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In this paper, we introduce the concept of fuzzy ordered hyperideals of ordered semihyperrings, which is a generalization of the concept of fuzzy hyperideals of semihyperrings to ordered semihyperring theory, and we investigate its related properties. We show that every fuzzy ordered quasi-hyperideal is a fuzzy ordered bi-hyperideal, and, in a regular ordered semihyperring, fuzzy ordered quasi-hyperideal and fuzzy ordered bi-hyperideal coincide.

The theory of algebraic hyperstructures is a well-established branch of classical algebraic theory which was initiated by Marty [

The notion of semiring was introduced by Vandiver [

After the introduction of fuzzy sets by Zadeh [

Inspired by the study on ordered semihyperrings, we study the concept of fuzzy ordered hyperideals, fuzzy ordered quasi-hyperideals, and fuzzy ordered bi-hyperideals of an ordered semihyperring and we present some examples in this respect. The rest of this paper is organized as follows. In the second section, we review basic concepts regarding the ordered hyperstructures. The third section is dedicated to the fuzzy ordered hyperideals and some properties. In Section

In what follows, we summarize some basic notions and facts about semihypergroups, semihyperrings, and ordered semihyperrings.

Let

Let

A semihyperring

A nonempty subset

A nonempty subset

An

Note that the concept of ordered semihyperring is a generalization of the concept of ordered semiring.

Semihyperrings are viewed as ordered semihyperrings under the equality order relation [

Let

Let

If

It is clear that

Let

Let

If

Let

If

Consider the semihyperring defined in Example

The concept of a fuzzy subset of a nonempty set first was introduced by Zadeh in 1965 [

Notice that the relationships between fuzzy sets and algebraic hyperstructures have been already considered by many researchers [

Let

Let

Any hyperideal of an ordered semihyperring

Proof is similar to Lemma 4.2 in [

Notice that the characteristic function of a nonempty subset

A fuzzy subset

Let

Conversely, let

Finally, let

Let

Each level subset

Each strong level subset

Each level subset

Each strong level subset

Each nonempty level subset of

Each nonempty strong level subset of

Let

Let

If

If

It is straightforward.

In this section, we define the concepts of fuzzy ordered bi-hyperideal and fuzzy ordered quasi-hyperideal in ordered semihyperrings and give relationships between them.

Let

Let

A fuzzy subset

The proof is similar to the proof of Theorem

Let

The following theorem can be proved in a similar way in the proof of Theorem 4.8 of [

A fuzzy subset

Let

It is straightforward.

Let

Every fuzzy ordered hyperideal of

Every fuzzy ordered quasi-hyperideal of

(i) Only we show that the condition (ii) of Definition

Let

(ii) Assume that

The following example shows that the converse of Theorem

Consider the ordered semihyperring

An ordered semihyperring

Let

“

“

(1) If

(2) If

Now, we have to prove the main characterization theorem for regular semihyperrings. We denote by

Let

Let

In the structural theory of fuzzy algebraic systems, fuzzy ideals with special properties always play an important role. In this paper, we study fuzzy ordered hyperideals, fuzzy ordered quasi-hyperideals, and fuzzy ordered bi-hyperideals of an ordered semihyperring. We characterize regular ordered semihyperrings by the properties of these fuzzy hyperideals. As a further work, we will also concentrate on characterizations of different classes of ordered semihyperrings in terms of fuzzy interior hyperideals.

All data generated or analysed during this study are included in this article.

The authors declare that they have no conflicts of interest.