Let M and N be two Neutrosophic sets. One can say, by language abuse, that any element Neutrosophically belongs to any set, due to the percentage of truth/indeterminacy/falsity, which varies between 0 and 100. For example: x (50, 20, 30) ∈ M (Which means, with a probability of 50% x is in M, with a probability of 30% x is not in M, and the rest is undecidable) y (O, O, I 00) ∈ M (Which normally means y is not for sure in M), or z (O, 100, 0) ∈M (Which means one doesn't know absolutely anything about z's affiliation with M). Definition 1. A Single-Valued Neutrosophic Set (SVNS) A in the universe U is of the form:

A = {(x, T(x), I(x), F(x)): x  U, T(x), I(x), F(x)  [0,1], 0  T(x)+I(x)+F(x)  3}. (1)

In Eq. (1), T(x), I(x), F(x) are the degrees of truth (Or membership), indeterminacy (Or neutrality) and falsity (Or non-membership) of x in A respectively, called the Neutrosophic components of x. For simplicity, we write A. Indeterminacy is defined to be in general everything that exists between the opposites of truth and falsity.

Definition 2. Let (t1, i1, f1), (t2, i2, f2) be in A and let k be a positive number. Then

I. The sum (t1, i1, f1) + (t2, i2, f2) = (t1 + t2, i1+i2, f1+ f2).

II. The scalar product k(t1, i1, f1) = (kt1, ki1, kf1).

Definition 3. Let A be a SVNS and let (t1, i1, f1), (t2, i2, f2), …., (tk, ik, fk) be a finite number of elements of A. Assume that (ti, ii, fi) appears ni times in an application, i = 1,2,…, k. Set n = n1+n2+…+nk. Then the mean value of all these elements of A is defined to be the element of A.

(tm, im, fm) = 1/n [n1(t1, i1, f11)+n2(t2, i2, f2)+….+nk(tk, ik, fk)].

A pair (A, −) where A is a nonempty set and ‘−‘ is a binary operation on A is called a subtraction algebra if I. x − (y − x) = x.

II. x − (x − y) = y − (y − x).

III. (x − y) − z = (x − z) − y for all x, y, z ∈ A.

Axiom 3 permits us to omit parentheses in expressions of the form (x − y) − z. The subtraction determines an order relation on A: x ≤ y if and only if x − y = 0, where 0 = x − x is an element that does not depend on the choice of x ∈ A. The ordered set (A, ≤) is a semi-Boolean algebra, that is, it is a meet semi lattice with zero 0 in which every interval [0, x] is a Boolean algebra with respect to the induced order. Here x ∧ y = x − (x − y); the complement of an element y ∈ [0, x] is x − y; and if y, z ∈ [0, x], then

y∨ z = (y 0 ∧ z 0) 0 = x − ((x − y) ∧ (x − z)) = x − ((x − y) − ((x − y) − (x − z))).

In a subtraction algebra, the following hold:

I. x − 0 = x and 0 − x = 0.

II. x − (x − y) ≤ y.

III. x ≤ y if and only if x = y − w for some w ∈ X.

IV. x ≤ y implies x − z ≤ y − z and z − y ≤ z − x for all z ∈ X.

V. x − (x − (x − y)) = x – y.

Definition 4. Let X be a nonempty set. A set X(I) =< X ∪ I > generated by X and I is called a Neutrosophic set. The elements of X(I) are of the form (x, yI) where x and y are elements of X. I is called an indeterminate and it has the property I = I for all positive integer n.

Definition 5. Let (X, −) be any classical subtraction algebra and let X(I) =< X ∪ I > be a set generated by X and I. Consider the Neutrosophic algebraic structure (X(I), −N) where for all (a, bI),(c, dI) ∈ X(I), −N is defined by

(a, bI) −N (c, dI) = (a − c,(b − d)I), for all a, b, c, d ∈ X.

We call (X(I), −N) a Neutrosophic subtraction algebra.

An element x ∈ X is represented by (x, 0) ∈X(I) and (0, 0) represents the constant element in X(I).

Neutrosophic Probability studies the chance that a particular event E will occur, where that chance is represented by three coordinates (Variables): t% true, i% indeterminate, and f% false, with t+i+f = 100 and f, i, t ∈ [0,100]. Neutrosophic Statistics is the analysis of such events.

In Imprecise Probability (IP), the probability of an event A.

IP(A) = (a, b) ⊆ [0,1],

is an interval included into [0,1], not a crisp number.

The Neutrosophic Probability that an event A occurs is

NP(A) = (ch(A), ch(neutA), ch(antiA)) = (T, I, F),

but sometimes instead of “Neuta” we say “Indeterminacy related to A” and we denote it by “indetermA”; also we denote “Antia” by A ̅; where T, I, F are standard or nonstandard subsets of the nonstandard unitary interval, and the T is the chance that A occurs, denoted ch(A); I is the indeterminate chance related to A, ch(indetermA); and F is the chance that A does not occur, ch(A ̅).

So, NP is a generalization of the IP as well. Therefore, using other notations we have

NP(A) = (ch(A), ch(indeterm ), ch(A ̅)).

The universal set, endowed with a Neutrosophic probability defined for each of its subset, forms a Neutrosophic probability space.

Definition 6. Let (G, *) be any group, the Neutrosophic group is generated by I and G under * denoted by N(G) = {〈G ∪ I〉, *}.

Example 1. Let Z7 = {0, 1, 2,…, 6} be a group under addition modulo 7. N(G) = {〈Z7∪ I〉, ‘+’ modulo 7} is a Neutrosophic group which is in fact a group. For N(G) = {a + bI / a, b ∈Z7} is a group under ‘+’ modulo 7. Thus this Neutrosophic group is also a group.

Example 2. Consider the set G= Z5\{0}, G is a group under multiplication modulo 5. N(G) = {〈G ∪ I〉, under the binary operation, multiplication modulo 5}. N(G) is called the Neutrosophic group generated by G ∪ I. Clearly N(G) is not a group, for I = I and I is not the identity but only an indeterminate, but N(G) is defined as the Neutrosophic group.

Theorem 1. Let (G, *) be a group, N(G) = {〈G ∪ I〉, *} be the Neutrosophic group.

I. N(G) in general is not a group.

II. N(G) always contains a group.

Proof: To prove N(G) in general is not a group it is sufficient we give an example; consider 〈Z5 \ {0} ∪ I〉 = G = {1, 2, 4, 3, I, 2 I, 3 I, 4 I}; G is not a group under multiplication modulo 5. In fact {1, 2, 3, 4} is a group under multiplication modulo 5.

N(G) the Neutrosophic group will always contain a group because we generate the Neutrosophic group N(G) using the group G and I. So G ≠ ⊂N(G); hence N(G) will always contain a group.

Definition 7. Let N(G) be a Neutrosophic group. The number of distinct elements in N(G) is called the order of N(G). If the number of elements in N(G) is finite we call N(G) a finite Neutrosophic group; otherwise we call N(G) an infinite Neutrosophic group, we denote the order of N(G) by o(N(G)) or |N(G)|.

Definition 8. Let N(G) be a Neutrosophic group. An element x ∈ N(G) is said to be a Neutrosophic element if there exists a positive integer n such that xn = I, if for any y a Neutrosophic element no such n exists then we call y to be a Neutrosophic free element.

Definition 9. Let Gbe a set equipped with a binary, G: G × G → G. A Neutrosophicpseudogroup (G, . )is defined when it satisfies the following properties:

I. Neutrosophic closure property: For all elements a, b ∈ G, the result of the operation a . b is a Neutrosophic element in G.

II. Neutrosophic associativity property: For all elements a, b, c ∈ G, the operation satisfies a form of associativity that accounts for indeterminacy. That is, (a . b) . c and a . (b . c) are indeterminately associative. III. Neutrosophic identity: There exists an element e ∈ G (The Neutrosophic identity element) such that for all elements a ∈ G, the operation satisfies a . e ≈ e . a ≈ a, where ≈ denotes approximate equality under the Neutrosophic framework.

IV. Neutrosophic inverses: For each element a ∈ G, there exists an element b ∈ G (The Neutrosophic inverse of a) such that a . b ≈ b . a ≈ e, where ≈ denotes approximate equality under the Neutrosophic framework. V. Neutrosophic elements: Each element a ∈ G is represented as a Neutrosophicnumber a = (Ta, Ia, Fa), where Ta, Ia and Fa are the truth, indeterminacy, and falsehood components of a, respectively.

Definition 10. Let N(G) = 〈G ∪ I〉 be a neutrosophic group generated by G and I. A proper subset P(G) is said to be a Neutrosophic subgroup if P(G) is a Neutrosophic group i.e. P(G) must contain a (Sub) group.

Example 3. Let N(Z2) = 〈Z2∪ I〉 be a Neutrosophic group under addition. N(Z2) = {0, 1, I, 1 + I}. Now we see {0, I} is a group under + in fact a NEUTROSOPHIC group {0, 1 + I} is a group under ‘+’ but we call {0, I} or {0, 1 + I} only as pseudo Neutrosophic groups for they do not have a proper subset which is a group. So {0, I} and {0, 1 + I} will be only called as pseudo Neutrosophic groups (Subgroups).

We can thus define a pseudo Neutrosophic group as a Neutrosophic group, which does not contain a proper subset which is a group. Pseudo Neutrosophic subgroups can be found as a substructure of Neutrosophic groups. Thus a pseudo Neutrosophic group though has a group structure is not a Neutrosophic group and a Neutrosophic group cannot be a pseudo Neutrosophic group. Both concepts are different.

Definition 11. Let S be a semigroup, the semigroup generated by S and I i.e. S ∪ I denoted by 〈S ∪ I〉 is defined to be a Neutrosophic semigroup.

Note: All Neutrosophic semigroups contain a proper subset, which is a semigroup.

Definition 12. Let N(S) be a Neutrosophic semigroup. The number of distinct elements in N(S) is called the order of N(S), denoted by o(N(S)).

Definition 13. Let N(S) be a Neutrosophic semigroup. A proper subset P of N(S) is said to be a Neutrosophic subsemigroup, if P is a Neutrosophic semigroup under the operations of N (S). A Neutrosophic semigroup N(S) is said to have a subsemigroup if N(S) has a proper subset, which is a semigroup under the operations of N(S).

Theorem 2. Let N(S) be a Neutrosophic semigroup. Suppose P1, P2 be any two Neutrosophic subsemigroups of N(S) then P1∪P2 (i.e. the union) the union of two Neutrosophic subsemigroups in general need not be a Neutrosophic subsemigroup.

Multiple-Criteria Decision Making (MCDM) or Multiple-Criteria Decision Analysis (MCDA) is a sub-discipline of operations research1 that explicitly evaluates multiple conflicting criteria2 in DM3 (Both in daily life and in settings such as business, government and medicine). It is also known as multiple attribute utility theory, multiple attribute value theory, multiple attribute preference theory, and multi-objective decision analysis.It is particularly useful in complex decision-making scenarios where various criteria must be considered simultaneously. Common methods of MCDM includes:

I. Analytic Hierarchy Process (AHP): Developed by Thomas Saaty, AHP involves decomposing a complex decision problem into a hierarchy of simpler sub-problems. Each level of the hierarchy represents different criteria or sub-criteria, and pairwise comparisons are used to establish priorities.

II. Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS): TOPSIS ranks alternatives based on their distance from an ideal solution (The best possible scenario) and a nadir solution (The worst possible scenario). The alternative closest to the ideal solution and farthest from the nadir solution is preferred.

III. Elimination and Choice Expressing Reality (ELECTRE): ELECTRE is a family of MCDM methods that uses pairwise comparisons to eliminate less favorable alternatives. It is particularly useful for dealing with qualitative criteria.

IV. Multi-Attribute Utility Theory (MAUT): MAUT involves assigning utility values to different levels of criteria and then calculating a total utility score for each alternative. The alternative with the highest utility score is chosen V. Simple Additive Weighting (SAW): Also known as the weighted sum method, SAW involves assigning weights to criteria, multiplying the scores of alternatives by these weights, and summing the results to obtain a final score for each alternative.

VI. VlseKriterijumskaOptimizacija I KompromisnoResenje (VIKOR): VIKOR focuses on ranking and selecting from a set of alternatives, and determining compromise solutions, with respect to conflicting criteria.

I. Multi-refined Neutrosophic set has also been utilized in sentiment analysis for Data Mining. The model consisted of two positive (T), two negative (F), and three indeterminate (I) membership functions. Their

https://en.wikipedia.org/wiki/Operations_research https://en.wikipedia.org/wiki/Decision_making https://en.wiktionary.org/wiki/criterion model showed outperforming results when analysing tweets on ten different topics related to the Indian scenario and other international scenarios.

II. A utilized single-valued and interval-valued Neutrosophic graphs in Blockchain and bitcoin application was proposed. They also listed the advantages and limitations of Blockchain graphs. Another model was also proposed and their main focus was to select the most appropriate Blockchain model for providing a secure and trustworthy healthcare Blockchain solution. Their proposed model was well-presented yet had a set of limitations, including its generalization.

III. Neutrosophic PROMETHEE method in one-to-one marketing. They proved the necessity of analysing different aspects of potential buyers, including their emotional and physiological states. One of the main obstacles of their model was the collection, and governance of the customers’ emotions related data, especially under the existence of the data privacy rules exist in many companies. Consequently, collecting the proper data needed time and cost.

IV. From 2020 until publishing this research, the whole world is struggling with different generations of Coronavirus (i.e., COVID-19). Many researchers tried to contribute to developing models for dealing with COVID-19 using a different algorithm, and Neutrosophic logic was on top of these utilized algorithms. A frameworkwas developed that combined COVID-19’s disruptive technologies for analyzing this pandemic virus. Their framework had many advantages, including restricting COVID-19’s outbreaks and ensuring the healthcare team’s safety. The power of their model was applying it in an empirical case study. Meanwhile, it would be great to be generalized on other use cases.While some other researchers, differentiated between COVID-19 and other four chest diseases that had some common symptoms. They utilized Neutrosophic logic for this purpose to diagnose COVID-19 using only the CT scan and the primary symptoms. They also studied the effect of the Internet of Things (IOT) in helping the medical staff monitor the spread of COVID-19. Their proposed model achieved 98% detection accuracy. Yet, it was recommended to update their study by including other COVID-19’s symptoms added by the World Health Organization (WHO) related to the virus's evolution.

V. Some researchers combined Neutrosophic techniques with AHP method. Their main goal was to support enterprise decision-making in the Internet of Things (IoT) era. Their proposed algorithm was applied in different enterprises, including Smart Village in Egypt and Smart City in U.K. and China. Meanwhile, involving more companies in their model validation would enrich its results. Another set of researchers designed an interval complex Neutrosophic set and listed its characteristics. To prove their model's practicality, they applied it in supplier selection related to a transportation company. Meanwhile, it still needed to be applied to more real-life datasets to prove its robustness.Then, researchers developed a supply chain-related multi-criteria group Decision-Making Method. Their proposed model combined analytical network process method to VIKOR Method. Their model that was developed for a Neutrosophic environment that had incomplete information, utilized triangular membership function was applied in a real case study. It showed outperforming results. Its main limitation was the dependency on experts’ opinions and it was hard to find those experts fulfilled the researchers’ predefined requirements. Also, its dependency on a forecasting phase that needed to have large input data to have robust results. Meanwhile, it was recommended to generalize it to other applications. VIKOR Method was also utilized to guide the decision-making process in an uncertain environment in another research. For this purpose, they combined VIKOR method to the cubic Neutrosophic number. This combination was illustrated through an example but was recommended to be applied in real applications.

Example 6. A soccer club wants to choose a new player among 6 candidates, say P1, P2, P3, P4, P5 and P6. The desired qualifications of the new player are to be fast, younger than 30 years, higher than 1.70 m and experienced. Assume that P1, P2, P6 are the fast players, P2, P3, P5,P6 are the players being younger than 30 years, P3, P5 are the players with heights greater than 1.70 m and P4 is the unique experienced player. Assume that the technical manager of the soccer club, being not sure about the qualitative grades assigned to each of the 6 candidate players, he decided to proceed by replacing them by Neutrosophic triplets, in the way that we have previously described. As a result, the tabular matrix of the DM process took the form of the following table below. Which is the best decision for the club in this case ?

Solution: The choice value of each player in this case is defined to be the mean value of the Neutrosophic triplets of the line of the table in which he belongs. Thus, by

(tm, im, fm) = 1/n [n1(t1, i1, f11)+n2(t2, i2, f2)+….+nk(tk, ik, fk)].

The choice value of P1 is equal to

[(1, 0, 0)+2(0, 0, 1)+(0.6, 0.3, 0.1)] = (1.6, 0.3, 2.1) = (0.4, 0.075, 0.525).

In the same way one finds that the choice values of P2, P3, P4, P5 and P6 are (0.55, 0.005, 0.4), (0.775, 0.15, 0.075), (0.375, 0.05, 0.575), (0.775, 0.1, 0.125) and (0.6, 0.1, 0.3) respectively.

In this case, the club’s technical manager could use either an optimistic criterion by choosing the player with the greatest truth degree, or a conservative criterion by choosing the player with the lower falsity degree. Consequently, using the optimistic criterion he must choose one of the players P3 and P5, whereas using the conservative criterion he must choose the player P3. A combination of the two criteria leads to the final choice of player P3. Observe, however, that, since the indeterminacy degree of P3 is 0.15 and of P5 is 0.1, there is a slightly greater doubt about the qualifications of P3 with respect to the qualifications of P5. In other words, the choice of P3 is connected with a slightly greater risk. In final analysis, therefore, all the Neutrosophic components assigned to each player give useful information about his qualifications.

Example 7. The probability that candidate C will win an election is say 25% true (Percent of people voting for him), 35% false (Percent of people voting against him), and 40% indeterminate (Percent of people not coming to the ballot box, or giving a blank vote - not selecting anyone, or giving a negative vote-cutting all candidates on the list). Dialectic and dualism do not work in this case anymore.

Example 8. Another example, the probability that tomorrow it will rain is say 50% true according to meteorologists who have investigated the past years' weather, 30% false according to today's very sunny and droughty summer and 20% undecided (Indeterminate).