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International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 876919, 7 pages
On Pure Hyperradical in Semihypergroups
Department of Mathematics ####^~^~^~^~^~^####x26; Computer Science, Faculty of Natural Sciences, University of Gjirokastra, Gjirokastra 6001, Albania
Received 24 March 2012; Accepted 23 May 2012
Academic Editor: Adolfo Ballester-Bolinches
Copyright © 2012 Kostaq Hila and Krisanthi Naka. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper deals with a class of algebraic hyperstructures called semihypergroups, which are a generalization of semigroups. In this paper, we introduce pure hyperradical of a hyperideal in a semihypergroup with zero element. For this purpose, we define pure, semipure, and other related types of hyperideals and establish some of their basic properties in semihypergroups.
1. Introduction and Preliminaries
The applications of mathematics in other disciplines, for example, in informatics, play a key role and they represent, in the last decades, one of the purpose, of the study of the experts of hyperstructures theory all over the world. Hyperstructure theory was introduced in 1934 by the French mathematician Marty , at the 8th Congress of Scandinavian Mathematicians, where he defined hypergroups based on the notion of hyperoperation, began to analyze their properties, and applied them to groups. In the following decades and nowadays, a number of different hyperstructures are widely studied from the theoretical point of view and for their applications to many subjects of pure and applied mathematics by many mathematicians. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. Several books have been written on hyperstructure theory, see [2####^~^~^~^~^~^####x2013;5]. A recent book on hyperstructures  points out on their applications in rough set theory, cryptography, codes, automata, probability, geometry, lattices, binary relations, graphs and hypergraphs. Another book  is devoted especially to the study of hyperring theory. Several kinds of hyperrings are introduced and analyzed. The volume ends with an outline of applications in chemistry and physics, analyzing several special kinds of hyperstructures: -hyperstructures and transposition hypergroups. Some principal notions about semihypergroups theory can be found in [6####^~^~^~^~^~^####x2013;18]. Recently, Davvaz et al. [19####^~^~^~^~^~^####x2013;23] introduced the notion of -semihypergroup as a generalization of a semigroup, a generalization of a semihypergroup, and a generalization of a -semigroup. They presented many interesting examples and obtained a several characterizations of -semihypergroups.
In this paper, we introduce pure hyperradical of a hyperideal in a semihypergroup with zero element. For this purpose, we define pure, semipure, and other related types of hyperideals, and we establish some of their basic properties in semihypergroups.
Recall first the basic terms and definitions from the hyperstructure theory.
A map is called hyperoperation or join operation on the set , where is a nonempty set and denotes the set of all nonempty subsets of .
A hyperstructure is called the pair where is a hyperoperation on the set .
A hyperstructure is called a semihypergroup if for all , , which means that If and are nonempty subsets of , then
A nonempty subset of a semihypergroup is called a subsemihypergroup of if and is called in this case supersemihypergroup of .
Let be a semihypergroup. Then, is called a hypergroup if it satisfies the reproduction axiom, for all , .
A nonempty subset of a semihypergroup is called a right (left) hyperideal of if for all and , A nonempty subset of is called a hyperideal (or two-sided hyperideal) if it is both a left hyperideal and right hyperideal.
An element in a semihypergroup is called scalar identity if An element in a semihypergroup is called zero element if
We call a semihypergroup a regular semihypergroup if for every , for some . Hence, every regular semigroup is a regular semihypergroup. Notice that if is a hypergroup, then for every . This implies that every hypergroup is a regular semihypergroup.
2. Main Results
In this section, we introduce pure hyperradical of a hyperideal in a semihypergroup with zero element. For this purpose, we define pure, semipure, and other related types of hyperideals, and we establish some of their basic properties. In what follows, will denote a semihypergroup with scalar identity 1, which contains a zero element.
Definition 2.1. Let be a semihypergroup. A right hyperideal of is called a right pure right hyperideal if for each , there is an element such that . If is a two-sided hyperideal with the property that for each , there is an element such that , then is called a right pure hyperideal.
Left pure left hyperideals and left pure hyperideals are defined analogously.
Definition 2.2. Let be a semihypergroup. A right hyperideal of is called a right semipure right hyperideal if for each , there is an element belonging to some proper right hyperideal of such that . If is a two-sided hyperideal with the property that for each , there is an element belonging to a proper hyperideal of such that , then is called a right semipure hyperideal.
Left semipure left hyperideals and left semipure hyperideals can be similarly defined.
Example 2.3. Let be a semihypergroup on with the hyperoperation given by the following: It is easy to see that , are right pure right hyperideals of .
Example 2.4. Let be a semihypergroup on with the hyperoperation given by the following: Clearly, , , and are right pure right hyperideals of . is a two-sided hyperideal of which is a right pure hyperideal but not a left pure hyperideal. Also, is a right and left semipure hyperideals. is a two-sided hyperideal of which is a right and left pure hyperideal.
Example 2.5. Let . Then, with the hyperoperation is a semihypergroup. Let and . Then, is a semihypergroup, and, moreover, is a two-sided hyperideal of which is neither right pure nor left pure, but it is left and right semipure.
Proposition 2.6. Let be a two-sided hyperideal of . Then is right pure if and only for any right hyperideal , .
Proof. Suppose is a right pure hyperideal of . Since is a right hyperideal of , . Also, since is a left hyperideal, . Hence, . Let . Since is a right pure hyperideal, there exists such that . As and , . Hence, . This implies that . Conversely, assume , for any right hyperideal of . We show that is right pure. Let . Then, . Since . Hence, there exists such that . This proves that is a right pure hyperideal.
Corollary 2.7. If is a right pure hyperideal, then .
Proposition 2.8. and are right pure hyperideals of . Any union and finite intersection of right pure (resp., semipure) hyperideals is right pure (resp., semipure).
Proof. (0) and are obviously right pure hyperideals. Let and be right pure hyperideals and let . Since and is right pure, exists such that . Similarly, exists such that . Thus, we have . Since , it follows that is right pure. The remaining cases of this proposition can be similarly proved.
It follows from the above proposition that if is any hyperideal of , then contains a largest pure hyperideal, which is in fact the union of all pure hyperideals contained in (such hyperideals exist, e.g., (0)) and hence a pure hyperideal. The largest pure hyperideal contained in is denoted by . Similarly, each hyperideal contains a largest semipure hyperideal, denoted by . (resp., ) is called the pure (resp., semipure) part of .
Definition 2.9. Let be a right pure (resp., semipure) hyperideal of . Then, is called purely (resp., semipurely) maximal if is a maximal element in the set of proper right pure (resp., semipure) hyperideals.
In Example 2.4, , and are right pure hyperideals of the semihypergroup , and it is clear that is purely maximal hyperideal.
Definition 2.10. Let be a right pure (resp., semipure) hyperideal of . Then, is called purely (resp. semipurely) prime if it is proper and if for any right pure (resp., semipure) hyperideals and , or .
In Example 2.4, the hyperideal mentioned above is purely prime hyperideal.
The following propositions are stated for pure and semipure hyperideals simultaneously. However, the proofs are given only for one case, since the proofs are similar for the remaining cases.
Proposition 2.11. Any purely (resp., semipurely) maximal hyperideal is purely (resp., semipurely) prime.
Proof. Suppose is purely maximal, and are right pure hyperideals such that . Suppose . Then, . Now, .
Proposition 2.12. The pure (resp. semipure) part of any maximal hyperideal is purely (resp. semipurely) prime.
Proof. Let be a maximal hyperideal of . We show that , the pure part of , is purely prime. Suppose with pure. If , then and we are done. Suppose , then . Hence, . Hence, . This implies that , since is pure.
Proposition 2.13. If is right pure (resp., semipure) hyperideal of and , then there exists a purely (resp., semipurely) prime hyperideal such that and .
Proof. Consider the set ordered by inclusion: is a right semipure hyperideal, . Then, , since . Let be a totally ordered subset of . Clearly, is a semipure hyperideal with . Hence, is inductively ordered. Therefore, by Zorn's Lemma, has a maximal element . We will show that is semipurely prime. Suppose are right semipure hyperideals such that and . Since and are semipure, is a semipure hyperideal such that . We then claim that an . Because, if , then by the maximality of , we have . But this contradicts the assumption . Hence, . Since , it follows that . Hence, by contrapositivity, we conclude that is semipurely prime.
Proposition 2.14. Let be a proper right pure (resp. semipure) hyperideal of . Then, is contained in a purely (resp., semipurely) maximal hyperideal.
Proof. Consider the set, ordered by inclusion, is a proper right semipure hyperideal and . Clearly, , since . Moreover, any is a proper hyperideal because . Hence, any directed union of elements in is still in . So, is inductively ordered. Hence, by Zorn's Lemma, contains a maximal element and any proper semipure hyperideal containing also contains and so it belongs to . But such an hyperideal will be itselft, since is maximal in . Therefore, is semipurely maximal.
Proposition 2.15. Let be a right pure (resp., semipure) hyperideal of . Then, is the intersection of purely (resp., semipurely) prime hyperideals of containing .
Proof. By Propositions 2.14 and 2.11, there exists a set is a semipurely prime hyperideal containing . Hence . To prove , assume there exists an element such that . Then by the above proposition, there exists a semipurely prime hyperideal such that , but . This implies that . This proves the proposition.
Definition 2.16. Let be a right hyperideal of and let be the set of right pure hyperideals containing . Then, we define and call it the pure hyperadical of . Note that the set is a right pure hyperideal containing is nonempty, since itself belongs to this set.
Proposition 2.17. If is the pure hyperradical of the hyperideal , then each of the following statement holds true: (1) is either pure or semipure hyperideal containing . (2) is contained in every right pure hyperideal which contains . (3)If are those purely prime hyperideals which contain , then .
Proof. (1) If the set is a right pure hyperideal of containing consists of alone, then . Hence, is pure in this case. In case the set is a right pure hyperideal containing has only a finite number of elements, then is pure by Proposition 2.8. In general, is semiprime.
(2) This is obvious.
(3) Since every pure hyperideal is contained in a purely maximal hyperideal (Proposition 2.14) and every purely maximal hyperideal is purely prime (Proposition 2.11), the set is purely prime containing is nonempty. Hence, from Part (2), it follows that . We prove that . To prove this, assume that . Since , where each is a right pure hyperideal containing . Hence, for some . Thus is a proper pure hyperideal which contains but misses . Hence, by Proposition 2.13, there exists a purely prime hyperideal , such that and . Hence, , where 's are purely prime hyperideals containing . From this we conclude that .
Definition 2.18. Let be a hyperideal of . Then the semipure part of , that is, the union of all semipure hyperideals contained in is called the semipure hyperradical of .
Proposition 2.19. For each hyperideal , is the intersection of semipurely prime hyperideals.
Proof. It follows from Proposition 2.15.
Note that and are distinct in general. For example, if of Example 2.4, then
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