The associated hyperringoid to a Krasner hyperring

ABSTRACT

“Ends Lemma” is used to construct a hypergroupoid from a (quasi) partially ordered groupoid. But this lemma does not work well for creating a hyperringoid from a (partially) ordered ringoid. In this paper, we plan to gain that by modifying this lemma, called modified “Ends Lemma”. Then we construct a EL²-hyperring, as a generalization of a “EL-hyperring”, by applying on a (partially) ordered Krasner hyperring.

KEYWORDS:

• (Partially) ordered Krasner hyperring
• EL²-hyperring
• modified Ends Lemma

1. Introduction

Hyperstructure theory as a natural generalization of algebraic structure theory was born by F. Marty at the 8th Congress of Scandinavian Mathematicians in 1934 [1]. He defined the concept of hypergroups based on the notion of hyperoperation. Since then, many mathematicians have widely studied a number of different hyperstructures. For instance, P. Corsini wrote one of the first books about hypergroups in 1993 [2], and a recent book on hyperstructures was written by B. Davvaz in 2012 [3].

The applications of hyperstructures to other areas have been extensively studied such as optimization theory, graph theory, physics, chemistry, theory of discrete event dynamical systems, generalized fuzzy computation, automata theory, formal language theory, coding theory and analysis of computer programs, for example, see [4–6].

In this paper, a relationship between (partially) ordered sets and algebraic hyperstructures would be studied. This topic was first studied by Vougiouklis in 1987 [7]. Since then, many researchers, such as Vougiouklis [8–10], Corsini [2, 11, 12], Hoskova [13] and Heidari and Davvaz [14], have analysed the connection between hyperstructures and (partially) ordered sets. One special aspect of this issue, known as EL-hyperstructures, was touched upon by Chvalina [15]. Also, Rosenberg in [16], Hoskova in [13], Rackova in [17] and Novak in [18–23] extended some results on the ordered semigroups and ordered groups connected with EL-hyperstructures. M. Novak mainly studied EL-hyperstructures that constructed from a (partially) quasi-ordered (semi) groups. He considered subhyperstructures of EL-hyperstructures in [21]. Also, he discussed some interesting results of important elements in this family of hyperstructures [19]. Then, in [20] Novak studied some basic properties of El-hyperstructures such as invertibility, normality, being closed (ultra closed) and so on.

In 2015, El-(semi)hypergroups constructed based on a (partially) quasi-ordered (semi)hypergroups were studied [24]. This paper helps us construct El-(semi)hyperrings based on a given partially ordered Krasner hyperrings. These hyperstructures are called EL²-hyperstructures.

1.1. Definitions and preliminaries

In the following, we present some basic definitions and ideas from the hyperstructure theory. The hyperstructures are algebraic structures equipped with at least one multi-valued operation, called a hyperoperation. A non-empty set H, endowed with a hyperoperation, is called a hypergroupoid. denotes the set of all non-empty subsets of H. In this definition, if A and B are two non-empty subsets of H and , then we define An element is called an identity of if , for all and it is called a scalar identity of if , for all . If e is a scalar identity of , then e is the unique identity of . A hypergroupoid which verifies the condition , for all , is called a semihypergroup. If the semihypergroup H satisfies , for all , it is called a hypergroup [3]. This condition is known as reproduction axiom.

Because of dealing with the theory of ordered structures, we recall that an ordered semihypergroup is a semihypergroup together with a partial order ≤ such that satisfies the monotone condition as follows:which is here means that for all , there exists and for all there exists such that . The case is defined similarly. Indeed, the concept of ordered semihypergroups is a generalization of the concept of ordered semigroups. To read the concept and properties of ordered semigroups, we refer the reader to [25].

There are several definitions for a hyperring, if we replace at least one of the two operations by a hyperoperation. In general case, is a good hyperring (hyperring), if + and . are two hyperoperations such that is a hypergroup, is a semihypergroup and the hyperoperation . is distributive (weak distributive) over the hyperoperation +, which means that for all x,y,z of R we have and ( and ). We call a (good) hyperfield if is a (good) hyperring and is a hypergroup. We say that is an additive hyperring, if the addition + is a hyperoperation and the multiplication . is a usual operation. A special case of this type is the Krasner hyperring. To read more details about hyperrings, see [26].

1.2. Modified Ends Lemma

In the following of this section, we present the concept of modified Ends Lemma and some of theorems related to it.

Definition 1.1: ([27])

Let R be a ring. We say that R is partially ordered when there exists a partial order ≤ on the underlying set R that it satisfies:

1. implies ,

2. and imply that , for any a,b,c in R.

If any two arbitrary elements a,b of R are comparable, then R is ordered.

Note: In any (partially) ordered ring with unit element , we have 0<1.

Example 1.2:

1. The ring with the ordinary addition and multiplication operation and the natural order relation is an ordered ring.

2. The ring of integral-valued functions on set I, with pointwise order, is partially ordered (when I has at least two elements).

Remark 1.1:

1. There is no (partially) ordered and finite non-trivial group , unless at least two distinct elements are in relation ≤ (i.e. ≤ is not trivial). Suppose to the contrary there exists such that . Now, if , then there exists such that and . So, according to the monotone condition, we have Now, due to the transitivity of ≤, we have x < 0. It is a contradiction.

2. There is no (partially) ordered ring with unit element , that is finite. Consider (partially) ordered ring . Due to Definition 1.1, for every n>0, we have Thus, the identity 1, as an element of the group , has infinite order. As a result, R with an identity 1 is an infinite ring.

In any (partially) ordered ring R, the absolute value of an element x can be defined as follows:

By applying original “Ends Lemma”, hypergroupoids are created from (quasi, partially) ordered groupoids. But about ringoids, we are faced with structures that are equipped with two addition and multiplication operations. After applying original “Ends Lemma” to (partially) ordered ringoids, the distribution multiplication hyperoperation by the ratio of addition hyperoperation on the right and left in creating hyperringoids will not be established. To accomplish this important, we will have the following well-defined hyperoperations:

Definition 1.3:

Let R be an ordered ring. For all , we define (1) (2)

Note: If is a (partially) ordered set and , then the subset of R is called principal end generated by and denoted by .

In the following, we present another statements of the original “Ends Lemma”.

Lemma 1.4:

Let R be an ordered ring. By the definitions which are presented in (1) and (2), is a commutative semihypergroup and is a semihypergroup.

Proof:

There are eight cases to show the associativity of the defined hyperoperation. But since proofs of all cases are similar, we only consider the case in which a,b,c<0. In this case, we show that . Consider . Then there are such that and . Due to associativity of operation +, we have . Now by attention to , . The proof of the other side is similar. At the end we have It is easy to see the commutativity of the hyperoperation ⊕. For the multiplication hyperoperation ⊙, we will have the same proof.

Theorem 1.5:

Let R be an ordered ring. Then is a good semihyperring.

Proof:

It is sufficient to prove the distribution multiplication hyperoperation by the ratio of addition hyperoperation on the right (or left). So, let and . Then On the other hand, The proofs of other cases are similar.

Notice that if R is not ordered, then there exists an element such that , so is meaningless. Therefore, the definitions of hyperoperations ⊕ and ⊙ in (1) and (2), respectively, are not efficient for or when at least one of a or b is not belonging to . So, we modify definitions of ⊕ and ⊙ in (1) and (2), respectively, in the following way.

Definition 1.6:

Let be a partially ordered ring. For , we define (3) (4)

With the hyperoperations and presented in (3(3) ) and (4(4) ), for any two elements a,b of partially ordered ring R, we have and .

Example 1.7:

The hyperstructure is defined as follows: is a good hyperring. For instance, since the unions of all rows and columns of the table ⊕ are equal to "> is a good hyperring. For instance, since the unions of all rows and columns of the table ⊕ are equal to R, so the reproduction principle is hold.

Theorem 1.8:

The following propositions are hold:

1. Let be a partially ordered group. Then is a hypergroup.

2. Let be a partially ordered group. Then is a hypergroup.

Proof:

1. It is sufficient to show the associative property of hyperoperation and the reproduction principle for the case in which at least one element is not belonging to . Let , and . Then we have On the other hand, It is easy to see the reproduction principle is hold.

2. The proof of this proposition is similar to the proof of Proposition (1).

Also, we can see that

Theorem 1.9:

The following propositions are hold.

1. Let be an ordered ring. Then is a good hyperring.

2. Let be a partially ordered ring. Then is a hyperring.

Proof:

• We only show that the distribution ⊙ by the ratio of of the left. Then the case in which and is considered. So we have On the other hand, The distribution ⊙ by the ratio of of the right is proved similarly. Therefore, is a good hyperring.

• We show the distribution multiplication hyperoperation by the ratio of addition hyperoperation on the right (or left), only in the case that and . On the other hand, The proofs of the other cases are done, similarly. Therefore, is a hyperring.

Example 1.10:

Let with ordinary addition of complex numbers. Put . Then is a partially ordered group.

Example 1.11:

Consider the partially ordered ring of integral-valued functions on a set , with pointwise order. Then is a good hyperring.

Definition 1.12:

[26] A canonical hypergroup is a non-empty set H endowed with an additive hyperoperation, satisfying the following properties:

1. for all , ,

2. for all , x+y=y+x,

3. there exists such that 0+x=x+0=x, for all ,

4. for every , there exists a unique element , such that (we shall write for and we call it the opposite of x),

5. implies that and , that is is reversible.

Definition 1.13:

[26] A Krasner hyperring is an algebraic hyperstructure which satisfies the following axioms:

1. is a canonical hypergroup,

2. is a semigroup having zero as a bilaterally absorbing element, i.e. x.0=0.x=0,

3. The multiplication . is distributive with respect to the hyperoperation +.

We say that a Krasner hyperring is commutative (with unit element) if is a commutative semigroup (with unit element). Also, we say that a Krasner hyperring R is a Krasner hyperfield, if is a group. A Krasner hyperring R is called a hyperdomain, if R is a commutative hyperring with unit element and a.b=0 implies that a=0 or b=0 for all .

Example 1.14:

Let be a semigroup with zero 0 such that is a group. Define the hyperoperation + on A by Then is a Krasner hyperring ([2]).

Example 1.15

Consider the unit interval and define the hyperoperation + on it byThen, is a Krasner hyperring where . is the usual multiplication ([28]).

We now give an example of a finite hyperfield with two elements 0 and 1, as follows.

Example 1.16:

Let be the finite set with two elements. Then becomes a Krasner hyperfield with the following hyperoperation + and binary operation [29].

In the following, we are trying to create new hyperstructures of (partially) ordered hyperstructures by aid of modified Ends Lemma.

Definition 1.17:

[30] Let be a (partially) ordered hypergroupoid. For , we define the new hyperoperation as follows:

Remark 1.2:

We name as the EL²-hypergroupoid associated to (partially) ordered hypergroupoid .

2. (Partially) ordered Krasner hyperrings

2.1. Ordered Krasner hyperring

In this section, we introduce the notion of ordered Krasner hyperring and present several examples that illustrate the significance of this hyperstructure. Then we create the new hyperrigoids from that by the modified “Ends Lemma”.

Definition 2.1:

An algebraic hyperstructure is called a (partially) ordered Krasner hyperring if is a Krasner hyperring with a (partial) order relation ≤, such that for all a, b and c in R:

1. If , then , meaning that for any , there exists and for any , there exists such that [30].

2. If and , then and .

Indeed, the concept of ordered Krasner hyperrings is a generalization of the concept of ordered rings.

Remark 2.1:

If a Krasner hyperring is arranged in a (partial) order, then the existence of a positive ordered Krasner hyperring is excluded. Indeed, for an element x in an (partially) ordered Krasner hyperring, if , then . On the other hand, by attention to , we have inevitably .

Example 2.2

Let be a set with the hyperoperation + defined as follows: Then, is a canonical hypergroup. Triple is a partially ordered canonical hypergroup where the order relation ≤ is defined by

Example 2.3

Let be a set with the hyperaddition + and the multiplication . defined as follows: Then, is a Krasner hyperring [31]. is a quasi-ordered Krasner hyperring where the quasi-order relation ≤ is defined by

Example 2.4:

In Example 2.2, if we define the multiplication on the canonical hypergroup as for any , , the resulting algebraic hyperstructure is a Krasner hyperring. Also, if the order relation is defined by then is an ordered Krasner hyperring.

Example 2.5:

Let be a set with the hyperaddition + and the multiplication . defined as follows: and Then, is a Krasner hyperring. is a quasi-ordered Krasner hyperring where the quasi-order relation ≤ is defined by

In the following, we try to modify Definition 1.17. for (partially) ordered hyperringoids. First of all, we present some necessary lemmas.

Lemma 2.6:

Let be a (partially) ordered set and . Then we have

2. the set is the maximum element of the family of subsets of R such that .

Proof:

1. We have . So, according to the reflexivity of the (partial) order relation ≤, for every , there exists such that . On the other hand, if is an arbitrary element, then there exists such that , or equivalently, there exists such that .

2. Suppose to the contrary, there exists such that and . Since , there exists such that . As a result for every , . On the other hand, from and , it follows that there exists such that . It is a contradiction.

Lemma 2.7:

Let be a (partially) ordered set and . Then we have .

Proof:

According to the maximality of among all the subsets X of R satisfying the property , it is sufficient that prove . Since , from , it follows that , and so by the reflexivity, for every , there exists such that . On the other hand, if , then there exists such that . Also, from , it follows that there exists such that . As a result, according to the transitivity of the (partial) order relation, there exists such that .

Due to the maximality of among all the subsets X of satisfying the property , the following corollary is easy to see.

Corollary 2.8

For non-empty subsets A and B of a (partially) ordered canonical hypergroup we have

1. and ;

2. ;

3. .

Lemma 2.9:

Let be a (partially) ordered Krasner hyperring, and . Then we have . If R is a(partially) ordered Krasner hyperfield, then .

Proof:

If , then there exists such that y=ax. Since , so there exists such that . Therefore and . Thus . Now, if R is a (partially) ordered Krasner hyperfield and , then there exists such that . Since a and therefore are positive, we have . Also, we have . Now, by putting , we have , and as a result .

2.2. EL²- hyperringoids

In the following, we are going to construct new hyperringoids from (partially) ordered Krasner hyperrings, through what we have achieved so far.

Definition 2.10:

Let is an ordered Krasner hyperring. For , we define the new hyperoperation as follows:

Remark 2.2

1. It is easy to see that Definition 2.10 is not useful for satisfying the reproduction principle. Indeed, there exists no positive-ordered Krasner hyperring. On the other hand, this definition is meaningless for partially ordered Krasner hyperrings which are not ordered. So we improve Definition 2.10 as follows:

2. If . is the multiplicative operation in a (partially) ordered Krasner hyperring, then due to Definition 1.1, we define the new multiplicative operation as follows:

Theorem 2.11:

Let is an ordered canonical hypergroup. Then the hyperoperation ⊕ is associative.

Proof:

Suppose that and . Then Other states are proved in the same way.

Theorem 2.12:

Let is a partially ordered canonical hypergroup. Then -hypergroupoid is a hypergroup.

Proof:

Suppose that and . Then Now, if , and , then we have on the other hand, Other states are proved in the same way. Due to the definition , it is easy to see that reproduction principles are established.

Theorem 2.13:

Let is an ordered Krasner hyperring. Then the algebraic structure is a semigroup.

Proof

Suppose that and . Then

Theorem 2.14:

Let is an ordered Krasner hyperring. Then the hyperstructure is an additive hyperring.

Proof:

Due to the Theorems 2.7, 2.8 and 2.9, it is sufficient that we show the distribution by the ratio of on the left. So, suppose that and : on the other hand, The distribution by the ratio of on the right is proved, similarly.

Theorem 2.15:

Let is an ordered Krasner hyperfield. Then the hyperstructure is a good additive hyperring.

Proof:

Due to Lemma 2.5, it is easy to prove.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

H. Mirabdollahi http://orcid.org/0000-0002-7098-424X

S. Mirvakili http://orcid.org/0000-0002-9474-0709