In general, ordered algebraic structures, particularly ordered semigroups, play an important role in fuzzification in many applied areas, such as computer science, formal languages, coding theory, error correction, etc. Nowadays, the concept of ambiguity is important in dealing with a variety of issues related to engineering modeling problems, network theory, decision-making problems in real-life situations, and so on. Several theories have been developed by various researchers to overcome the difficulties that arise from uncertainty, including fuzzy sets, intuitionistic fuzzy sets, probability, soft sets, neutrosophic sets, and many more. In this paper, we focus solely on neutrosophic set theory. In ordered semigroups, we define and investigate the properties of neutrosophic

In [

Smarandache proposed the notions of neutrosophic sets to handle uncertainty that arises everywhere. It is the generalization of fuzzy sets and intuitionistic fuzzy sets. Using these three attributes such as a truth (

In [

In [

The aim of this paper is to define the concepts of neutrosophic

An non-empty set

For _{1}_{2}

For any _{1}_{2}: for all _{1}∈ _{2}∈

For _{1}∈

For all _{2}≤ _{1} imply _{2}∈

Let

Let

The set

We refer to [

A neutrosophic

It is evident that all the neutrosophic

.
_{1}_{2}_{3}_{4}_{5}
_{1}_{1}_{4}_{1}_{4}_{4}
_{2}_{1}_{2}_{1}_{4}_{4}
_{3}_{1}_{4}_{3}_{4}_{5}
_{4}_{1}_{4}_{1}_{4}_{4}
_{5}_{1}_{4}_{3}_{4}_{5}

Let

Definition 2.10. Let

It is evident that neutrosophic

Let

If

In this section, we study some properties of neutrosophic

_{1}≤ _{2}. Then

If _{2}∈ _{2}) = −1.

If _{2}

(ii)⇒(i) Assume _{2}≤ _{1}. Then _{2}∈

_{1}≤ _{2}. Then

If _{2}∈

If _{2}

(ii)⇒(i) Assume _{2}≤ _{1}. Then _{2}∈

Proof. The proof is a routine procedure.

_{1}≤ _{1}_{1}. Now,

Therefore

left (resp., right) simple if it does not contain any proper left (resp., right) ideal of

simple if it does not contain any proper ideal of

_{1}∈ _{k}_{1}_{2}∈ _{k}_{1}∈ _{k}_{1}_{2}∈ _{k}_{2}≤ _{1}. Then _{1}∈ _{k}_{2}∈ _{k}_{k}

Conversely, let _{k1} is an ideal. As _{2}∈ _{k1}, we have

_{1}≤ _{2}

Conversely, suppose

_{1}≤ _{2}. Then

_{1}≤ _{2}. Then _{α}∈ [ −1, 0). So

Let _{1}≤ _{2}. Then _{β}∈ [ −1, 0). So

Let _{1}≤ _{2}. Then _{γ} ∈ [ −1, 0). So

Hence by Theorem 3.12,

Let

_{T}, _{I}, _{F}∈ [ −1, 0] such that −3≤ _{T} +_{I} +_{F}≤ 0.

_{T}, _{I}, _{F}∈ [ −1, 0] such that −3≤ _{T} +_{I} +_{F}≤ 0.

where _{T}, _{I}, _{F}∈ [ −1, 0] such that −3≤ _{T} +_{I} +_{F}≤ 0.

_{T}, _{I}, _{F}, _{T}, _{I}, _{F} ∈ [ −1, 0] with −3≤ _{T} +_{I} +_{F}≤ 0 and −3≤ _{T} +_{I} +_{F}≤ 0, then _{T}, _{I}, _{F}) = (_{T}∨ _{T}, _{I}∧ _{I}, _{F}∨ _{F}).

_{T}, _{I}, _{F}, _{T}, _{I}, _{F} ∈ [ −1, 0] with −3≤ _{T} +_{I} +_{F}≤ 0 and −3≤ _{T} +_{I} +_{F}≤ 0, then _{T}, _{I}, _{F}) = (_{T}∨ _{T}, _{I}∧ _{I}, _{F}∨ _{F}).

For any _{1}≤ _{2}, we have

Therefore

_{1}, _{2}∈ Ω, then

Now for any

So _{1}_{2}_{3}∈ Ω.

Let _{1}≤ _{2} and _{2}∈ Ω. Then _{1}∈ Ω and hence Ω is an interior ideal of

Following [_{1}) ≼ _{2}). _{1} ≤ _{2} [each inverse isotone mapping is (1–1)]. _{1}._{2}) = _{1}) * _{2}) for all

For a map _{T}, _{I}, _{F}) with −3≤ _{T} +_{I} +_{F}≤ 0, define a neutrosophic

Let _{1}≤ _{2}. Then

Therefore

Let _{1}≤ _{2}. Then

Hence by Theorem 3.17,

Let

Let _{1}≤ _{2}. Then

Therefore

Let _{1}≤ _{2}. Then

Therefore, by Theorem 3.20,

Let

If

If

^{−1}(_{1}) ≠ ^{−1}(_{2}) ≠ ^{−1}(_{1}) and ^{−1}(_{2}) such that

Now,

Let _{1}, _{2}∈ _{1}≤ _{2}. Then

Therefore

If

^{−1}(_{1}) ≠ ^{−1}(_{2}) ≠ ^{−1}(_{3}) ≠ ^{−1}(_{1}), ^{−1}(_{2}) and ^{−1}(_{3}) such that

Now,

Let _{1}≤ _{2}. Then

Hence, by Theorem 3.23,

In ordered semigroups, the concepts of neutrosophic

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.