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Neutrosophic N-interior ideals in semigroups

Neutrosophic ℵ −interior ideals in semigroups

K. Porselvi, B. Elavarasan, F. Smarandache

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K. Porselvi, Balasubramanian Elavarasan, F. Smarandache (2020). Neutrosophic N-interior ideals in semigroups. Neutrosophic Sets and Systems 36, 70-80

Abstract: We define the concepts of neutrosophic ℵ -interior ideal and neutrosophic ℵ−characteristic interior ideal structures of a semigroup. We infer different types of semigroups using neutrosophic ℵ-interior ideal structures. We also show that the intersection of neutrosophic ℵ-interior ideals and the union of neutrosophic ℵ-interior ideals is also a neutrosophic ℵ-interior ideal.

Keywords: Semi group, neutrosophic ℵ − ideals, neutrosophic ℵ -interior ideals, neutrosophic ℵ−product.

1. Introduction

Nowadays, the theory of uncertainty plays a vital role to manage different issues relating to modelling engineering problems, networking, real-life problem relating to decision making and so on. In 1965, Zadeh[24] introduced the idea of fuzzy sets for modelling vague concepts in the globe. In 1986, Atanassov [1] generalized fuzzy set and named as Intuitionistic fuzzy set. Also, from his viewpoint, there are two degrees of freedom in the real world, one a degree of membership to a vague subset and the other is a degree of non-membership to that given subset.

Smarandache generalized fuzzy set and intuitionistic fuzzy set, and named as neutrosophic set (see [4, 7, 8, 14, 19, 22-23]). These sets are characterized by a truth membership function, an indeterminacy membership function and a falsity membership function. These sets are applied to many branches of mathematics to overcome the complexities arising from uncertain data. A Neutrosophic set can distinguish between absolute membership and relative membership. Smarandache used this in non-standard analysis such as the result of sports games (winning/defeating/tie), decision making and control theory, etc. This area has been studied by several authors (see [3, 11, 12, 16-18]).

For more details on neutrosophic set theory, the readers visit the website http://fs.gallup.unm.edu/FlorentinSmarandache.htm

In [2], Abdel Basset et al. designed a framework to manage scheduling problems using neutrosophic theory. As the concept of time-cost tradeoffs and deterministic project scheduling disagree with the real situation, some data were changed during the implementation process. Here fuzzy scheduling and time-cost tradeoffs models assumed only truth-membership functions dealing

with uncertainties of the project and their activities duration which were unable to treat indeterminacy and inconsistency.

In [6], Abdel Basset et al. evaluated the performance of smart disaster response systems under uncertainty. In [5], Abdel Basset et al. introduced different hybrid neutrosophic multi-criteria decision-making framework for professional selection that employed a collection of neutrosophic analytical network process and order preference by similarity to the ideal solution under bipolar neutrosophic numbers.

In [21], Prakasam Muralikrishna1 et al. presented the characterization of MBJ – Neutrosophic �� – Ideal of �� – algebra. They analyzed homomorphic image, pre–image, cartesian product and related results, and these concepts were explored to other substructures of a �� – algebra. In [9], Chalapathi et al. constructed certain Neutrosophic Boolean rings, introduced Neutrosophic complement elements and mainly obtained some properties satisfied by the Neutrosophic complement elements of Neutrosophic Boolean rings.

In [14], M. Khan et al. presented the notion of neutrosophic ℵ-subsemigroup in semigroup and explored several properties. In [11], Gulistan et al. have studied the idea of complex neutrosophic subsemigroups and introduced the concept of the characteristic function of complex neutrosophic sets, direct product of complex neutrosophic sets.

In [10], B. Elavarasan et al. introduced the notion of neutrosophic ℵ-ideal in semigroup and explored its properties. Also, the conditions for neutrosophic ℵ-structure to be neutrosophic ℵ-ideal are given, and discussed the idea of characteristic neutrosophic ℵ-structure in semigroups and obtained several properties. In [20], we have introduced and discussed several properties of neutrosophic ℵ-bi-ideal in the semigroup. We have proved that neutrosophic ℵ-product and the intersection of neutrosophic ℵ-ideals were identical for regular semigroups. In this paper, we define and discuss the concepts of neutrosophic ℵ-interior ideal and neutrosophic ℵ-characteristic interior ideal structures of a semigroup.

Throughout this paper, �� denotes a semigroup. Now, we present the important definitions of semigroup that we need in sequel.

Recall that for any ��1, ��2 ⊆ ��, ��1��2 = {����|�� ∈ ��1 ������ �� ∈ ��2} , multiplication of ��1 and ��2.

Let �� be a semigroup and ∅ ≠ ��1 ⊆ ��. Then (i) ��1 is known as subsemigroup if ��1 2 ⊆ ��1. (ii) A subsemigroup ��1 is known as left (resp., right) ideal if ��1�� ⊆ ��1(resp., ����1 ⊆ ��1). (iii) ��1 is known as ideal if ��1 is both a left and a right ideal. (iv) �� is known as left (resp., right) regular if for each �� ∈��, there exists �� ∈�� such that �� = ����2(resp., �� =��2��) [13]. (v) �� is known as regular if for each ��1 ∈��, there exists �� ∈�� such that ��1 =��1����1 (vi) �� is known as intra-regular if for each ��1 ∈��, there exist ��,�� ∈�� such that ��1 = ����1 2�� [15].

2. Definitions of neutrosophic ℵ - structures

We present definitions of neutrosophic ℵ−structures namely neutrosophic ℵ−subsemigroup, neutrosophic ℵ−ideal, neutrosophic ℵ−interior ideal of a semigroup ��

The set of all the functions from �� to [−1,0] is denoted by ℑ(��,[−1,0]). We call that an element of ℑ(��,[−1,0]) is ℵ−function on ��. A ℵ−structure means an ordered pair (��,��) of �� and an ℵ−function �� on ��.

Definition 2.1.[14] A neutrosophic ℵ − structure of �� is defined to be the structure: ����: = where ����, ���� and ���� are the negative truth, negative indeterminacy and negative falsity membership function on X (ℵ − functions).

It is evident that −�� ≤ ����(��)+ ����(��)+ ����(��)≤ �� for all �� ∈ ��. Definition 2.2.[14] A neutrosophic ℵ − structure ���� of �� is called a neutrosophic ℵ −subsemigroup of �� if the following assertion is valid: ����(��������) ≤ ����(����) ˅ ����(����) (∀ ����, ���� ∈ ��) (����(��������) ≥ ����(����) ∧ ����(����)). ����(��������) ≤ ����(����) ˅ ����(����) .Let ���� be a neutrosophic ℵ −structure and ��, ��, �� ∈ [−��, ��] with −�� ≤ �� + �� + �� ≤ ��. Consider the sets:

���� �� = {���� ∈ ��|����(����) ≤ ��} ���� �� ={���� ∈ ��|����(����) ≥ ��} ���� �� = {���� ∈ ��|����(����) ≤ ��}.

The set ����(��, ��, ��)≔{���� ∈ �� |����(����)≤ ��, ����(����)≥ ��, ����(����) ≤ ��} is known as (��, ��, ��)-level set of ����. It is easy to observe that ����(��, ��, ��)= ���� �� ∩ ���� �� ∩ ���� �� .

Definition 2.3.[10] A neutrosophic ℵ −structure ���� of �� is called a neutrosophic ℵ −left (resp., right) ideal of �� if

����(��������) ≤ ����(����) (��������. , ����(��������) ≤ ����(����)) (∀ ����, ���� ∈ ��) ( ����(��������) ≥ ����(����) (��������., ����(��������) ≥ ����(����)) ). ����(��������) ≤ ����(����) (��������., ����(��������) ≤ ����(����)) ���� is neutrosophic ℵ−ideal of �� if ���� is neutrosophic ℵ−left and ℵ−right ideal of ��.

Definition 2.4. A neutrosophic ℵ −subsemigroup ���� of �� is known as neutrosophic ℵ −interior ideal if

����(������)≤ ����(��) (∀ ��, ��, �� ∈ ��)(����(������)≥ ����(��)). ����(������)≤ ����(��)

It is easy to observe that every neutrosophic ℵ−ideal is neutrosophic ℵ−interior ideal, but neutrosophic ℵ−interior ideal need not be a neutrosophic ℵ− ideal, as shown by an example. Example 2.5. Let �� be the set of all non-negative integers except 1. Then �� is a semigroup with usual multiplication.

Let ���� = { 0 (−0.9,−0.1,−0.7),

2 (−0.4 −0.6,−0.5),

5 (−0.3,−0.8,−0.3),

10 (−0.3,−0.8,−0.1),

����ℎ������������ (−0.7,−0.4,−0.6)}. Then ���� is

neutrosophic ℵ −interior ideal, but not neutrosophic ℵ− ideal with ����(2.5)=−0.3≰����(2).

Definition 2.6.[14] For any �� ⊆ ��, the characteristic neutrosophic ℵ −structure is defined as

where

����(��)��: X→[−��, ��], �� → { −�� ���� �� ∈ �� �� ������������������,

����(��)��: X→[−��, ��], �� → { �� ���� �� ∈ �� −�� ������������������, ����(��)��: X→[−��, ��], �� → {−�� ���� �� ∈ �� �� ������������������.

Definition 2.7.[14] Let ����: =

�� (����, ����, ����) and ����: =

�� (����, ����, ����) be neutrosophic ℵ −structures of

��. Then (i) ���� is called a neutrosophic ℵ − substructure of ����, denote by ���� ⊆ ���� , if ����(��)≥ ����(��), ����(��) ≤ ����(��), ����(��) ≥ ����(��) for all r ∈ ��. (ii) If ���� ⊆ ���� and ���� ⊆ ����, then we say that ���� = ����. (iii) The neutrosophic ℵ − product of ���� and ���� is defined to be a neutrosophic ℵ −structure of ��,

where ���� ʘ ���� ∶=

For �� ∈ ��, the element �� (����∘��(��), ����∘��(��), ����∘��(��))

is simply denoted by (���� ʘ ����)(��) =

(iii) The union of ���� and ����, a neutrosophic ℵ −structure over �� is defined as

where

(���� ∪ ����)(����)= ����∪��(����)= ����(����) ˄ ����(����), (���� ∪ ����)(����)= ����∪��(����)= ����(����) ˅ ����(����), (���� ∪ ����)(����)= ����∪��(����)= ����(����) ˄ ����(����) ∀���� ∈ ��. (iv) The intersection of ���� and ����, a neutrosophic ℵ −structure over �� is defined as

where

3. Neutrosophic ℵ −interior ideals

We study different properties of neutrosophic ℵ−interior ideals of ��. It is evident that neutrosophic ℵ− ideal is a neutrosophic ℵ−interior ideal of ��, but not the converse. Further, for a regular and for an intra-regular semigroup, every neutrosophic ℵ−interior ideal is neutrosophic ℵ−ideal.

All throughout this part, we consider ���� and ���� are neutrosophic ℵ−structures of ��. Theorem 3.1. For any L ⊆ X, the equivalent assertions are: (i) �� is an interior ideal, (ii) The characteristic neutrosophic ℵ−structure χL(XN) is a neutrosophic ℵ−interior ideal. Proof: Suppose �� is an interior ideal and let ��, ��, �� ∈ ��.

If �� ∈ ��, then ������ ∈ ��, so ����(��)��(������) = −1 = ����(��)��(��), ����(��)��(������) = 0 = ����(��)��(��) and ����(��)��(������) = −1 = ����(��)��(��).

If �� ∉ ��, then ����(��)��(������) ≤ 0 = ����(��)��(��), ����(��)��(������)≥ −1 = ����(��)��(��) and ����(��)��(������) ≤ 0 = ����(��)��(��). Therefore ����(����) is a neutrosophic ℵ−interior ideal.

Conversely, assume that ����(����) is a neutrosophic ℵ− interior ideal. Let �� ∈ �� and ��,�� ∈��. Then

So ������ ∈ ��. ����(��)��(������)≤����(��)��(��)=−1, ����(��)��(������) ≥����(��)��(��)=0, ����(��)��(������) ≤ ����(��)��(��)= −1 .

Theorem 3.2. If ���� and ���� are neutrosophic ℵ− interior ideals, then ����∩�� is neutrosophic ℵ− interior ideal. Proof: Let ���� and ���� be neutrosophic ℵ− interior ideals. For any ��,��,�� ∈��, we have ����∩��(������)=����(������)˅����(������) ≤����(��)˅����(��)=����∩��(��), ����∩��(������)=����(������)˄����(������) ≥����(��)˄����(��)=����∩��(��), ����∩��(������)=����(������)˅����(������) ≤����(��)˅����(��)=����∩��(��). Therefore ����∩�� is neutrosophic ℵ− interior ideal. □

Corollary 3.3. The arbitrary intersection of neutrosophic ℵ− interior ideals is a neutrosophic ℵ− interior ideal.

Theorem 3.4. If ���� and ���� are neutrosophic ℵ− interior ideals, then ����∪�� is neutrosophic ℵ− interior ideal. Proof: Let ���� and ���� be neutrosophic ℵ− interior ideals. For any ��,��,�� ∈��, we have ����∪��(������)=����(������)˄����(������) ≤����(��)˄����(��)=����∪��(��), ����∪��(������)=����(������)˅����(������) ≥����(��)˅����(��)=����∪��(��), ����∪��(������)=����(������)˄����(������) ≤����(��)˄����(��)=����∪��(��). Therefore ����∪�� is neutrosophic ℵ− interior ideal. □

Corollary 3.5. The arbitrary union of neutrosophic ℵ− interior ideals is neutrosophic ℵ− interior ideal.

Theorem 3.6. Let �� be a regular semigroup. If ���� is neutrosophic ℵ− interior ideal, then ���� is neutrosophic ℵ− ideal.

Proof: Assume that ���� is an interior ideal, and let ��,�� ∈��. As �� is regular and �� ∈��, there exists �� ∈�� such that �� =������. Now, ����(����) =����(��������) ≤����(��), ����(����) =����(��������)≥����(��) and ����(����) =����(��������)≤����(��). Therefore ���� is neutrosophic ℵ− right ideal.

Similarly, we can show that ���� is neutrosophic ℵ− left ideal and hence ���� is neutrosophic ℵ− ideal. □ Theorem 3.7. Let �� be an intra-regular semigroup. If ���� is neutrosophic ℵ− interior ideal, then ���� is neutrosophic ℵ− ideal. Proof: Suppose that ���� is neutrosophic ℵ− interior ideal, and let ��,�� ∈��. As �� is intra regular and �� ∈��, there exist ��,�� ∈�� such that �� =����2��. Now, ����(����)=����(����2����) ≤����(��), ����(����) =����(����2����)≥����(��) ����(����) =����(����2����)≤����(��). Therefore ���� is neutrosophic ℵ− right ideal. similarly, we can show that ���� is neutrosophic ℵ− left ideal and hence ���� is neutrosophic ℵ− ideal. □

Definition 3.8. A semigroup �� is left simple (resp., right simple) if it does not contain any proper left ideal (resp., right ideal) of ��. A semigroup �� is simple if it does not contain any proper ideal of ��.

Definition 3.9. A semigroup �� is said to be neutrosophic ℵ−simple if every neutrosophic ℵ− ideal is a constant function i.e., for every neutrosophic ℵ−ideal ���� of ��, we have ����(��) =����(��), ����(��) =����(��) and ����(��)=����(��) for all ��,��∈��.

Notation 3.10. If �� is a semigroup and �� ∈ ��, we define a subset, denoted by ���� as follows: ���� ≔ {�� ∈��| ����(��)≤����(��), ����(��)≥����(��) ������ ����(��)≤����(��)}.

Proposition 3.11. If ���� is neutrosophic ℵ− right (resp., ℵ− left, ℵ− ideal) ideal, then ���� is right (resp., left, ideal) ideal for every �� ∈��. Proof: Let �� ∈ ��. Then it is clear that φ ≠ Is  ��. Let �� ∈ Is and �� ∈ ��. Then ���� ∈ Is. Indeed; Since ���� is neutrosophic ℵ− right ideal and ��,�� ∈��, we get ����(����)≤����(��), ����(����)≥����(��) and ����(����)≤����(��). Since �� ∈ Is, we get ����(��)≤����(��), ����(��)≥����(��) and ����(��)≤����(��) which imply ���� ∈ Is. Therefore ���� is a right ideal for every �� ∈��. □

Theorem 3.12.[4] For any �� ⊆ ��, the equivalent assertions are: (i) L is left (resp., right) ideal, (ii) Characteristic neutrosophic ℵ−structure χL(XN) is neutrosophic ℵ−left (resp., right) ideal.

Theorem 3.13. Let �� be a semigroup. Then �� is simple if and only if �� is neutrosophic ℵ−simple.

Proof: Suppose �� is simple. Let ���� be a neutrosophic ℵ− ideal and ��,�� ∈��. Then by Proposition 3.11, ���� is an ideal of ��.As�� is simple, we have ���� =��. Since �� ∈����, we have

����(��) ≤����(��), ����(��) ≥����(��) and ����(��) ≤����(��).

Similarly, we can prove that ����(��)≤ ����(��), ����(��)≥ ����(��) and ����(��)≤ ����(��). So ����(��)= ����(��), ����(��)= ����(��) and ����(��)= ����(��). Hence �� is neutrosophic ℵ− simple.

Conversely, assume that �� is neutrosophic ℵ− simple and �� is an ideal of ��. Then by Theorem 3.12, χI(XN) is a neutrosophic ℵ− ideal. We now claim that �� =��. Let �� ∈��. Since �� is neutrosophic ℵ− simple, we have χI(XN) is a constant function and χI(XN)(w) =χI(XN)(y) for every �� ∈��. In particular, we have χI(TN)(w)=χI(TN)(d) =−1, χI(IN)(w) =χI(IN)(d) =0 and χI(FN)(w)=χI(FN)(d) =−1 for any �� ∈�� which implies �� ∈��. Thus �� ⊆�� and hence �� =��. □

Lemma 3.14. Let �� be a semigroup. Then �� is simple if and only for every �� ∈ ��, we have �� = ������. Proof: Suppose �� is simple and let �� ∈ ��. Then ��(������) ⊆ ������ and (������)�� ⊆ ������ imply that ������ is an ideal. Since �� is simple, we have ������ = ��.

Conversely, let �� be an ideal and let �� ∈ ��. Then �� = ������, ������ ⊆ ������ ⊆ �� which implies �� = ��. Therefore �� is simple. □

Theorem 3.15. Suppose �� is a semigroup. Then �� is simple if and only every neutrosophic ℵ− interior ideal of �� is a constant function. Proof: Suppose �� is simple and ��, �� ∈ ��. Let ���� be neutrosophic ℵ− interior ideal. Then by Lemma 3.14, we get �� =������ =������. As �� ∈������, we have �� =������ for ��,�� ∈��. Since ���� is neutrosophic ℵ− interior ideal, we have ����(��) =����(������)≤����(��), ����(��) =����(������)≥����(��) and ����(��)=����(������) ≤����(��). Similarly, we can prove that ����(��)≤����(��), ����(��)≥����(��) and ����(��)≤ ����(��). So ���� is a constant function.

Conversely, suppose ���� is neutrosophic ℵ− ideal. Then ���� is neutrosophic ℵ− interior ideal. By hypothesis, ���� is a constant function and so ���� is neutrosophic ℵ−simple. By Theorem 3.13, �� is simple. □

Theorem 3.16. Let ���� be neutrosophic ℵ − structure and let ��, ��, �� ∈ [−1, 0] with−3 ≤ �� + �� + �� ≤ 0. If ���� is neutrosophic ℵ −interior ideal, then (��, ��, ��)-level set of ���� is neutrosophic ℵ −interior ideal whenever ����(��, ��, ��) ≠ ∅. Proof: Suppose ����(��,��,��)≠ ∅ for ��,��,�� ∈ [−��,��] with −��≤ ��+��+��≤��.

Let ���� be a neutrosophic ℵ −interior ideal and let ��, ��, �� ∈ ����(��, ��, ��). Then ����(������) ≤ ����(��)≤ ��; ����(������)≥ ����(��)≥ �� and ����(������) ≤ ����(��)≤ �� which imply ������ ∈ ���� (��, ��, ��). Therefore ����(��, ��, ��) is a neutrosophic ℵ −interior ideal of ��. □

Theorem 3.17. Let ���� be neutrosophic ℵ − structure with ��, ��, �� ∈ [−1, 0] such that −3 ≤ �� + �� + �� ≤ 0. If ���� ��, ���� �� and ���� �� are interior ideals, then ���� is neutrosophic ℵ− interior ideal of �� whenever it is non-empty.

Proof: Suppose that for ��, ��, �� ∈ �� with ����(������) > ����(��). Then ����(������) > ���� ≥ ����(��) for some ���� ∈[−��, ��). So �� ∈ ���� ����(��) but ������ ∉ ���� ����(��), a contradiction. Thus ����(������)≤ ����(��).

Suppose that for ��, ��, �� ∈ �� with ����(������) < ����(��). Then ����(������) < ���� ≤ ����(��) for some ���� ∈ [−��, ��). So �� ∈ ���� ����(��) but ������ ∉ ���� ����(��), a contradiction. Thus ����(������) ≥ ����(��).

Suppose that for ��, ��, �� ∈ �� with ����(������) > ����(��). Then ����(������) > ���� ≥ ����(��) for some ���� ∈[−��, ��). So �� ∈ ���� ����(��) but ������ ∉ ���� ����(��), a contradiction. Thus ����(������) ≤ ����(��).

Thus ���� is neutrosophic ℵ − interior ideal. □

Theorem 3.18. Let ���� be neutrosophic ℵ − structure over ��. Then the equivalent assertions are: (i) ���� is neutrosophic ℵ −interior ideal, (ii) ����ʘ ����ʘ���� ⊆ ���� for any neutrosophic ℵ − structure ����. Proof: Suppose ���� is neutrosophic ℵ − interior ideal. Let �� ∈ ��. For any ��, ��, �� ∈ �� such that �� = ������. Then ����(��)= ����(������) ≤ ����(��) ≤ ����(��)˅����(��)˅����(��) which implies ����(��)≤ ����∘��∘��(��). Otherwise �� ≠ ������. Then ����(��)≤ �� = ����∘��∘��(��). Similarly, we can prove that ����(��)≥ ����∘��∘��(��) and ����(��)≤ ����∘��∘��(��). Thus ����ʘ ����ʘ���� ⊆ ����.

Conversely, assume that ����ʘ ����ʘ���� ⊆ ���� for any neutrosophic ℵ −structure ����.

Let ��, ��, �� ∈ ��. If �� = ������, then ����(������)= ����(��) ≤ (����(��)�� ∘ ���� ∘ ����(��)�� )(��) = ⋀ {����(��)�� ∘ ����)

and ����(������) =����(��) ≤ (����(��)�� ∘ ���� ∘ ����(��)�� )(��) = ⋀ {����(��)�� ∘ ����)

≤ ����(��)��(��)˅ (��)��(��)˅ ����(��)��(��) = ����(��). Therefore ���� is neutrosophic ℵ −interior ideal. □

Notation 3.19. Let �� and �� be semigroups. A mapping ��: �� → �� is said to be a homomorphism if ��(����) = ��(��)��(��) for all ��, �� ∈ ��. Throughout this remaining section, we denote ������(��), the set of all automorphisms of ��.

Definition 3.20. An interior ideal �� of a semigroup �� is called a characteristic interior ideal if ��(��)= �� for all �� ∈ ������(��).

Definition 3.21. Let �� be a semigroup. A neutrosophic ℵ − interior ideal ���� is called neutrosophic ℵ − characteristic interior ideal if ����(��(��)) = ����(��), ����(��(��)) = ����(��) and ����(��(��)) = ����(��) for all �� ∈ �� and all �� ∈ ������(��).

Theorem 3.22. For any L ⊆ X, the equivalent assertions are: (i) L is characteristic interior ideal, (ii) The characteristic neutrosophic ℵ− structure χL(XM) is neutrosophic ℵ− characteristic interior ideal. Proof: Suppose �� is characteristic interior ideal and let �� ∈ ��. Then by Theorem 3.1, χL(XM) is neutrosophic ℵ−interior ideal. If �� ∈��, then ����(��)��(��)=−1, ����(��)��(��)=0, and ����(��)��(��)= −1. Now, for any ℎ∈������(��), ℎ(��)∈ℎ(��)=�� which implies ����(��)��(ℎ(��))=−1,����(��)��(ℎ(��))= 0, and ����(��)��(ℎ(��))=−1. If �� ∉��, then ����(��)��(��)=0,����(��)��(��)=−1, and ����(��)��(��) =0. Now, for any ℎ ∈������(��), ℎ(��)∉ℎ(��) which implies ����(��)��(ℎ(��))=0,����(��)��(ℎ(��))=−1, and ����(��)��(ℎ(��))=0. Thus ����(��)��(ℎ(��))=����(��)��(��), ����(��)��(ℎ(��))=����(��)��(��), and ����(��)��(ℎ(��))=����(��)��(��) for all �� ∈�� and hence χL(XM) is neutrosophic ℵ− characteristic interior ideal.

Conversely, assume that χL(XM) is neutrosophic ℵ−characteristic interior ideal. Then by Theorem 3.1, �� is an interior ideal . Now, let ℎ ∈������(��) and �� ∈��. Then ����(��)��(��)= −1, ����(��)��(��) =0 and ����(��)��(��)=−1. Since χL(XM) is neutrosophic ℵ−characteristic interior ideal , we have ����(��)��(ℎ(��))=����(��)��(��), ����(��)��(ℎ(��))=����(��)��(��) and ����(��)��(ℎ(��))= ����(��)��(��) which imply ℎ(��)∈��. So ℎ(��)⊆�� for all ℎ∈������(��). Again, since ℎ ∈������(��) and �� ∈��, there exists �� ∈�� such that ℎ(��) =��.

Suppose that �� ∉ ��. Then ����(��)��(��) = 0, ����(��)��(��) = −1 and ����(��)��(��) = 0. Since ����(��)��(ℎ(��))= ����(��)��(��), ����(��)��(ℎ(��))= ����(��)��(��) and ����(��)��(ℎ(��))= ����(��)��(��), we get ����(��)��(ℎ(��))= 0, ����(��)��(ℎ(��))= −1 and ����(��)��(ℎ(��))= 0 which imply ℎ(��) ∉ ��, a contradiction. So �� ∈ �� i.e., ℎ(��) ∈ ��. Thus �� ⊆ ℎ(��) for all ℎ ∈ ������(��) and hence �� is characteristic interior ideal. □

Theorem 3.23. For a semigroup ��, the equivalent statements are: (i) �� is intra-regular, (ii) For any neutrosophic ℵ −interior ideal ����, we have ����(��) = ����(��2) for all �� ∈ ��. Proof: (��) ⇒ (����) Suppose �� is intra-regular, and ���� is neutrosophic ℵ − interior ideal and �� ∈ ��. Then there exist ��, �� ∈ �� such that �� = ����2��. Now ����(��) = ����(����2��)≤ ����(��2)≤ ����(��) and so ����(��) = ����(��2), ����(��) = ����(����2��)≥ ����(��2)≥ ����(��) and so ����(��)= ����(��2), and ����(��)= ����(����2��)≤ ����(��2)≤ ����(��) and so ����(��) = ����(��2). Therefore ����(��)= ����(��2) for all �� ∈ ��. (����) ⇒(��) Let (����) holds and �� ∈ ��. Then ��(��2) is an ideal of ��. By Theorem 3.5 of [4], ����(��2)(����) is neutrosophic ℵ −ideal. By assumption, ����(��2)(����)(��) = ����(��2)(����)(��2). Since ����(��2)(��)��(��2)= −1 = ����(��2)(��)��(��2) and ����(��2)(��)��(��2)= 0, we get ����(��2)(��)��(��)= −1 = ����(��2)(��)��(��) and ����(��2)(��)��(��2)= 0 which imply �� ∈ ��(��2). Hence �� is intra-regular. □

Theorem 3.24. For a semigroup ��, the equivalent statements are: (i) �� is left (resp., right) regular,

(ii) For any neutrosophic ℵ −interior ideal ����, we have ����(��) = ����(��2) for all �� ∈ ��. Proof: (��) ⇒ (����) Let �� be left regular. Then there exists �� ∈ �� such that �� = ����2 . Let ���� be a neutrosophic ℵ −interior ideal. Then ����(��)= ����(����2)≤ ����(��) and so ����(��)= ����(��2), ����(��)= ����(����2)≥ ����(��) and so ����(��) = ����(��2), and ����(��) = ����(����2)≤ ����(��) and so ����(��) = ����(��2). Therefore ����(��)= ����(��2) for all �� ∈ ��. (����) ⇒ (��) Suppose (����) holds and let ���� be neutrosophic ℵ −interior ideal. Then for any �� ∈ ��, ����(��2)(��)��(��)= ����(��2)(��)��(��2)= −1, ����(��2)(��)��(��) = ����(��2)(��)��(��2)= 0 and ����(��2)(��)��(��)= ����(��2)(��)��(��2)= −1 which imply �� ∈ ��(��2). Thus �� is left regular. □

Conclusions

In this paper, we have introduced the concepts of neutrosophic ℵ − interior ideals and neutrosophic ℵ − characteristic interior ideals in semigroups and studied their properties, and characterized regular and intra-regular semigroups using neutrosophic ℵ-interior ideal structures. We have also shown that R is a characteristic interior ideal if and only if the characteristic neutrosophic ℵ−structure χR(XN) is neutrosophic ℵ−characteristic interior ideal. In future, we will define neutrosophic ℵ−prime ideals in semigroups and study their properties.

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