Neutrosophic ℵ –Ideals ( ℵ- Subalgebras) of Subtraction Algebra

The connection between neutrosophy and algebra has been of great interest with respect to many researchers. The objective of this paper is to provide a connection between neutrosophic ℵ−structures and subtraction algebras. In this regard, we introduce the concept of neutrosophic ℵ−ideals in subtraction algebra. Moreover, we study its properties and find a necessary and sufficient condition for a neutrosophic ℵ−structure to be a neutrosophic ℵ−ideal.


Introduction
Neutrosophic sets were introduced by Florentin Smarandache [11] as a new mathematical tool for dealing with uncertainity. They can be viewed as a generalization of the fuzzy sets that were introduced in 1965 by Lotfi Zadeh [14]. Where Zadeh defined fuzzy sets as mathematical model of vagueness in which an element belongs to a given set to some degree that is a number between 0 and 1 (both inclusive). Neutrosophy is a base of neutrosophic logic which is an extension of fuzzy logic where indeterminacy is included [13]. In neutrosophic logic [10], each proposition is estimated to have the degree of truth in a subset , the degree of indeterminacy in a subset , and the degree of falsity in a subset . The study of neutrosophic sets and their properties have a great importance in the sense of applications as well as in understanding the fundamentals of uncertainty. Some related work can be found in [1,2,3,12].
A crisp set in a universe can be defined in the form of its membership function : → {0,1} where ( ) = 1 if ∈ and ( ) = 0 if ∉ . A single valued neutrosophic set is an example of neutrosophic set which has many applications [10]. A new function, which is called negative-valued function, was introduced by Jun et al. [5] and they used it to construct ℵ-structures. Some work related to neutrosophic ℵ-structures can be found in [6,7]. Schein [9] considered systems of the form ( ,•,\), where is a set of functions closed under the composition "•" of functions and the set theoretic subtraction "\" and hence ( , \) is a subtraction algebra. Jun et al. [4] introduced the concept of ideals in subtraction algebras and discussed the properties of these ideals. Some researchers worked on combining the notions of neutrosophic sets and subtraction algebra. For example, Ibrahim et al. introduced neutrosophic subtraction algebra (semigroups) and presented some results about them. Moreover, Park [8] discussed neutrosophic ideals of subtraction algebras by using single valued neutrosophic sets.
In this paper, we apply the concept of neutrosophic ℵ-structures in subtraction algebras. And it is organized as follows: After an Introduction, in Section 2 and Section 3, we present some basic results about neutrosophic ℵ-strructures as well as about subtraction algebras that are used throughout the paper. In Section 4, we introduce neutrosophic ℵ-ideals (ℵ-subalgebras) of subtraction algebra and prove that the intersection, the product, the homomorphic preimage, and onto homomrphic image are neutrosophic ℵ-ideals. Finally, in Section 5, we prove a necessary and sufficient condition for ℵ-structures to be neutrosophic ℵ-ideals by introducing the ( , , )− level sets.

Neutrosophic ℵ-structures
In this section, we present some basic results about neutrosophic ℵ-structures. For more details about neutrosophy, we refer to [5,6,7]. Definition 2.1. [5] Let be a non-empty set. A function from → [−1,0] is called a negative-valued function (ℵfunction) from to [−1,0]. Definition 2.2. [7] Let be a non-empty set. A neutrosophic ℵ-structure over is defined as follows: where N , N , N are ℵ-functions on which are called the negative truth membership function, the negative indeterminacy membership function and the negative falsity membership function, respectively, on . It is clear that for any ℵ-structure N over , −3 ≤ N ( ) + N ( ) + N ( ) ≤ 0 for all ∈ .  : ∈ } over is defined as follows:

Subtraction algebra
In this section, we present some results related to subtraction algebra that are used throughout the paper. For more details, we refer to [4,9,15].   Table 1.
is a subtraction algebra defined in Table 2.

Definition 3.3. [4]
A non-empty subset of a subtraction algebra is called a subalgebra of if for all , ∈ , − ∈ .

Definition 3.4. [4]
A non-empty subset of a subtraction algebra is called an ideal of if it satisfies the following conditions.
Remark 3.1. Every ideal of a subtraction algebra is a subalgebra. But the converse may not hold.
is an isomorphism if is a bijective homomrphism. In this case, we say that and are isomorphic subtraction algebras and we write ≅ .

Operations on neutrosophic ℵ-ideals (ℵ-subalgebra) of subtraction algebra
In this section, we introduce neutrosophic ℵ-ideals (ℵ-subalgebras) of subtraction algebra, present some examples, and study different operations on them.     Then 2 N is a neutrosophic ℵ-subalgebra of 2 that is not neutrosophic ℵ-ideal of 2 .

Remark 4.2.
The results in this section are also valid for neutrosophic ℵ-subalgebras. But we restrict our proof to neutrosophic ℵ-ideals.   Therefore, N⋂M is a neutrosophic ℵ-ideal (ℵ-subalgebra) of .    Proof. The proof is straightforward.  Table 4.

Level sets and neutrosophic ℵ-ideals (ℵ-subalgebra) of subtraction algebra
In this section, we define ( , , )− level sets of N and study their relation with ℵ-ideals of .
Remark 5.1. The results in this section are also valid for neutrosophic ℵ-subalgebras (instead of ideal we have subalgebra). But we restrict our proof to neutrosophic ℵ-ideals. Therefore, N is a neutrosophic ℵ-ideal of .    And it is either an empty set or an ideal of . Therefore, by Lemma 5.1, N is a neutrosophic ℵ-ideal of .

Conclusion
In this paper, we combined the notions of ℵ−structures, neutrosophy, and subtraction algebra to introduce ℵ−ideals (ℵ-subalgebras) of subtraction algebras. Some operations on the defined notions were discussed. Moreover, the ( , , )− level sets were introduced and used to find a necessary and sufficient condition for ℵ-structures to be neutrosophic ℵ-ideals (ℵ-subalgebras).
For future work, it would be interesting to check whether there is a relation between our results about ℵ−ideals (ℵsubalgebras) of subtraction algebras and the results related to single valued neutrosophic subtraction algebras discussed by Chul Hwan Park in [8].