On Neutrosophic Offuninorms

Uninorms comprise an important kind of operator in fuzzy theory. They are obtained from the generalization of the t-norm and t-conorm axiomatic. Uninorms are theoretically remarkable, and furthermore, they have a wide range of applications. For that reason, when fuzzy sets have been generalized to others—e.g., intuitionistic fuzzy sets, interval-valued fuzzy sets, interval-valued intuitionistic fuzzy sets, or neutrosophic sets—then uninorm generalizations have emerged in those novel frameworks. Neutrosophic sets contain the notion of indeterminacy—which is caused by unknown, contradictory, and paradoxical information—and thus, it includes, aside from the membership and non-membership functions, an indeterminate-membership function. Also, the relationship among them does not satisfy any restriction. Along this line of generalizations, this paper aims to extend uninorms to the framework of neutrosophic offsets, which are called neutrosophic offuninorms. Offsets are neutrosophic sets such that their domains exceed the scope of the interval [0,1]. In the present paper, the definition, properties, and application areas of this new concept are provided. It is necessary to emphasize that the neutrosophic offuninorms are feasible for application in several fields, as we illustrate in this paper.


Introduction
Uninorms extend the t-norm and t-conorm axiomatic in fuzzy theory.They retain the axioms of commutativity, associativity, and monotony.Alternatively, they generalize the boundary condition, where the neutral element is any number lying in [0,1].Thus, t-norm and t-conorm are special cases of uninorms, t-norms have 1 as their neutral element and the neutral element of t-conorms is 0, see [1][2][3].
Uninorms are theoretically important, and moreover they have also been used as operators in several areas of application; for example, in image processing, to aggregate group decision criteria, among others, see [4][5][6][7][8].An exhaustive search on uninorm applications made by the authors of this paper yielded more than six hundred scientific articles that have been written in the last five years devoted to this subject.
Rudas et al. in [9] report that uninorms have been applied in diverse applications ranging, e.g., from defining Gross Domestic Product index in economics, to fusing sequences of DNA and RNA or combining information on taxonomies or dendograms in biology, and in the fusion of data provided by sensors of robotics in data mining, and in knowledge-based and intelligent systems.Particularly, they offer many examples in Decision Making, Utility Theory, Fuzzy Inference Systems, Multisensor Data Fusion, network aggregation in sensor networks, image approximation, Symmetry 2019, 11, 1136 4 of 26 Definition 1.Let X be a space of points (objects), with a generic element in X denoted by x.A Neutrosophic Set A in X is characterized by a truth-membership function T A (x), an indeterminacy-membership function I A (x), and a falsity-membership function F A (x). T A (x), I A (x), and F A (x) are real standard or nonstandard subsets of ] -0, 1 + [.There is no restriction on the sum of T A (x), I A (x), and F A (x), thus, -0≤ inf T A (x)+ inf I A (x) + inf F A (x) ≤ sup T A (x)+ sup I A (x) + sup F A (x)≤ 3 + (see [26]).
The neutrosophic sets are useful in their nonstandard form only in philosophy, in order to make a distinction between absolute truth (truth in all possible worlds-according to Leibniz) and relative truth (truth in at least one world), but not in technical applications, thus the Single-Valued Neutrosophic Sets are defined, see Definition 2. Definition 2. Let X be a space of points (objects), with a generic element in X denoted by x.A Single-Valued Neutrosophic Set A in X is characterized by a truth-membership function T A (x), an indeterminacy-membership function I A (x), and a falsity-membership function F A (x). T A (x), I A (x), and F A (x) are elements of [0,1].There is no restriction on the sum of T A (x), I A (x), and F A (x), thus, 0 ≤T A (x)+I A (x) + F A (x) ≤ 3 (see [38]).
The domain of the single-valued neutrosophic sets does not surpass the limits of the interval [0,1].This is a classical condition imposed in previous theories such as probability and fuzzy sets.Despite the past, Smarandache in 2007 proposed the membership >1 and <0 and illustrated this proposal; see [39] (pp.92-93) and the example given in the introduction of this paper.In the following, the Single-Valued Neutrosophic Oversets, Single-Valued Neutrosophic Undersets, and Single-Valued Neutrosophic Offsets are formally defined.Definition 3. Let X be a universe of discourse and the neutrosophic set A 1 ⊂X.Let T(x), I(x), F(x) be the functions that describe the degree of membership, indeterminate-membership, and non-membership respectively, of a generic element x∈X, with respect to the neutrosophic set A 1 : T, I, F: X→[0, Ω], where Ω> 1 is called overlimit, T(x), I(x), F(x)∈[0, Ω].A Single-Valued Neutrosophic Overset A 1 is defined as A 1 = (x, T(x), I(x), F(x)), x ∈ X , such that there exists at least one element in A 1 that has at least one neutrosophic component that is bigger than 1, and no element has neutrosophic components that are smaller than 0 (see [31]).Definition 4. Let X be a universe of discourse and the neutrosophic set A 2 ⊂X.Let T(x), I(x), F(x) be the functions that describe the degree of membership, indeterminate-membership, and non-membership, respectively, of a generic element x∈X, with respect to the neutrosophic set A 2 : T, I, F: X→[Ψ, 1], where Ψ< 0 is called underlimit, T(x), I(x), F(x)∈[Ψ, 1].A Single-Valued Neutrosophic Underset A 2 is defined as A 2 = (x, T(x), I(x), F(x)), x ∈ X , such that there exists at least one element in A 2 that has at least one neutrosophic component that is smaller than 0, and no element has neutrosophic components that are bigger than 1 (see [31]).Definition 5. Let X be a universe of discourse and the neutrosophic set A 3 ⊂X.Let T(x), I(x), F(x) be the functions that describe the degree of membership, indeterminate-membership, and non-membership respectively, of a generic element x∈X, with respect to the neutrosophic set A 3 : T, I, F: X→[Ψ, Ω], where Ψ< 0 < 1 <Ω, Ψ is called underlimit, while Ω is called overlimit, T(x), I(x), F(x)∈[Ψ, Ω].A Single-Valued Neutrosophic Offset A 3 is defined as A 3 = (x, T(x), I(x), F(x)), x ∈ X , such that there exists at least one element in A 3 that has at least one neutrosophic component that is bigger than 1, and at least another neutrosophic component that is smaller than 0 (see [31]).
Let us note that the oversets, undersets, and offsets cover the three possible cases to characterize.Now, the logical operations over these kinds of sets have to be redefined, in view that the classical ones cannot always be straightforwardly extended to these domains.This is the case of complement given by Smarandache in [31], whereas the union and intersection definitions do not change with respect to those of single-valued neutrosophic sets.This is summarized below: Let X be a universe of discourse, A = (x, T A (x), I A (x), F A (x) ), x ∈ X and B = (x, T B (x), I B (x), F B (x) ), x ∈ X be two single-valued neutrosophic oversets/undersets/offsets.T A , I A , F A , T B , I B , F B : X→[Ψ, Ω], where Ψ≤ 0< 1 ≤Ω, Ψ is the underlimit, whilst Ω is the overlimit, T A (x), I A (x), F A (x),T B (x), I B (x), F B (x)∈[Ψ, Ω].Let us remark that the three cases are here comprised, viz., overset when Ψ = 0 and Ω>1, underset when Ψ< 0 and Ω = 1, and offset when Ψ< 0 and Ω> 1.
Then, the main operators are defined as follows: Let us remark that when Ψ = 0 and Ω = 1, the precedent operators convert in the classical ones.With regard to logical operators, e.g., n-norms and n-conorms, their redefinitions in the offsets framework are not so evident.Below, definitions of offnegation, neutrosophic component n-offnorm, and neutrosophic component n-offconorm are provided.
One offnegation can be defined as in Equation (1).
To simplify the notation, sometimes we use Let us remark that the definition of the neutrosophic component n-offnorm is valid for every one of the components, thus, we have to apply it three times.Also, Definition 6 contains the definition of n-norm when Ψ = 0 and Ω = 1.
To simplify the notation sometimes we use Proof.The proof is equivalent to the proof of Proposition 1.
In this paper, we use the notion of lattice, based on the poset denoted by ≤ O , where T 1 , I 1 , F 1 ≤ O T 2 , I 2 , F 2 if and only if T 2 ≥ T 1 , I 2 ≤ I 1 and F 2 ≤ F 1 , where the infimum and the supremum of the set are Ψ, Ω, Ω and Ω, Ψ, Ψ , respectively.
One property that is preserved of n-norms is that the minimum is the biggest neutrosophic component n-offnorm for T O , as it is demonstrated in Proposition 1. Proposition 2 proved that the maximum is the smallest neutrosophic component n-offconorm for I O and F O when we consider ≤ O .
Evidently, the minimum is a neutrosophic component n-offnorm and the maximum is a neutrosophic component n-offconorm; see Example 1. Example 2 extends the Łukasiewicz t-norm and t-conorm to the neutrosophic offsets.Let us remark that the simple product t-norm and its dual t-conorm cannot be extended to this new domain.
Finally, we recall the definition of neutrosophic uninorms that appeared in [25], see Definition 8.

Definition 8.
A neutrosophic uninorm U N is a commutative, increasing, and associative mapping, U N : (] − 0, 1 F x , y T y , I y , F y = U N T(x, y), U N I(x, y), U N F(x, y) , where U N T means the degree of membership, U N I the degree of indeterminacy, and U N F the degree of non-membership of both x and y.Additionally, there exists a neutral element e ∈ ] − 0, 1 Let us observe that this definition can be restricted to single-valued neutrosophic sets.Neutrosophic uninorms generalize n-norms, n-conorms, uninorms in L*-fuzzy set theory, and fuzzy uninorms.

On Neutrosophic Offuninorms
This section contains the core of the present paper.It is devoted to exposing the definitions and properties of the neutrosophic offuninorms.
The definition of a neutrosophic uninorm is an especial case of neutrosophic offuninorm when Ψ = 0 and Ω = 1 (see Definition 8) and, additionally, we are dealing with single-valued neutrosophic sets.
It is easy to prove that the neutral element e is unique.Let c be a neutrosophic component (T O , I O or F O ). c: where Ψ≤ 0 and Ω≥1.Let us define four useful functions, ), Ω], defined in Equations ( 2)-( 5), respectively.
where, the superscript -1 means it is an inverse mapping.If the condition c(e) ∈ (Ψ, Ω) is fulfilled, then the degenerate cases Ω = Ψ, c(e) = Ψ and c(e) = Ω are excluded.Therefore, ϕ 1 (c(x)) and ϕ 2 (c(x)) are well-defined non-constant linear functions.Thus, they are bijective and have inverse mappings defined in Equations ( 3) and ( 5), respectively, in the sense that for c(x) . These properties can be easily verified.Also, it is trivial that they are non-decreasing mappings.
, min and max are non-decreasing mappings, thus both To prove c(e) is the neutral element, we have two cases, which are the following: Therefore, identity is satisfied.6) and (7) for c(e) ∈ (Ψ, Ω).They are associative.
Proof.Four cases are possible: ).These proofs are also valid for U D . iii.
Thus, U C satisfies the associativity.Similarly, associativity of U D can be proved.
Let us remark that we applied the properties, c(x) Proof.Since Lemma 1, they are commutative, non-decreasing operators, and c(e) is the neutral element.Since Lemma 2, they are associative operators.Moreover, it is easy to verify that U C (Ψ, Ω) = Ψ and U D (Ψ, Ω) = Ω.
Example 4. Two neutrosophic component n-offuninorms can be defined as where ∧ LO and ∨ LO were defined in the Example 2; c(e)∈(Ψ, Ω).
Proof.Evidently, both operators are commutative, since N u O is.Also, it is non-decreasing since N u O and the functions in Equations ( 2)-( 5) are.They are associative because of the associativity of N u O .
It is easy to verify that the overbounding conditions are also satisfied.
Additionally, we have Proof.Let us define the function , expressed in Equations ( 10) and (11), respectively.
Evidently, they are increasing bijective mappings.
Conversely, if we have Nu O (•, •), we can define ÛN (•, •) as follows: Let us remark that we maintain the definition of inverse mapping that we explained in Equations ( 3) and (5).
In agreement with Proposition 5, many predefined neutrosophic uninorms can be used to define n-offuninorms.In turn, fuzzy uninorms can be used to define neutrosophic uninorms, thus, it is simply necessary to find examples in the field of fuzzy uninorms; see further Section 4.1.First, let us make reference to some properties of n-offuninorms.
Given the neutrosophic component n-offuninorm If there exists y = T O (y), Proof.
See that we applied the commutativity and associativity of Then, according to the previous results we have Let us assume without loss of generality that satisfying that at least one of Ψ 1 and Ψ 2 is smaller than 0, or at least one of Ω 1 and Ω 2 is bigger than 1, then, a neutrosophic component n-offuninorm aggregates both of them, according to the interpretation we have to obtain.

Applications
In the following, we illustrate the applicability of the present investigation aided by three areas of application.

N-Offuninorms and MYCIN
Let us start with the parameterized Silvert uninorms, see [40]: where λ > 0 and c N (e λ ) = 1 λ+1 .To convert this family to the equivalent one defined into [−1, 1] we have to apply the Equations in Proposition 5.Then, it is obtained Let us note that lim An additional consequence of these assertions is that inequalities 0<λ 1 <λ 2 imply Applying Equations ( 2)-( 5) to the conditions of the present example, the following transformations are obtained: 1+λ .Then, a neutrosophic component n-offnorm and a neutrosophic component n-offconorm are defined from Equations ( 8) and ( 9), as follows: , respectively.Other properties of u Oλ (•, •) are the following: To prove those inequalities are strict, let us suppose the equation Then, we conclude it is Archimedean.
u O1 (•, •) means the combination of the CFs of two independent experts about the hypothesis H. CF = -1.0means expert has 100% evidence against H and CF = 1.0 means he or she has 100% evidence to support H. The smaller the CF, the greater the evidence against H; the larger the CF, the greater the evidence supporting H; whereas evidence with degree close to 0 means a borderline degree of evidence.Here, u O1 (c O (x), −c O (x)) = 0, where u O1 (−1, 1) = u O1 (1, −1) = −1 for meaning that the 100% contradiction is assessed as 100% against H.The original u O1 (•, •) in [32] accepts they are undefined.
Another function is the Modified Combining Function C(x,y), see [34], defined as The components n-offnorm and n-offconorm obtained from the PROSPECTOR are the following: ) − 1, respectively, see Figures 1 and 2. respectively, see Figures 1 and 2.  Hitherto we mostly calculated on neutrosophic components, nevertheless n-offuninorms have to be defined for the three components altogether.For example, given x, y Conjunctive and disjunctive neutrosophic component n-offuninorms were illustrated in Example 3; see also Example 5. Example 6 is a hypothetical example to explain the use of this theory in a real-life situation.Example 6.Three physicians, denoted by A, B, and C, have to emit a criterion about a patient's disease which suffers from somewhat confusing symptoms.They agree that the Certainty Factor is the better way to express their opinions.They use single-valued neutrosophic offsets, instead of a simple CF to increase the accuracy of the criteria.
Conjunctive and disjunctive neutrosophic component n-offuninorms were illustrated in Example 3; see also Example 5. Example 6 is a hypothetical example to explain the use of this theory in a real-life situation.Physician A thinks that the probability they are dealing with a thyroid disease is A T = <−0.6,0.4, 0.6> and that it is an infectious disease is A I = <0.8,−0.5, −0.8>, thus, A is 60% against H T and 40% undecided about it; however, A is 80% in favor of H I and 50% sure about it.

Example 6. Three physicians, denoted by A, B, and C, have to emit a criterion about a patient's disease which suffers from somewhat confusing symptoms. They agree that the Certainty Factor is the better way to express their opinions. They use single-valued neutrosophic offsets, instead of a simple CF to increase the accuracy of the criteria.
Similarly To decide what is the strongest hypothesis, H T or H I , they select the well-known PROSPECTOR function used in MYCIN (see Equation ( 12)) for each component.
Despite we proved in Proposition 5 that neutrosophic uninorms are mathematically equivalent to offuninorms, it is worthwhile to remark that the reason for using an interval different of [0, 1] is that it could be useful to model real-life problems.The present example is a good one to explain that reason.The advantages arise from the accuracy and compactness of an expert's information.In this example, from an expert's viewpoint, it is easier to express opinions in the scale [−1, 1] with the aforementioned meaning than in the scale [0, 1], which is less clear.Information compactness is given because of only a single offset is semantically equivalent to at least two neutrosophic sets.
Additionally, because of the significance of functions like u O1 (•, •) and C(x,y), which were used as aggregation functions in that well-known expert system, some authors have extended the domain of fuzzy uninorms to any interval [a, b], not necessarily restricted to a = 0 and b = 1; see [33,34].
This fact supports the usefulness of the present work, where for the first time the precedent ideas on extending the truth values beyond the scope of [0, 1] naturally associate with the offset concept maintaining the original definitions of the aggregation functions used in MYCIN.
Another powerful reason is the applicability of u O1 (•, •) and C(x,y), and hence of the fuzzy uninorms defined in [a, b], as threshold functions of artificial neurons in Artificial Neural Networks, as well as to Fuzzy Cognitive Maps, which are used in fields like decision making, forecasting, and strategic planning [33].
Such applications of uninorms in the fuzzy domain can be explored in the framework of neutrosophy theory, e.g., in Artificial Neural Networks based on neutrosophic sets, in Neutrosophic Cognitive Maps, among others [36,37].

N-Offuninorms and Implicators
Fuzzy uninorms are used to define implicators (see [41], pp.151-160).This application was extended to neutrosophic uninorms ( [25]).To extend the implication operator in the offuninorm framework, first, we need to consider the notion of offimplication, which has been defined symbolically.
The Symbolic Neutrosophic Offlogic Operators or briefly the Symbolic Neutrosophic Offoperators extend the Symbolic Neutrosophic Logic Operators, where every one of T, I, F has an under and an over version (see [31], pp.132-139).
T O = Over Truth, T U = Under Truth; I O = Over Indeterminacy, I U = Under Indeterminacy; F O = Over Falsehood, F U = Under Falsehood.Let S N = {T O , T, T U , I O , I, I U , F O , F, F U } be the set of neutrosophic symbols, an order is defined in S N as follows: if '<' denotes "more important than", we have the following order, T U < 3. Let us note that the proposed order is not the unique one, it depends on the decision maker's objective.Let us observe that I is the center of the elements according to <.For every α ∈ S N , the symbolic neutrosophic offcomplement is denoted by C O (α) and it is defined as the symmetric element respect to the median centered in I, e.g., C SO (F O ) = F U and C SO (F) = T, hence, given α ∈ S N its symbolic neutrosophic offnegation is   α = C SO (α).Additionally, for any α, β ∈ S N the symbolic neutrosophic offconjunction is defined as α ∧  β = min(α, β), the symbolic neutrosophic offdisjunction is defined as α ∨  β = max(α, β), whereas the symbolic neutrosophic offimplication is defined in Equation (13).
The neutrosophic offnegation satisfies the following properties: 1.It is a non-increasing operator, which extends the classical negation operator in fuzzy logic theory.It is strictly decreasing when Ω + Ψ = 1. 2. It extends the notion of symbolic neutrosophic offnegation because satisfies the following properties: 2.1.It is centered in 0.5, i.e.,   0.5 = 0.5, therefore I = 0.5.Let us observe that I is the center of the elements according to <.For every α ∈ S N , the symbolic neutrosophic offcomplement is denoted by C O (α) and it is defined as the symmetric element respect to the median centered in I, e.g., C SO (F O ) = F U and C SO (F) = T, hence, given α ∈ S N its symbolic neutrosophic offnegation is Additionally, for any α, β ∈ S N the symbolic neutrosophic offconjunction is defined as α ∧ SO β = min(α, β), the symbolic neutrosophic offdisjunction is defined as α ∨ SO β = max(α, β), whereas the symbolic neutrosophic offimplication is defined in Equation (13).
The neutrosophic offnegation satisfies the following properties: It is a non-increasing operator, which extends the classical negation operator in fuzzy logic theory.It is strictly decreasing when Ω + Ψ = 1.

2.5
When

3.
If The precedent properties are easy to demonstrate.
Hence, the definition of offimplication where, N and O is the offnegation defined in Equation (14).
Example 7. One illustrative example of Equation ( 16) is obtained revisiting Section 4.1, by defining the following neutrosophic component n-offnorm: This is the transformation of Silvert uninorms to the domain [−1, 2] 2 applying the functions in Equations ( 10) and (11), and the transformation in Proposition 5. Also, let us take U ZD (c(x), c(y)) of Example 3. See that [−1, 2] is symmetric respect to 0.5, and the neutral element is 0.5.
Then, we study the offuninorm defined in the following equation: Thus, we define the offimplication generated by U O (•, •) according to Equation ( 16) as follows: where in this case we have U ZD T O (α), This is the transformation of Silvert uninorms to the domain [−1, 2] 2 applying the functions in Equations ( 10) and (11), and the transformation in Proposition 5. Also, let us take U ZD ((), ()) of Example 3. See that [−1, 2] is symmetric respect to 0.5, and the neutral element is 0.5.
Thus, we define the offimplication generated by   (•,•) according to Equation ( 16) as follows: where in this case we have    This is the transformation of Silvert uninorms to the domain [−1, 2] 2 applying the functions in Equations ( 10) and (11), and the transformation in Proposition 5. Also, let us take U ZD ((), ()) of Example 3. See that [−1, 2] is symmetric respect to 0.5, and the neutral element is 0.5.
Thus, we define the offimplication generated by   (•,•) according to Equation ( 16) as follows: where in this case we have    This offimplicator satisfies the overbounding conditions It is easy to check that substituting u O (•, , we obtain the more classical equations 0, 1, 1

N-Offuninorms and Voting Games
The applicability of uninorms to solve group decision problems is evident.However, the use of them as part of a game theory solution is not so obvious.This subsection is devoted to solving voting games based on n-offuninorms.
A cooperative game with transferable utility consists of a pair (N,v), where N = {1, 2, . . .,n} is a non-empty set of players,n ∈ N and v: 2 N →R, i.e., v(•) is a function of the power set of N such that each coalition or S⊆ N is associated with a real number.v is called characteristic function and v(S) represents the conjoint payoff of players in S. Additionally, v(∅) = 0 (see [42], p. 2).
A simple game models voting situations.It is a cooperative game such that for every coalition S, either v(S) = 0 or v(S) = 1, and v(N) = 1 (see [42], p. 7).
One solution is the Shapley-Shubik index, which is the Shapley value to simple games (see [42], pp.6-7).The equation of Shapley value is the following: where |S| is the cardinality of coalition S, |N| is the cardinality of the set of players or grand coalition and φ i (v) is the value assigned to player i in the game.This is the unique solution which satisfies the following axioms: If i is such that for every coalition S the equation v(S ∪ {i}) = v(S) holds, then φ i (v) = 0 (Dummy), • Given v and w two games over N, then This value is the sum of the terms [v(S ∪ {i}) − v(S)], which mean the marginal contribution of player i to the coalitions S, multiplied by which is the probability that |S| − 1 players precede player i in the game and |N| − |S| players follow him or her.Thus, the Shapley value of i is the expected marginal contribution of i to the game (see [42], p. 7).The result of the Shapley-Shubik index is interpreted as a measure of each player's power.
The n-offgame is interpreted in the following way: 1.
Experts forecast that voters will rank coalition S in the k th position of their preference, also they cannot decide if S will be ranked in the l th position.The first place or k = 1 corresponds to the preferred coalition of all and so on.Additionally, the n-offgame must satisfy the following rules: 2.
Given any two coalitions S 1 and S 2 , S 1 S 2 , we have the first component that both v(S 1 ) and v(S 2 ) are different.Thus, every coalition is associated with a unique number in the order of preference.

3.
v(S) = (k,k,2 n -k+1) means experts have no doubt that coalition S will be voted in the k th position.
Let us observe that it is not a simple game.This game can be interpreted as a multicriteria decision-making problem, where its solution is a measure of every player's power in the game Shapley value can be the solution to voting n-offgames, in the form given in Equation ( 18): Let us note that the minus sign in the expression was taken for convenience because the rank we applied is decreasing respect to the coalition´s significance.Additionally, v(S ∪ {i}) − v(S) is the difference between two 3-tuple values, thus the operation (k 1 , l 1 , 18) means the expected number of places won or lost in voter preference, as predicted by experts.
Apparently, Shapley value cannot be the solution to this problem because v(∅) 0 and v(•) is not a game.However, if we take in that v(S) = (k,l,2 n -k+1) in fact represents three games, namely, v 1 (S)=k, v 2 (S)= l, and v 3 (S) =2 n -k+1, one per component and additionally taking into account they are linear transformations of three games with characteristic functions w 1 , w 2, , and w 3 ; where w 1 (S)= 2 n − v 1 (S), w 2 (S)= 2 n − v 2 (S), and w 3 (S) = 1 − v 3 (S), then, the marginal contributions of the three pairs, w 1 (•) and v 1 (•), w 2 (•) and v 2 (•), w 3 (•) and v 3 (•), are the same except for the sign.Thus, these three pairs have the same Shapley value except for the sign and therefore this property is extended to v(•) and w(•).
Shapley value is a rational solution to the game, nevertheless, it can differ from actual human behavior, as Zhang et al. suggested in [43] to model restrictions in game decisions according to the human behavior based on fuzzy uninorms.Therefore, we propose n-offuninorms to explore other behaviors in human decision making by recursively applying an n-offuninorm to every pair of values Here we explore n-offuninorms defined on [−L, L], L = 2 n −1 and with the PROSPECTOR parameterized function with λ> 0 and neutral element e = L 1−λ 1+λ , see Equation (19).

3.
Let S j be the set of coalitions not containing i, and j = 1, 2, . . ., 2 n−1 .Let us take a i1 = v(S 1 ) and a i2 = v(S 2 ) and calculate a prev = U Oλ and go to step 4. Let us point out that in the precedent algorithm the associativity of n-offuninorms was used.Moreover, the algebraic sum in Shapley value and the n-offuninorms yield to somewhat similar results.Thus, for U oλ (•,•) with λ = 1, we have that x, y < 0 imply both U oλ (x,y)<min(x,y) and x+y< min(x,y), whereas when x, y > 0, we have U oλ (x,y)>max(x,y) and x+y> max(x,y).For x,y satisfying x•y<0, then both U oλ (x,y) and x+y are compensatory operators, and finally 0 is the neutral element of The solutions in Table 3 prove that the greater λ, the greater the solution values.Thus, when λ is increased, its associated solution models more optimistic behavior with respect to the first component, which is compensated with more pessimistic behavior with respect to the third component.
The advantages of the proposed approach are more evident when it is compared with a classical one restricted to {0, 1}.Here we used a semantic represented with natural numbers and we calculated directly on them.In contrast, for applying classical definitions in {0, 1}, we would need to define eight Boolean functions, one per element.What is more, some operations such as marginal contributions, which is an algebraic difference, cannot be directly applied in the logic sense.
In case we would need to extend the approaches to the continuous gradation, then a continuous ranking can be modeled with the identity line I d (x) = x, but in the classical approach, eight memberships functions would have to be considered, where the simplest ones are triangular (see Figure 6).From Figure 6 we can infer that there exists a transformation between both models; however, the proposed model is the simplest one.

5.Discussion
Neutrosophic oversets, undersets, and offsets are concepts of a novel and non-conventional theory of uncertainty.Historically, the convention of restricting logic to the interval [0, 1] has dominated fuzzy logic and its generalizations.Possibly this is a legacy of probability and mathematical logic, where, semantically speaking, 0 and 1 have been considered the two extreme opposite sides.Therefore, oversets, undersets, and offsets can be understood as controversial subjects.Nevertheless, Smarandache in [31] illustrates with some examples that such sets, of which their domains surpass the scope of [0, 1], could be useful to represent knowledge in a valid semantic.

Discussion
Neutrosophic oversets, undersets, and offsets are concepts of a novel and non-conventional theory of uncertainty.Historically, the convention of restricting logic to the interval [0, 1] has dominated fuzzy logic and its generalizations.Possibly this is a legacy of probability and mathematical logic, where, semantically speaking, 0 and 1 have been considered the two extreme opposite sides.Therefore, oversets, undersets, and offsets can be understood as controversial subjects.Nevertheless, Smarandache in [31] illustrates with some examples that such sets, of which their domains surpass the scope of [0, 1], could be useful to represent knowledge in a valid semantic.This is a recent theory that needs more developing and the scientific community's acknowledgment of its usefulness.One of our aims with this paper is to demonstrate that this theory can be useful.To achieve this end, we introduced the uninorm theory in the neutrosophic offset framework.This union is manifold advantageous, the most evident one being that we have provided a new aggregator operator to these sets.As we mentioned in the introduction, there exists a wide variety of fuzzy uninorm applications, namely, Decision Making [9,14,15], DNA and RNA fusion [9], logic [17], Artificial Neural Networks [16], among others.Uninorm is more flexible than t-norm and t-conorm because it includes the compensatory property in some cases, which is more realistic for modeling human decision making, as was experimentally proved by Zimmermann in [21].
Also, uninorms have enriched other theories when they were generalized to other frameworks.In L*-fuzzy set theory [23], uninorms also aggregate independent non-membership functions to achieve more precision.Moreover, neutrosophic uninorms aggregate the indeterminate-membership functions [25].
Additionally, some authors have associated uninorms with non-conventional theories.In [33,34] we can find some attempts to extend uninorm domains to an interval [a, b].The reason is that the PROSPECTOR function related to the MYCIN Expert System is one very important milestone in Artificial Intelligence history.The point is that the PROSPECTOR function is basically a uninorm except it is defined in the interval [−1, 1], thus, we can consider intervals greater than [0, 1].They have argued that there exist two reasons to maintain the interval [−1, 1]-the first one is the importance of the PROSPECTOR function, the second one is the facility to interchange information among users and decision makers in form of degrees to accept or reject hypotheses.
The second non-conventional approach is the bipolar or Multi-Polar uninorms defined in [24].The world is (and some people are) is evidently multi-polar; in case of bipolarity they are modeled in [−1, 1].Especially in [24], we have a multi-polar space consisting of an ordered pair of (k, x), where k∈{1, 2, . . ., n} represents a category or class and x∈(0, 1], with the convention 0 = (k, 0) for every category.This is a more complex representation that takes a unique interval [−n, n] where, for x∈[−n, n], the function round(x) represents the category and its fractional part represents the degree of membership to that category.This is a real extension of bipolarity in [−1, 1] to multi-polarity.In [31] (pp.127, 130) Tripolar offsets and Multi-polar offsets are defined.We illustrated in Example 8 that considering the semantic values belong to {−n,−n+1, . . ., 0, 1, . . ., n} could be advantageous.
The definition of uninorm-based implicators is not new in literature, they can be seen in [41] (pp.151-160) for fuzzy uninorms, in [17] it is extended for type 2 fuzzy sets, in [24] for L*-fuzzy set theory, and in [25] for neutrosophic uninorms.In the present paper, uninorm-based offimplicators are defined, however, we only counted on symbolic offimplication operators (see [31], p. 139).To extend this definition to a continuous framework, we had to extend the symbolic offnegation to a continuous one.
Finally, we preferred to illustrate a voting game solution instead of a group decision method because the relationship of offuninorms with the latter subject is predictable.However, to find any game theory associated with uninorms is uncommon in literature.One remarkable example can be seen in [43], where a behavioral approach has been made to certain kind of games, where uninorms model the humans' restrictions to make the division of gains among the players.
In the present paper, another approach is proposed where an indeterminacy component is taken into account.Also, we proved that modeling with a natural number semantic is simpler than to utilize the classical [0, 1] interval, because of the fact that n membership functions can be substituted by a linear identity function.We basically defined the voting game solution since the Shapley-Shubik index components (see [42], pp.6-7), where we only changed the algebraic sum by offuninorms.The classical approaches such as the Shapley-Shubik index are interested in a rational and fair solution; nevertheless, many times that does not occur in real negotiations and then behavioral solutions are needed.

Proposition 1 .Definition 7 .
Let N n O (•, •) be a neutrosophic component n-offnorm, then, for any elements x, y ∈M O we have N n O (c(x), c(y)) ≤ min(c(x), c(y)).Proof.Because of the monotonicity of the neutrosophic component n-offnorm and one of the overbounding conditions, we have N n O (c(x), c(y)) ≤ N n O (c(x), Ω) = c(x), hence N n O (c(x), c(y)) ≤ c(x) and similarly N n O (c(x), c(y)) ≤ c(y) can be proved, therefore, N n O (c(x), c(y)) ≤ min(c(x), c(y)).See that Proposition 1 maintains this property of the n-norms.Likewise to the definition of the neutrosophic component n-offnorm, in Definition 7 it is described the neutrosophic component n-offconorm.Let c be a neutrosophic component (T O , I O or F O ). c: M O →[Ψ, Ω], where Ψ≤ 0 and Ω ≥1.The neutrosophic component n-offconorm N co O : [Ψ, Ω] 2 → [Ψ, Ω] satisfies the following conditions for any elements x, y, and z ∈M O

Definition 9 .
Let c be a neutrosophic component (T O , I O or F O ). c: M O →[Ψ, Ω], where Ψ≤ 0 and Ω≥1.The neutrosophic component n-offuninorm N u O : [Ψ, Ω] 2 → [Ψ, Ω] satisfies the following conditions for any elements x, y, and z ∈M O : i.There exists c(e)∈M O , such that N u O

Lemma 2 .
Let c be a neutrosophic component (T O , I O , or F O ). c: M O →[Ψ, Ω], where Ψ≤ 0 and Ω≥1.Given ∧ O a neutrosophic component n-offnorm and ∨ O a neutrosophic component n-offconorm, let us consider U C (c(x), c(y)) and U D (c(x), c(y)) the operators defined in Equations (

Proposition 4 .
and only if the neutrosophic component n-offnorm and n-offconorm are Archimedean.Let us observe that < O is the order < defined in the real line when c(x) is T O (x) and it is > when c(x) is I O (x) or F O (x).Let c be a neutrosophic component (T O , I O or F O ). c: M O →[Ψ, Ω], where Ψ< 0 and Ω> 1, and let a neutrosophic component n-offuninorm N u O : [Ψ, Ω] 2 → [Ψ, Ω] .Then, for every x, y ∈ M O , a neutrosophic component n-offnorm and a neutrosophic component n-offconorm are defined by Equations (

Symmetry 2019, 11 , 1136 11 of 26 Proposition 5 .
Let (T O , I O , or F O ), c O : M O →[Ψ, Ω] and (T, I, or F), c N : MN→[0,1] be a neutrosophic component n-offset and a neutrosophic component, respectively.There exists a bijective mapping such that every neutrosophic component n-offuninormis transformed into a neutrosophic component uninorm and vice versa.
x) .Suppose x and e = T O (e), I O (e), F O (e) are ≤ O -incomparable, i.e., x O e and e O x.Then,

Symmetry 2019 ,
11, x FOR PEER REVIEW 16 of 25 <  <  < F O < I O <  O , where −∞ <  U < I U < F U < 0 , 0 ≤ F <  <  ≤ 1 and 1 < F O < I O <  O < +∞; see Figure3.Let us note that the proposed order is not the unique one, it depends on the decision maker's objective.
otherwise , see Figure 4, and u O (•, •) models the neutrosophic n-components I O and F O , see Figure 5.

Figure 4 .
Figure 4. Depiction of the neutrosophic n-offimplication generated by UZD for TO.

Figure 5 .
Figure 5. Depiction of the neutrosophic n-offimplication generated by uO for both, IO and FO.

Figure 4 .
Figure 4. Depiction of the neutrosophic n-offimplication generated by U ZD for T O .

Figure 4 .
Figure 4. Depiction of the neutrosophic n-offimplication generated by UZD for TO.

Figure 5 .
Figure 5. Depiction of the neutrosophic n-offimplication generated by uO for both, IO and FO.Figure 5. Depiction of the neutrosophic n-offimplication generated by u O for both, I O and F O .

Figure 5 .
Figure 5. Depiction of the neutrosophic n-offimplication generated by uO for both, IO and FO.Figure 5. Depiction of the neutrosophic n-offimplication generated by u O for both, I O and F O .
forecasted experts' ranking of the coalitions.Each coalition can represent a bloc of political parties.

Figure 6 .
Figure 6 .Depiction of two kinds of 3-person game modeling.Classical [0, 1] is represented in dashed lines and triangular membership functions, whereas the solid line represents the solution based on offsets.The points represent the Boolean restrictions.

Figure 6 .
Figure 6.Depiction of two kinds of 3-person game modeling.Classical [0, 1] is represented in dashed lines and triangular membership functions, whereas the solid line represents the solution based on offsets.The points represent the Boolean restrictions.