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.


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][5][6].

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, If e is a scalar identity of (H, •), then e is the unique identity of (H, •). A hypergroupoid which verifies the condition . This condition is known as reproduction axiom.
Because of dealing with the theory of ordered structures, we recall that an ordered semihypergroup (H, •, ≤) is a semihypergroup (H, •) together with a partial order ≤ such that satisfies the monotone condition as follows: The case x • z ≤ y • z 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, (R, +, .) is a good hyperring (hyperring), if + and . are two hyperoperations such that (R, +) is a hypergroup, (R, .) 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 x.(y + z) = x.y + x.z and (x + y).z = x.z + y.z (x.(y + z) # x.y + x.z and (x + y).z # x.z + y.z). We call (R, +, .) a (good) hyperfield if (R, +, .) is a (good) hyperring and (R, .) is a hypergroup. We say that (R, +, .) 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].

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

Example 1.2:
i) The ring (Z, +, ., ≤) with the ordinary addition and multiplication operation and the natural order relation is an ordered ring.
ii) The ring Z I of integral-valued functions on set I, with pointwise order, is partially ordered (when I has at least two elements).
Now, due to the transitivity of ≤, we have x < 0. It is a contradiction. (2) There is no (partially) ordered ring with unit element 1 = 0, that is finite. Consider (partially) ordered ring (R, +, ., ≤). Due to Definition 1.1, for every n>0, we have Thus, the identity 1, as an element of the group (R, +), has infinite order. As a result, R with an identity 1 is an infinite ring.
In any (partially) ordered ring R, the absolute value | x | 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: Note: If (R, ≤) is a (partially) ordered set and a [ R, then the subset {x [ R| |a| ≤ x} of R is called principal end generated by |a| [ R and denoted by [|a|) ≤ .
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), (R, ⊕) is a commutative semihypergroup and (R, ⊙) 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 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 (R, ⊕, ⊙) 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 a, b [ R − and c [ R + . Then On the other hand, The proofs of other cases are similar.
Notice that if R is not ordered, then there exists an element a 0 [ R such that a 0 Ó R + < R − , so |a 0 | is meaningless. Therefore, the definitions of hyperoperations ⊕ and ⊙ in (1) and (2), respectively, are not efficient for a ⊕ b or a ⊙ b when at least one of a or b is not belonging to R + < R − . So, we modify definitions of ⊕ and ⊙ in (1) and (2), respectively, in the following way.
Definition 1.6: Let (R, +, ., ≤) be a partially ordered ring. For a, b [ R, we define With the hyperoperations ⊕ 1 and ⊙ 1 presented in (3) and (4) (1) Let (R, +, ≤) be a partially ordered group. Then (R, ⊕ 1 ) is a hypergroup. (2) Let (R, ., ≤) be a partially ordered group. Then (R, ⊙ 1 ) is a hypergroup. Proof: (1) It is sufficient to show the associative property of hyperoperation ⊕ 1 and the reproduction principle for the case in which at least one element is not belonging to R + < R − . Let a Ó R + < R − , and b, c [ R + . 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.
(i) Let (R, +, ., ≤) be an ordered ring. Then (R, ⊕ 1 , ⊙) is a good hyperring. (ii) Let (R, +, ., ≤) be a partially ordered ring. Then (R, ⊕ 1 , ⊙ 1 ) is a hyperring. Proof: . We only show that the distribution ⊙ by the ratio of  The distribution ⊙ by the ratio of ⊕ 1 of the right is proved similarly. Therefore, (R, ⊕ 1 , ⊙) is a good hyperring. . We show the distribution multiplication hyperoperation ⊙ 1 by the ratio of addition hyperoperation ⊕ 1 on the right (or left), only in the case that a, c On the other hand, We now give an example of a finite hyperfield with two elements 0 and 1, as follows.
Example 1.16: Let F 2 = {0, 1} be the finite set with two elements. Then F 2 becomes a Krasner hyperfield with the following hyperoperation + and binary operation [29].
. 0 1 0 0 0 1 0 1 In the following, we are trying to create new hyperstructures of (partially) ordered hyperstructures by aid of modified Ends Lemma.

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". (1) If a ≤ b, then a + c ≤ b + c, meaning that for any x [ a + c, there exists y [ b + c and for any a ≤ b and 0 ≤ c, then a.c ≤ b.c and c.a ≤ c.b. Indeed, the concept of ordered Krasner hyperrings is a generalization of the concept of ordered rings. Proof: According to the maximality of [A) ≤ among all the subsets X of R satisfying the property A ≤ X, it is suf- Due to the maximality of [A) ≤ among all the subsets X of (R, ≤) satisfying the property A ≤ X, the following corollary is easy to see.

EL 2 -hyperringoids
In the following, we are going to construct new hyperringoids from (partially) ordered Krasner hyperrings, through what we have achieved so far.  (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: a. ′ b = |a|.|b|.
Proof: Suppose that 0 ≤ a, c and b ≤ 0. Then Other states are proved in the same way. Other states are proved in the same way. Due to the definition ⊕ 1 , it is easy to see that reproduction principles are established. The distribution . ′ by the ratio of ⊕ 1 on the right is proved, similarly.
Proof: Due to Lemma 2.5, it is easy to prove.

Disclosure statement
No potential conflict of interest was reported by the authors.