definitions not given in the present paper we refer to [7]. Lemma 1 [3] Let (S, ◦) be an hypersemigroup. If S is regular, then the right ideals and the left ideals of S are idempotent and for every right ideal A and every left ideal B of S, the product A ∗B is a quasi-ideal of S. Lemma 2 [4, 5] An hypersemigroup (S, ◦) is regular if and only if, for any nonempty subset A of S, we have A ⊆ A ∗ S ∗A. Lemma 3 Let (S, ◦) be an hypersemigroup, A a right ideal and B a left ideal of S. Then the intersection A ∩B is a quasi-ideal of S. DOI: https://doi.org/10.29020/nybg.ejpam.v13i2.3703 Email address: nkehayop@math.uoa.gr (N. Kehayopulu) http://www.ejpam.com 346 c© 2020 EJPAM All rights reserved. N. Kehayopulu / Eur. J. Pure Appl. Math, 13 (2) (2020), 346-350 347 Proof First of all, since A is a right ideal and B is a left ideal of S, the intersection A ∩ B is nonempty. Indeed: Take an element a ∈ A and an element b ∈ B (A,B 6= ∅); then a ◦ b ⊆ A ∗ B ⊆ A ∗ S ⊆ A and a ◦ b ⊆ A ∗ B ⊆ S ∗ B ⊆ B, so a ◦ b ⊆ A ∩ B. Since a ◦ b is a nonempty set, the set A ∩B is nonempty as well (see also [5]). We also have( (A ∩B) ∗ S ) ∩ ( S ∗ (A ∩B) ) ⊆ (A ∗ S) ∩ (S ∗B) ⊆ A ∩B, thus A ∩B is a quasi-ideal of S. � Lemma 4 [4, 5] An hypersemigroup (S, ◦) is regular if and only if, for every right ideal A and every left ideal B of S, we have A ∩B ⊆ A ∗B (equivalently, A ∩B = A ∗B). Lemma 5 If (S, ◦) is a regular hypersemigroup, then S ∗ S = S. Proof Since S is regular, for every nonempty subset A of S, by Lemma 2, we have A ⊆ A ∗ S ∗A . Thus we have S ⊆ (S ∗ S) ∗ S ⊆ S ∗ S ⊆ S and so S ∗ S = S. � A semigroup (S, ·) is called von Neumann regular (or just regular) if for each a ∈ S there exists x ∈ S such that a = axa [1, 8]. As always, P∗(S) denotes the set of all nonempty subsets of S. Theorem 6 An hypersemigroup (S, ◦) is regular if and only if the set Q of all quasi-ideals of S with the multiplication “∗” of P∗(S) is a von Neumann regular semigroup. Proof =⇒. First of all, for every quasi-ideal Q of S, we have Q = (Q ∗ S) ∩ (S ∗Q) (1) In fact: Since S is regular, R(Q) is a right ideal and L(Q) is a left ideal of (S, ◦), by Lemma 1, they are idempotent and we have Q ⊆ Q ∪ (Q ∗ S) = R(Q) = R(Q) ∗R(Q) = ( Q ∪ (Q ∗ S) ) ∗ ( Q ∪ (Q ∗ S) ) = Q ∗Q ∪Q ∗ S ∗Q ∪Q ∗Q ∗ S ∪Q ∗ S ∗Q ∗ S ⊆ Q ∗ S and Q ⊆ Q ∪ (S ∗Q) = L(Q) = L(Q) ∗ L(Q) = ( Q ∪ (S ∗Q) ) ∗ ( Q ∪ (S ∗Q) ) = Q ∗Q ∪ S ∗Q ∗Q ∪Q ∗ S ∗Q ∪ S ∗Q ∗ S ∗Q ⊆ S ∗Q. Thus we have Q ⊆ (Q ∗ S) ∩ (S ∗Q) ⊆ Q, then Q = (Q ∗ S) ∩ (S ∗Q) and property (1) is satisfied. In addition, since S is regular, A is a right ideal and B is a left ideal of S, by Lemmas 3 and 4, A ∗B is a quasi-ideal of S. So, by (1), we have A ∗B = (A ∗B ∗ S) ∩ (S ∗A ∗B) (2) We are ready now to prove that (Q, ∗) is a von Neumann regular semigroup. In this respect, we prove the following: N. Kehayopulu / Eur. J. Pure Appl. Math, 13 (2) (2020), 346-350 348 (Q, ∗) is semigroup. Indeed: First of all, in an hypersemigroup, the operation “∗” is associative (see [5], also [6; p. 22]). Let now Q1, Q2 be quasi-ideals of S. Then Q1 ∗Q2 is a quasi-ideal of S. Indeed: Since S is regular, Q1 ∗Q2 ∗ S is a right ideal and S ∗Q1 ∗Q2 is a left ideal of S, by Lemma 1, they are idempotent and we have( (Q1 ∗Q2) ∗ S ) ∩ ( S ∗ (Q1 ∗Q2) ) = (Q1 ∗Q2 ∗ S) ∗ (Q1 ∗Q2 ∗ S) ∩ (S ∗Q1 ∗Q2) ∗ (S ∗Q1 ∗Q2) = (Q1 ∗Q2 ∗ S ∗ S) ∗ (Q1 ∗Q2 ∗ S) ∩ (S ∗Q1 ∗Q2) ∗ (S ∗ S ∗Q1 ∗Q2) (since S ∗ S = S) = (Q1 ∗Q2 ∗ S) ∗ (S ∗Q1 ∗Q2) ∗ S ∩ S ∗ (Q1 ∗Q2 ∗ S) ∗ (S ∗Q1 ∗Q2) = (Q1 ∗Q2 ∗ S) ∗ (S ∗Q1 ∗Q2) (by (2)) ⊆ Q1 ∗ (Q2 ∗ S ∗Q2) ⊆ Q1 ∗ (Q2 ∗ S ∩ S ∗Q2) ⊆ Q1 ∗Q2 (since Q2 is a quasi-ideal of S). Hence Q1 ∗Q2 is a quasi-ideal of S. Thus (Q, ∗) is semigroup. The semigroup (Q, ∗) is a von Neumann regular semigroup. In fact: Let Q ∈ Q. Since (S, ◦) is regular, by Lemma 2, we have Q ⊆ Q ∗ S ∗Q ⊆ (Q ∗ S) ∩ (S ∗Q) ⊆ Q. Then Q = Q ∗ S ∗Q, where S ∈ Q and so (Q, ∗) is a von Neumann regular semigroup. ⇐=. We remark first that for each quasi-ideal Q of S, we have Q = Q ∗ S ∗Q (3) In fact: Let Q be a quasi-ideal of S. Since (Q, ∗) is von Neumann regular semigroup, there exists X ∈ Q such that Q = Q ∗X ∗Q. Then Q = Q ∗X ∗Q ⊆ Q ∗ S ∗Q ⊆ (Q ∗ S) ∩ (S ∗Q) ⊆ Q. Thus we have Q = Q ∗ S ∗Q and property (3) holds. We are ready now to prove that (S, ◦) is regular. For this, let A be a nonempty subset of S. By Lemma 2, it is enough to prove that A ⊆ A ∗ S ∗A. Since R(A) is a right ideal and L(A) is a left ideal of S, by Lemma 3, R(A) ∩ L(A) is a quasi-ideal of S. Then, by (3), we have A ⊆ R(A) ∩ L(A) = ( R(A) ∩ L(A) ) ∗ S ∗ ( R(A) ∩ L(A) ) ⊆ ( R(A) ∗ S ) ∗ L(A) ⊆ R(A) ∗ L(A) = ( A ∪ (A ∗ S) ) ∗ ( A ∪ (S ∗A) ) N. Kehayopulu / Eur. J. Pure Appl. Math, 13 (2) (2020), 346-350 349 = A ∗A ∪A ∗ S ∗A ∪A ∗ S ∗ S ∗A = A ∗A ∪A ∗ S ∗A, then A ∗A ⊆ A ∗A ∗A ∪A ∗ S ∗A ∗A ⊆ A ∗ S ∗A, thus we obtain A ⊆ A ∗ S ∗A and so the hypersemigroup (S, ◦) is regular. � Lemma 7 [4, 5] An hypersemigroup (S, ◦) is intra-regular if and only if, for every right ideal A and every left ideal B of S, we have A ∩B ⊆ B ∗A. An element a of a semigroup S is called idempotent if a2 = a. An idempotent semigroup or shorter a band is a semigroup in which all elements are idempotent. Theorem 8 Let (S, ◦) is an hypersemigroup. If (S, ◦) is both regular and intra-regular, then the set Q of all quasi-ideals of S with the operation “∗” is a band. “Conversely”, if the quasi-ideals of (S, ◦) are idempotent, then S is both regular and intra-regular. Proof =⇒. Let (S, ◦) be both regular and intra-regular. Since (S, ◦) is regular, by Theo- rem 6, (Q, ∗) is a semigroup. Moreover, the elements of the semigroup Q are idempotent. In fact: Let Q be a quasi-ideal of S. Since S is regular, we have Q = Q ∗ S ∗ Q (cf. the proof of Theorem 6). Hence we have Q = Q ∗ S ∗Q = (Q ∗ S ∗Q) ∗ S ∗ (Q ∗ S ∗Q) = (Q ∗ S ∗Q) ∗ S ∗ S ∗ (Q ∗ S ∗Q) (by Lemma 5) = (Q ∗ S) ∗ (Q ∗ S) ∗ (S ∗Q) ∗ (S ∗Q). Since S is intra-regular and Q ∗ S is a right ideal and S ∗Q is a left ideal of S, by Lemma 7, we have (Q ∗ S) ∩ (S ∗Q) ⊆ (S ∗Q) ∗ (Q ∗ S). Thus we have Q = (Q ∗ S) ∗ (Q ∗ S) ∗ (S ∗Q) ∗ (S ∗Q) ⊆ (Q ∗ S) ∗ (S ∗Q) ∗ (Q ∗ S) ∗ (S ∗Q) = (Q ∗ S ∗ S ∗Q) ∗ (Q ∗ S ∗ S ∗Q) = (Q ∗ S ∗Q) ∗ (Q ∗ S ∗Q) (by Lemma 5) = Q ∗Q ⊆ (Q ∗ S) ∩ (S ∗Q) ⊆ Q, and Q ∗Q = Q. Hence (Q, ∗) is an idempotent semigroup and so is a band. ⇐=. Let A be a right ideal and B a left ideal of S. By Lemma 3, A∩B is a quasi-ideal of S. By hypothesis, we have A∩B = (A∩B)∗(A∩B) ⊆ A∗B, B ∗A. Since A∩B ⊆ A∗B, by Lemma 4, S is regular. 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