A Note on Single Valued Neutrosophic Sets in Ordered Groupoids

The aim of this paper is to combine the notions of ordered algebraic structures and neutrosophy. In this regard, we deﬁne for the ﬁrst time single valued neutrosophic sets in ordered groupoids. More precisely, we study single valued neutrosophic subgroupoids of ordered groupoids, single valued neutrosophic ideals of ordered groupoids, and single valued neutrosophic ﬁlters of ordered groupoids. Finally, we present some remarks on single valued neutrosophic subgroups (ideals) of ordered groups.


Introduction
Neutrosophy [10], a new branch of philosophy that deals with indeterminacy, was launched by Smarandache in 1998. The theory of neutrosophy is based on the concept of indeterminacy (neutrality) that is neither true nor false. Smarandache [11] defined neutrosophic sets as a generalization of the fuzzy sets introduced by Zadeh [15] in 1965 and as a generalization of intuitionistic fuzzy sets introduced by Atanassov [4] in 1986. A special type of neutrosophic set is single valued neutrosophic set (SVNS) [14] which also can be considered as a generalization of fuzzy sets and intuitionistic fuzzy sets. In an SVNS, each element has a truth value "t", indeterminacy value "i", and a falsity value "f " where 0 ≤ t, i, f ≤ 1 and 0 ≤ t + i + f ≤ 3. When i = 0 and f = 1 − t, we get a fuzzy set and when 0 ≤ t + f ≤ 1 and i = 1 − t − f , we get an intuitionistic fuzzy set. Neutroosphic sets have many applications in different fields of Science and Engineering. In particular, they are connected to various fields of Mathematics and especially to Algebra. For example, many researchers [1,2,9,12,13] have worked on the connection between neutrosophy and algebraic structures.
Our paper introduces a new link between algebraic structures and neutrosophy. In particular, it is concerned about single valued neutrosophic sets in ordered groupoids and it is organized as follows: after an Introduction, in Section 2, we present some definitions related to neutrosophy that are used throughout the paper. In Section 3, we present some definitions about ordered groupoids (groups) and elaborate some examples that are used in Section 4 and Section 5. In Section 4, we define single valued neutrosophic subgroupoids (ideals) as well as single valued neutrosophic filters of ordered groupoids, present many non-trivial examples about the new defined concepts, and study some of their properties. Finally in Section 5, we apply the definition of SVNS in ordered groupoids to ordered groups and present some remarks and results.

Single valued neutrosophic sets
In this section, we present some definitions about neutrosophy that are used throughout the paper.
Definition 2.1. [14] Let X be a non-empty space of elements (objects). A single valued neutrosophic set (SVNS) A on X is characterized by truth-membership function T A , indeterminacy-membership function I A , and falsity-membership function F A . For each element x ∈ X, 0 ≤ T A (x), Then 1. A is called a single valued neutrosophic subset of B and denoted as If A is a single valued neutrosophic subset of B and B is a single valued neutrosophic subset of A then A and B are said to be equal single valued neutrosophic sets (A = B).
2. The union of A and B is defined to be the SVNS over X: Here, 3. The intersection of A and B is defined to be the SVNS over X: Here, Then the SVNS S ∩ M and S ∪ M over X are as follows.

Ordered groupoids and ordered groups
In this section, we present some examples on ordered groupoids and ordered groups that are used in Section 4 and Section 5. For more details about ordered algebraic structures, we refer to [5] and [6].
Definition 3.1. [5] Let (G, ·) be a groupoid (group) and "≤" be a partial order relation (reflexive, antisymmetric, and transitive) on G. Then (G, ·, ≤) is an ordered groupoid (ordered group) if the following condition holds for all z ∈ G.
If x ≤ y then z · x ≤ z · y and x · z ≤ y · z.
Definition 3.2. Let (G, ·, ≤) be an ordered groupoid (group). Then G is called a total ordered groupoid (group) if x and y are comparable for all x, y ∈ G. i.e., x ≤ y or y ≤ x for all x, y ∈ G.
An ordered groupoid (G, ·, ≤) is said to be commutative if x · y = y · x for all x, y ∈ G and an element e in an ordered groupoid (G, ·, ≤) is called an identity if e · x = x · e = x for all x ∈ G. If such an element exists then it is unique. Remark 3.3. Let (G, ·) be any groupoid (group). Then by defining "≤" on G as follows: For all x, y ∈ G, x ≤ y ⇐⇒ x = y.
Such an order is called the trivial order.
Ordered groups are a special case of ordered groupoids. We present some examples on infinite ordered groups.
Example 3.4. The groups of integers, rational numbers, real numbers under standard addition and usual order are ordered groups.
Example 3.5. Let Q + be the set of positive rational numbers. Then (Q + , ·, ≤) is an ordered group. Where "≤" is defined as follows: For all q, q ∈ Q + , We show that the partial order "≤" defines an order on Q + . Let q ≤ q and z ∈ Q + . Having q q ∈ N and z > 0 implies that q z qz ∈ N. Thus, qz ≤ q z. As an illustration for "≤" on Q + , we can say that 1 4 ≤ 1 2 as We present some examples on ordered groupoids that are not ordered groups.
Example 3.6. Let G be any non-empty set with a ∈ G and "≤" a partial order on G. Then by setting x · y = a for all x, y ∈ G, we get that (G, ·, ≤) is an ordered groupoid. Then (N, +, ≤ N ) is a commutative ordered groupoid. This is easily seen as ≤ N is a partial order on N and if x ≤ N y and z ∈ N then x + z ≥ y + z and hence x + z ≤ N y + z.
Finite groupoids can be presented by means of Cayley's table.
is a total ordered groupoid with an identity "e".
Definition 3.17. Let (G, ·, ≤) be an ordered groupoid and F ⊆ G. Then F is a filter of G if the following conditions are satisfied.
(1) x · y ∈ F for all x, y ∈ F ; (2) If x · y ∈ F then x, y ∈ F for all x, y ∈ G; (3) If x ∈ F, y ∈ G and x ≤ y then y ∈ F .

SVNS in ordered groupoids
In this section and inspired by the definition of fuzzy sets in ordered groupoids [7], we define for the first time single valued neutrosophic subgroupoids (ideals) (in Subsection 4.1) as well as single valued neutrosophic filters (in Subsection 4.2) of ordered groupoids and study some of their properties such as finding a relationship between subgroupoids/ideals/filters of ordered groupoids and single valued neutrosophic subgroupoids/ideals/filters of these ordered groupoids. Moreover, we construct many non-trivial examples on them.

Single valued neutrosophic subgroupoids (ideals) of groupoids
Definition 4.1. Let (G, ·, ≤) be an ordered groupoid and A an SVNS over G. Then A is single valued neutrosophic subgroupoid of G if for all x, y ∈ G, the following conditions hold: Definition 4.2. Let (G, ·, ≤) be an ordered groupoid and A an SVNS over G. Then A is single valued neutrosophic left ideal of G if for all x, y ∈ G, the following conditions hold: Definition 4.3. Let (G, ·, ≤) be an ordered groupoid and A an SVNS over G. Then A is single valued neutrosophic right ideal of G if for all x, y ∈ G, the following conditions hold:  Remark 4.6. Let (G, ·, ≤) be an ordered groupoid and α, β, γ ∈ [0, 1] be fixed values. Then is single valued neutrosophic ideal of G. Moreover, it is called the trivial single valued neutrosophic ideal.
Example 4.7. Let (N, +, ≤ N ) be the ordered groupoid defined in Example 3.7 and A be an SVNS over N defined as follows: For all n ∈ N, Then A is a single valued neutrosophic ideal of N. To prove that and by means of Remark 4.5, it suffices to show that A is a single valued neutrosophic right ideal of G. Let n, n ∈ N. Then n + n ≥ n and thus, Proof. If A is the trivial single valued neutrosophic ideal of G then we are done by Remark 4.6. Conversely, let A be a single valued neutrosophic left (right) ideal of G. We prove the case when A is a single valued neutrosophic right ideal of G and the case when A is a single valued neutrosophic left ideal of G is done similarly. Let A be a single valued neutrosophic right ideal of G. Then for all x ∈ G, we have: The latter implies that Therefore, A is the trivial single valued neutrosophic ideal of G.
We present an example on a single valued neutrosophic right ideal that is not a single valued neutrosophic left ideal and an example on a single valued neutrosophic subgroupoid that is neither a single valued neutrosophic left ideal nor a single valued neutrosophic right ideal.   Proof. Let x, y ∈ G. Then T Aα (x · y) ≥ T Aα (x) ∧ T Aα (y), I Aα (x · y) ≥ I Aα (x) ∧ I Aα (y), and F Aα (x · y) ≤ F Aα (x) ∨ F Aα (y) for all α. The latter implies that Let y ≤ x. Then T Aα (y) ≥ T Aα (x), I Aα (y) ≥ I Aα (x), and F Aα (y) ≤ F Aα (x) for all α. One can easily see that T α Aα (y) ≥ T α Aα (x), I α Aα (y) ≥ I α Aα (x), and F α Aα (y) ≤ F α Aα (x). Therefore, α A α is a single valued neutrosophic subgroupoid of G.
Remark 4.13. Let (G, ·, ≤) be an ordered groupoid and A α a single valued neutrosophic subgroupoid of G. Then α A α may not be a single valued neutrosophic subgroupoid of G.
We illustrate Remark 4.13 by the following example.    Proof. Let x, y ∈ G. Having A α a single valued neutrosophic right ideal of G implies that T Aα (x · y) ≥ T Aα (x), I Aα (x · y) ≥ I Aα (x), and F Aα (x · y) ≤ F Aα (x) for all α. The latter implies that . Similarly, having A α a single valued neutrosophic left ideal of G implies that T Aα (x · y) ≥ T Aα (y), I Aα (x · y) ≥ I Aα (y), and F Aα (x · y) ≤ F Aα (y) for all α. The latter implies that Theorem 4.18. Let (G, ·, ≤) be an ordered groupoid and A an SVNS over X. Then A is a single valued neutrosophic subgroupoid of G if and only if L (α,β,γ) is either the empty set or a subgroupoid of G for all 0 ≤ α, β, γ ≤ 1.
Proof. The proof is similar to that of Theorem 4.18.
Theorem 4.20. Let (G, ·, ≤) be an ordered groupoid and A an SVNS over X. Then A is a single valued neutrosophic ideal of G if and only if L (α,β,γ) is either the empty set or an ideal of G for all 0 ≤ α, β, γ ≤ 1.
Proof. The proof follows from Theorem 4.19 and having an ideal of an ordered groupoid is a left ideal and right ideal of it.

Single valued neutrosophic filters of groupoids
Definition 4.23. Let (G, ·, ≤) be an ordered groupoid and A an SVNS over G. Then A is single valued neutrosophic filter of G if for all x, y ∈ G, the following conditions hold:  Proof. The proof can be done in a similar way to that of Lemma 4.12.
Remark 4.27. Let (G, ·, ≤) be an ordered groupoid and A α a single valued neutrosophic filter of G. Then α A α may not be a single valued neutrosophic filter of G.
We illustrate Remark 4.13 by the following example.

Some remarks on SVNS in ordered groups
In this section, we apply the definition of SVNS in ordered groupoids to ordered groups and point out some remarks and results about SVNS in ordered groups. Ideas of this section can be considered as a base for a new possible research on SVNS in ordered groups.
Definition 5.1. Let (G, ·, ≤) be an ordered group and A an SVNS over G. Then A is single valued neutrosophic subgroup of G if for all x, y ∈ G, the following conditions hold: Proposition 5.2. Let (G, ·, ≤) be an ordered group with identity "e" and A a single valued neutrosophic subgroup of G. Then the following statements hold. Proof. The proof is straightforward. Proof. The proof follows from Proposition 4.8.
Proposition 5.4. Let (G, ·, ≤) be an ordered group with identity "e" and A an SVNS over G. If e and x are comparable for all x ∈ G then A is a single valued neutrosophic subgroup of G if and only if A is the trivial single valued neutrosophic subgroup of G.
Proof. If A is the trivial single valued neutrosophic subgroup of G then we are done.
Let A be a single valued neutrosophic subgroup of G. Since e, x are comparable, it follows that x ≤ e or e ≤ x. If x ≤ e then T A (x) ≥ T A (e), I A (x) ≥ I A (e), and F A (x) ≤ F A (e). Proposition 5.2, 2. implies that A is the trivial single valued neutrosophic subgroup of G. If e ≤ x then x −1 ≤ e. The latter implies that Thus, A is the trivial single valued neutrosophic subgroup of G.
Corollary 5.5. Let (G, ·, ≤) be a total ordered group. Then G has no non-trivial single valued neutrosophic subgroups.
Proof. Since (G, ·, ≤) is a total ordered group, it follows that "e" (the ideantity of G) and x are comparable for all x ∈ G. Proposition 5.4 completes the proof.
Proposition 5.6. Let (G, ·, ≤) be an ordered cyclic group with identity "e" and generator a, and A an SVNS over G. If e ≤ a then A is a single valued neutrosophic subgroup of G if and only if A is the trivial single valued neutrosophic subgroup of G.
Proof. If A is the trivial single valued neutrosophic subgroup of G then we are done.
Let A be a single valued neutrosophic subgroup of G. Since e ≤ a, it follows that a −1 ≤ e and hence T A (e) ≤ T A (a −1 ) = T A (a), I A (e) ≤ I A (a −1 ) = I A (a), and F A (e) ≥ F A (a −1 ) = F A (a). The latter and Proposition 5.2, 2. imply that T A (e) = T A (a), I A (e) = I A (a), and F A (e) = F A (a). Having e ≤ a and (G, ·, ≤) an ordered group implies that e ≤ a k for all k = 1, 2, . . . and hence, a −k ≤ e. The latter implies that T A (e) = T A (a k ), I A (e) = I A (a k ), and F A (e) = F A (a k ) for all k ∈ Z. Therefore, A is the trivial single valued neutrosophic subgroup of G.
Example 5.7. Using Proposition 5.6, we get that the ordered group of integers under standard addition and usual order has no non-trivial single valued neutrosophic subgroups.
Proposition 5.8. Let (G, ·, ≤) be a finite ordered group with identity "e" and A an SVNS over G. If e ≤ a or a ≤ e then a = e.
Proof. Let |G| = n. Then a n = e. If e ≤ a then a ≤ a k for all k = 1, 2, . . .. By setting k = n, we get that a ≤ e. And if a ≤ e then e ≤ a −1 and hence a −1 ≤ (a −1 ) n = e. In both cases, we get that a = e. Proposition 5.9. Let (G, ·, ≤) be a finite ordered group with identity "e". Then "≤" is the trivial order on G.
Proof. Suppose that there exist x, y ∈ G such that x ≤ y. Then e ≤ yx −1 . Proposition 5.8 asserts that yx −1 = e and hence, x = y.
From Proposition 5.9, we deduce that studying single valued neutrosophic subgroups of finite ordered group is the same as studying single valued neutrosophic subgroups of groups as the order is the trivial order. As a result, studying single valued neutrosophic subgroups of ordered groups should start with infinite groups.

Conclusion and discussion
This paper contributed to the study of neutrosophic algebraic structures by introducing, for the first time, SVNS in ordered algebraic structures. Several new concepts were defined and studied like single valued neutrosophic subgroupoids, single valued neutrosophic ideals, and single valued neutrosophic filters of ordered groupoids and many interesting examples were presented. Finally, an application of this study to ordered groups was discussed. The latter can be considered as a base for a new possible research on SVNS over ordered groups.
For future work, we will work on SVNS in ordered groups and elaborate more properties about it. Also we will work on SVNS in ordered semigroups.
Funding: "This research received no external funding." Conflicts of Interest: "The authors declare no conflict of interest."