PROPERTIES OF HYPERIDEALS IN ORDERED SEMIHYPERGROUPS. Thawhat Changphas. Bijan Davvaz

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1 italian journal of pure and applied mathematics n ( ) 425 PROPERTIES OF HYPERIDEALS IN ORDERED SEMIHYPERGROUPS Thawhat Changphas Department of Mathematics Faculty of Science Khon Kaen University Khon Kaen Thailand Bijan Davvaz Department of Mathematics Yazd University Yazd Iran Abstract. An ordered semihypergroup is a semihypergroup (S, ) together with a partial order on S such that the monotone condition holds, i.e., for all x, y, a S, if x y, then for all u x a there exists v y a such that u v, and similarly, for all u a x there exists v a y such that u v. Indeed, the concept of ordered semihypergroups is a generalization of the concept of ordered semigroups. In this paper, we study some aspects of hyperideals of ordered semihypergroups. We give a necessary and sufficient condition of a subset of Cartesian product of two ordered semihypergroups to be a prime hyperideal. Also, we study right simple element ordered semihypergroups containing right simple elements. Keywords: algebraic hyperstructure, ordered semigrouip, ordered semihypergroup, hyperideal, prime hyperideal, simple element. AMS Mathematics Subject Classification: 20N20, 06F Introduction and basic definitions The concept of algebraic hyperstructures was introduced in 1934 by Marty [11] and has been studied in the following decades and nowadays by many mathematicians. Let S be a nonempty set. A mapping : S S P (S), where P (S) denotes the family of all nonempty subsets of S, is called a hyperoperation on S. The couple (S, ) is called a hypergroupoid. In the above definition, if A and B are two nonempty subsets of S and x S, then we denote A B = a A b B a b, x A = {x} A and A x = A {x}.

2 426 t. changphas, b. davvaz A hypergroupoid (S, ) is called a semihypergroup if for every x, y, z S, x (y z) = (x y) z, that is x u = v z. u y z v x y A nonempty subset A of S is called a subsemihypergroup if x y A for all x, y in A. Semihypergroups are studied by many authors, for example, Bonansinga and Corsini [2], Davvaz [4], [5], De Salvo et al. [6], Freni [7], Hila et al. [9], Leoreanu [14], and many others. The concept of ordering hypergroups investigated by Chvalina [3] as a special class of hypergroups and studied by him and many others. In [8], Heidari and Davvaz studied a semihypergroup (S, ) besides a binary relation, where is a partial order relation such that satisfies the monotone condition. Indeed, an ordered semihypergroup (S,, ) is a semihypergroup (S, ) together with a partial order that is compatible with the hyperoperation, meaning that for any x, y, z in S, x y z x z y and x z y z. Here, z x z y means for any a z x there exists b z y such that a b. The case x z y z is defined similarly. Example 1 We have (S,, ) is an ordered semihypergroup where the hyperoperation and the order relation are defined by: a b c a a {a, b} {a, c} b a {a, b} {a, c} c a {a, b} c = {(a, a), (b, b), (c, c), (a, b)}. The covering relation and the figure of S are given by: = {(a, b)} b a c Example 2 We have (S,, ) is an ordered semihypergroup where the hyperoperation and the order relation are defined by: a b c d a a {a, b} {a, c} {a, d} b a {a, b} {a, c} {a, d} c a b c d d a b c d = {(a, a), (b, b), (c, c), (d, d), (a, b)}.

3 the homogeneous balance method and its applications The covering relation and the figure of S are given by: = {(a, b)} b a c d Note that the concept of ordered semihypergroups is a generalization of the concept of ordered semigroups [1], [10], [13]. Indeed, every ordered semigroup is an ordered semihypergroup. For a nonempty subset A of an ordered semihypergroup (S,, ), we write (A] = {x S x a for some a A}. The following is easy to see for nonempty subsets A, B of an ordered semihypergroup (S,, ): (1) A (A]; (2) A B (A] (B]; (3) (A] (B] (A B]; (4) ((A] (B]] = (A B]; (5) (A] (B] = (A B]. Let (S,, ) be an ordered semihypergroup. A subset A of S is called a hyperideal of S if it satisfies the following conditions: (1) x y A and y x A for all x in A, y in S; (2) for x A, y S, y x implies y A. Let (S,, S ) and (T,, T ) be two ordered semihypergroups. coordinatewise multiplication, i.e., Under the (s 1, t 1 ) (s 2, t 2 ) = s 1 s 2 t 1 t 2 where (s 1, t 1 ), (s 2, t 2 ) S T, the Cartesian product S T of S and T forms a semihypergroup. Define a partial order on S T by (s 1, t 1 ) (s 2, t 2 ) if and only if s 1 S s 2 and t 1 T t 2 where (s 1, t 1 ), (s 2, t 2 ) S T. Then, (S T,, ) is an ordered semihypergroup. 2. Prime ideals of the Cartesian product of two ordered semihypergroups A hyperideal P of an ordered semihypergroup (S,, ) is said to be prime if S \ P is a subsemihypergroup of S. Note that if a hyperideal P of S is prime, then P S. In this section we accept the empty set to be a prime hyperideal. Similar to the method of Petrich [15], we give a necessary and sufficient condition of a subset of Cartesian product of two ordered semihypergroups to be a prime hyperideal.

4 428 t. changphas, b. davvaz Theorem 2.1 Let (S,, S ) and (T,, T ) be ordered semihypergroups. Then, a subset L of S T is a prime hyperideal of S T if and only if there exist a prime hyperideal I of S and a prime hyperideal J of T such that L = (I T ) (S J). Proof. Assume that there exist a prime hyperideal I of S and a prime hyperideal J of T such that L = (I T ) (S J). If I = and J =, then L = ; hence L is a prime hyperideal of S T. Suppose that I or J. Then, L. We will show that L is a prime hyperideal of S T. Let (x, u) L and (y, v) S T. If x I, then x y I and y x I; hence and (x, u) (y, v) = x y u v I T (y, v) (x, u) = y x v u I T. Similarly, if u J, then (x, u) (y, v) S J and (y, v) (x, u) S J. Let (x, u) L and (y, v) S T be such that (y, v) (x, u), i.e., y S x, v T u. If x I, then y I; hence (y, v) I T. Thus, (y, v) L. Similarly, if u J, then (y, v) L. Therefore, L is a hyperideal of S T. Next, we assert that (S T ) \ L is a subsemihypergroup of S T. Since S \ I and T \ J, it follows that S \ I and T \ J are semihypergroups of S and of T, respectively. We have (S T ) \ L = (S \ I) (T \ J). Then, (S \ I) (T \ J) is a subsemihypergroup of S T. Hence, L is a prime hyperideal of S T. Conversely, assume that L is a prime hyperideal of S T. If L =, then L = ( T ) (S ). Assume that (x, u) L. We assert that {x} T L or S {u} L. Suppose that {x} T L and S {u} L. Then, there exist v T and y S such that (x, v) / L and such that (y, u) / L. We have and (x, v) (y, u) (x, v) (y, u) = x y x y v u v u (x y, v) (x, u) (y, v u) = x y x y v u v u. Since (x, v) (y, u) (x, v) (y, u) (S T ) \ L, we have (x y, v) (x, u) (y, v u) (S T ) \ L. But, since (x, u) L, we have (x y, v) (x, u) (y, v u) L. contradiction. Hence,, {x} T L or S {u} L. Let and let A = {x S {x} T L} and B = {u T S {u} L}, I = (A] and J = (B]. This is a

5 the homogeneous balance method and its applications Let (x, u) L. Then, {x} T L or S {u} L. Thus, x I or u J. Hence, (x, u) (I T ) (S J). Thus, L (I T ) (S J). The reverse inclusion is clear. Hence, L = (I T ) (S J). We will show that I is a prime hyperideal of S. That J is a prime hyperideal of T can be proved similarly. If I =, then I is a prime hyperideal of S. Assume that I. If I = S, then L = S T. This is a contradiction since L is a prime hyperideal of S T. Hence, S \ I. Similarly, T \ J. Let x, y S \ I and u T \ J. Then, (x, u), (y, u) (S \ I) (T \ J). Since L is prime, we have (S \ I) (T \ J) is a subsemihypergroup of S T. Since x y u u = (x, u) (y, u) (S \ I) (T \ J) we get x y S \ I. Thus, S \ I is a subsemihypergroup of S. Let x I, y S and u T \ J. Since x I, (x, u) L. Since L is a hyperideal of S T, we have and (x, u) (y, u) = x y u u L (y, u) (x, u) = y x u u L. Since T \ J is a subsemihypergroup, so u u T \ J. Since we obtain x y u u, y x u u L x y u u, y x u u I T and hence x y, y x I. It is clear that if x I and y S such that y x, then y I. Therefore, I is a prime hyperideal of S. Suppose that (S, ) and (T, ) are semihypergroups. Then, the Cartesian product S T is a semihypergroup under the coordinatewise multiplication. Define a partial order S on S by x S y if and only if x = y for all x, y S. Then, S forms an ordered semihypergroup. Similarly, T forms an ordered semihypergroup with a partial order T defined in a similar way. Using Theorem 2.1, we have the following result proved in [15]. Corollary 2.2 Let (S, ) and (T, ) be semihypergroups. Then, a subset L of S T is a prime hyperideal of S T if and only if L = (I T ) (S J) for some prime hyperideals I and J of S and of T, respectively.

6 430 t. changphas, b. davvaz 3. Right simple ordered semihypergroups Let (S,, ) be an ordered semihypergroup. An element a of S is said to be right simple if S = (a S]. If S contains a right simple element then it is called a right simple element ordered semihypergroup. If every element of S is right simple, then S is called a right simple ordered semihypergroup. Theorem 3.1 If (S,, S ) and (T,, T ) are two right simple element ordered semihypergroups, then S T is a right simple element ordered semihypergroup, too. Moreover, if A and B are the sets of all right simple elements of S and of T, respectively, then A B is the set of all right simple elements of S T. Proof. Assume that (S,, S ) and (T,, T ) are right simple element ordered semihypergroups with the sets of all right simple elements A and B, respectively. If (a, b) A B, then S T = (a S] (b T ]. If (s, t) S T, then s a s for some s in S, and t b t for some t in T. Since (s, t) a s b t, it follows that (s, t) a s b t = ((a, b) (S T )]. (s,t) S T Hence, (a, b) is a right simple element of S T. If (a, b) is a right simple element of S T, then S T = ((a, b) (S T )] = a s b t. (s,t) S T If (s, t) S T, then (s, t) (u, v) for some (u, v) a s b t where s S, t T. Since s u a s (a S], we have s (a S], and so S = (a S]. Similarly, T = (b T ]. Hence, (a, b) A B. It is well known the following result in semigroup theory. Let S be a right simple element semigroup and let R denote the set of all right simple elements of S. Then, the following conditions holds: (1) R is a subsemigroup of S; (2) If S \ R is nonempty, then it is the maximal right ideal of S and is prime, too ([12]). In the following, we extend the above result based on ordered semihypergroups. Theorem 3.2 Let (S,, ) be right simple element ordered semihypergroup with the set of all right simple elements R. The following statements hold: (1) R is a subsemihypergroup of S. (2) If S \ R is nonempty, then it is the maximal right hyperideal of S and is prime, too.

7 the homogeneous balance method and its applications Proof. If (S,, ) is right simple ordered semihypergroup, then it is clear that (1) and (2) hold. Then, we assume that (S,, ) is not a right simple ordered semihypergroup. (1) Let a, b R. Since S = (a S] and S = (b S], we have S = (a S] = (a (b S]] ((a] (b S]] = (a b S], and so a b R. (2) Assume that S \ R. Let x S and a S \ R. If a x R, then S = (a x S] (a S]; hence a R. This is a contradiction. Thus, a x S \R. Let x S \ R and y S be such that y x. If y R, then S = (y S] (x S]; hence x R. This is a contradiction. Thus, S \ R is a right hyperideal of S. Let A be a right hyperideal of S such that S \ R A. Then, there is an element a in A \ (S \ R). Since S = (a S] A, so A = S. By (1), it follows directly that S \ R is prime. Theorem 3.3 If an ordered semihypergroup (S,, ) has a unique maximal right hyperideal A such that S \ A (b] for all b in S \ A, then S \ A is the set of all right simple elements of S. Proof. Let R denote the set of all right simple elements of S. Let a R. If a A, then S = (a S] A. This is a contradiction. Thus, R S \ A. Let b S \ A. We have (b S] is a right hyperideal of S. If (b S] S, then by assumption we have (b S] A; hence (A {b}] is a right hyperideal of S. By A (A {b}], S = (A {b}]. Hence, S \ A = (b]. This is a contradiction. Hence, (b S] = S. Let (S,, ) be an ordered semihypergroup. An equivalence relation R is defined on S by arb if and only if (a a S] = (b b S] for any a, b in S. An element a of an ordered semihypersroup (S,, ) is said to be right regular if a (a 2 S]. Theorem 3.4 Let (S,, ) be a right simple element ordered semihypergroup with set of all right simple elements R. Then, (1) R is an R-class of S; (2) every element of R is right regular. Proof. (1) If a, b R, then S = (a S] and S = (b S]; hence (a a S] = (b b S]. This shows that arb. Let x S be such that xra for some a in R. Then, (x x S] = (a a S] = S. If S \ R =, then x R. If S \ R and x S \ R, then S = (x x S] S \ R; hence S = S \ R. This is a contradiction. Hence, x R.

8 432 t. changphas, b. davvaz (2) If a R, then a (a S] (a (a S]] (a 2 S]; hence a is right regular. Acknowledgment. The paper was essentially prepared during the second author s stay at the Department of Mathematics, Khon Kaen University in The second author is greatly indebted to Dr. T. Changphas for his hospitality. References [1] Birkhoff, G., Lattice theory, 25, Rhode Island, American Mathematical Society Colloquium Publications, Am. Math. Soc., Providence, [2] Bonansinga, P., Corsini, P., On semihypergroup and hypergroup homomorphisms, Boll. Un. Mat. Ital., B (6), 1 (2) (1982), [3] Chvalina, J., Commutative hypergroups in the sense of Marty and ordered sets, General algebra and ordered sets (HornãLipova, 1994), [4] Davvaz, B., Some results on congruences in semihypergroups, Bull. Malays. Math. Soc., (2), 23 (2000), [5] Davvaz, B., Characterizations of sub-semihypergroups by various triangular norms, Czechoslovak Mathematical Journal, 55 (4) (2005), [6] De Salvo, M., Freni, D., Lo Faro, G., Fully simple semihypergroups, J. Algebra, 399 (2014), [7] Freni, D., Minimal order semihypergroups of type U on the right, II, J. Algebra, 340 (2011), [8] Heidari, D., Davvaz, B., On ordered hyperstructures, Politehn. Univ. Bucharest Sci. Bull., Ser. A, Appl. Math. Phys., 73 (2) (2011), [9] Hila, K., Davvaz, B., Naka, K., On Quasi-hyperideals in Semihypergroups, Communications in Algebra, 39 (2011), [10] Khayopulu, N., On intra-regular ordered semigroups, Semigroup Forum, 46 (1993), [11] Marty, F., Sur une generalization de la notion de groupe, 8 iem congres Math. Scandinaves, Stockholm, (1934) [12] Masat, F.E., A generalization of right simple semigroups, Fund. Math., 101 (2) (1978), [13] Lee, D.M., Lee, S.K., On intra-regular ordered semigroups, Kangweon- Kyungki Math. Jour., 14 (2006), [14] Leoreanu, V., About the simplifiable cyclic semihypergroups, Ital. J. Pure Appl. Math., 7 (2000), [15] Petrich, M., Prime ideals of the cartesian product of two semigroups, Czechoslovak Mathematical Journal, 12 (1) (1962), Accepted: