Fuzzy Sets and Algebraic Hyperoperations to Model Interpersonal Relations Sarka Hoskova-Mayerova1 and Antonio Maturo2 1
University of Defence Brno, Department of Mathematics and Physics, Kounicova 65, 66210 Brno, Czech Republic
[email protected]
2
University of Pescara-Chieti, Department of Humanities, Arts and Social Sciences, Via dei Vestini 31, 66100 Chieti, Italy
[email protected]
Abstract: In Social Sciences propositions and descriptions of interpersonal relationships are given by means of linguistic expressions, which cannot be formalized with the classic binary logic. Then a necessary tool are fuzzy sets that give the possibility to measure the degree of belonging of an element to a set described by a linguistic property or the degree of a relation between individuals. Moreover many times there is uncertainty on the result of an aggregation operation and we can have the necessity to consider together many possible results of the interaction of any ordered pair of elements, e.g. individuals. We propose the algebraic hyperoperations as very useful instruments to manage these types of uncertainty. We show as fuzzy relations and hypergroupoids permit to have an efficient representations of many aspects of social phenomena. Keywords: Social Relations, fuzzy relations, hypergroupoids, mathematical models for social sciences.
1.
Introduction
Social Relations are described with mathematical models by many authors (e.g. Moreno, 1945, 1951) and al. A more complete description can be obtained if we utilize instruments to manage semantic uncertainty. At this aim we propose to introduce models based on fuzzy sets and algebraic hyperoperations. The first one give the possibility to measure the degree of belonging of an element to a set described by a linguistic property or the degree of a relation between individuals, the second one permit to consider together many possible results of the interaction of any ordered pair of elements. In Social Sciences these aspects are very important and the instrument we propose can be very efficient for modelling the complexity of social phenomena. The paper is organized as follows: Firstly, in Section 2, we review some basic definitions on fuzzy sets and fuzzy relations and discuss the modelling of social relations with fuzzy sets. In Section 3, we give an introduction to the theory of algebraic hyperstructures, its history and motivation, and discuss the formalizing social relations with this mathematical tool. In particular we introduce some particular social relations and the correspondent associated algebraic hyperstructures for modelling these relations. In Section 4, we study some mathematical properties of the introduced relations and hypergroupoids and their social meanings. Finally, in Section 5 we present some conclusions and perspective of research in modelling problems of social behaviors, and coalitions, with fuzzy relations and hypergroupoids.
2.
Modelling social relations with fuzzy sets
2.1 Basic definition on fuzzy sets and fuzzy relations Let us recall some basic results on the fuzzy relations that we use in the following.
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Let U be a nonempty set (in Social Science it can be, e.g., a set of individuals, or a set of media, or a set of social conditions, etc.). A fuzzy set on U is a function : U [0,1]. The meaning of such a function is that it represents a linguistic property in the set U, (e.g., the property of individuals to be clever, old, educated, etc.), and for every individual xU, (x) is the degree to which the property holds. If (x)=1, then the individual x has the property in the maximum possible degree. If (x)=0 then the individual does not meet the property at all. Let us denote by F(U) the set of all fuzzy sets on U. Zadeh, in many papers (Zadeh, 1965, 1975), managed to impose the theory of fuzzy sets, showing significant and consistent applications. Some important books on fuzzy sets and fuzzy logic are (Klir, Yuan, 1995; Ross, 1995; Dubois, Prade, 1988). Some interesting applications of fuzzy sets to Social Sciences are considered by Ragin in (Ragin, 2000). A fuzzy relation R on U is a fuzzy set on UU, i.e., a function R: UU [0,1]. For every (x,y)U we write also xRy to denote R(x, y). The meaning of a fuzzy relation is the mathematical representation of a linguistic relation in the set U, (e.g., the property of the individual x to be friend, trustful, relative, to the individual y, or dominated by y etc.). If xRy = 1, then the individual x has the relation with y in the maximum possible degree. On the contrary, if xRy = 0 then the individual x does not have the relation with y at all. We denote by R(U) the set of all fuzzy relations on U.
2.2 Fuzzy relations in Social Sciences One of the bases of Social Sciences are the relations among persons. Usually a relation is described by a linguistic proposition and fuzzy sets are a tool to represent in mathematical form concepts expressed in a imprecise form. If U is a set of individuals (the universal set to consider), usually it is not possible to affirm that for an ordered pair (x,y) of persons belonging to U a relation R given in a linguistic form holds or not at all, as happens in a binary context. For instance, let us consider one of the linguistic relations “friendship”, “cooperation between colleagues”, “affection for a person”, “confidence in a person for a job” and other interpersonal relations. A correct and overall complete modeling of each of these relations is obtained only if we assign to every pair (x,y)U×U a real number R(x,y) = xRy belonging to interval [0,1] that measures the extent to which a suitable decision maker (e.g. a mediator) believes the relation holds. This leads us to model the relation, as a fuzzy set with domain U×U, that is a function: R: (x,y) U × U R(x,y) [0,1].
(2.1)
Another important aspect consists in a fuzzy relation between a set U of persons and a set of objects. It is formalized by a fuzzy set with domain U × , i.e. a function: : (x,) U × (x,) [0,1].
(2.2)
For instance, let U be a set of children and a set of media. Some example of media that transmit information to children and then are in relation with children are books, television, internet, cinemas, parents, teachers, and so on. Some possible fuzzy relations are: “confidence in the media”, “dependence on the media”, “concentration in the use of the media”, and so on. If is one of these fuzzy relations, (x,) means the degree to which x is in the relation with .
3.
Modelling social relations with algebraic hyperstructures
3.1. Basic definition on hyperstructures The first paper on the hyperstructures date back to 1934 (Marty, 1934). Thereafter there are some works on the subject (Eaton, 1940; Griffit, 1938), but the theory has developed especially since the book (Prenowitz, Jantosciak, 1979) and was consolidated with the publication of the book (Corsini, 1993). Since then he has developed impetuous. In order to propose some applications for Social Sciences let us introduce the necessary basic concepts. A hypergroupoid is a non-empty set G, together with a multivalued function : G × G G, sometimes called the multiplication on G, i.e. a function : G×G P*(G) = P(G) – {}, that to every ordered pair of elements of
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G associates a nonempty subset of G. Usually, we write ab, or sometime simply ab (if there is no possibility of misunderstanding), instead of (a, b). Furthermore, if ab = {c}, then we use the abbreviation ab = c. An example in the Euclidean plane can be the function that to every pair of points associates the segment joining them (Dresher, 1938); an example in a group of students is the function that to every pair (a, b) of students associates the students with an age between the ages of a and b (including a and b). Given a hypergroupoid (G, ), the multiplication “” induces a binary operation (also written ) on P(G), the powerset of G, given by AB = {ab | aA, b B}. As a result, we have an induced groupoid on P(G). Instead of writing {a}B, A{b}, and {a}{b}, we write {aB}, {Ab}, and {ab}, respectively. From now on, when we write (ab)c, we mean “first, take the product of a and b via the multivalued binary operation on G, then take the product of the set ab with the element c, under the induced binary operation on P(G)”. A hypergroupoid (G, ) is said to be: commutative if ab=ba for all a, b G; a quasi-hypergroup G if aG = Ga=G for all aG; associative if (ab)c = a(bc). A hypergroup is an associative quasi-hypergroup. Remark 3.1. Obtaining associativity of “” on G, however, is trickier. Then, given a hypergroupoid (G, ), always we have to consider some feeble types of associativity. We may define: right associativity (ab)c a(bc); left associativity a(bc) (ab)c; weak associativity (ab)c a(bc) ≠. Example 3.2 Let G be a group and H a subgroup of G. Let M be the collection of all left cosets of H in G. For aH, bH M, set aH bH:={cH | c = ahb, hH}. Marty has proved that M is a hypergroup with multiplication “”. Let G be a hypergroupoid. An element eG is said to be an identity if for all aG, aea and a ae. Moreover, if ea=ae=a for all a G, e is said to be an absolute identity. Let us remark that if e, fG are both absolute identities, then e = ef = f, so G can have at most one absolute identity. Suppose G is a hypergroupoid with an identity e. An element bG is said to be an inverse of a with respect to e if eba and eab. If e is an absolute identity we simply say that b is an inverse of a. Sometimes it is necessary to generalize the concept of hypergroupoid allowing the multiplication result will be the empty set. Then the following definition is introduced. A partial hypergroupoid is a nonempty set G together with a partial hyperoperation , i.e. a function : G×G P(G), where P(G) is the set of all subsets of G. Of course, a partial hypergroupoid is a hypergroupoid if the image of the function is contained in P*(G), the set of all nonempty subsets of G.
3. 2. Modelling social relations with algebraic hyperstructures Some important issues in Social Science are the “formalization” of concept as: “are between”, “are individuate by two elements”, “are in particular subset containing two elements”, “are in a coalition individuate by two elements” and so on. These concepts can be considered as “geometric concepts“ because they generalize traditional geometric constructions as “the segment of a line as the set of points that are between two points", “the line as the set of points individuate by two points”, “the vector space generated by two independent vectors”, and similar. These “geometric concepts” can be efficacious formalized by introducing models based on algebraic hyperstructures, that are functions that to every ordered pair of elements of a set U associates a subset of U. There are many links between the concepts of fuzzy relations and hyperstructures. From a fuzzy relation R we can obtain in many ways hyperstructures called “associated” to R or “generated" by R. In the sequel we deal with a social context. We show how some social relations can be formalized as fuzzy relations, we introduce some associated hyperstructures and finally we interpret their meaning from a social point of view. Moreover, we propose a modeling with algebraic hyperoperations of coalitions of individuals. Let R be a fuzzy relation on a set U of individuals (e.g. students, professors, players, etc.). In various social contexts xRy may make sense that x "likes (or prefers) for something" y, for example, x likes y for friendship, x
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prefers y for an activity, x choose y to a party. Namely x plays an active role in decision-making, and y a passive role. If R is a crisp relation, then x R y =1 (x like y) or x R y = 0 (x does not like y): for example x chooses or does not choose y for a party. If R is a fuzzy relation, then x R y[0,1] and represents the extent to which x likes y for something, such as, for example, friendship or to perform an activity together. We have the following possible intersection: R(x, z) R(y, z) is the extent to which x and y agree in liking or preference of z, for example, {x, y} form a committee and z is a candidate; R(z, x) R(z, y) is the measure wherein z considers x and y are similar; R(x, z) R(z, y) could be the extent to which x agrees y, assuming z as an intermediary. In this context, we can introduce the following meaningful partial hypergroupoids associated to R: liking partial hypergroupoid (U, ) defined as : (x, y) U U x y = {z UU: R(x, z) R(y, z) > 0}; receiving partial hypergroupoid (U, ) defined as : (x, y) U U x y = {z U U: R(z, x) R(z, y) > 0}; intermediate partial hypergroupoid (U, ) defined as : (x, y) U U x y = {z U U: R(x, z) R(z, y) > 0}. The conditions for these partial hypergroupoids to be hypergroupoids have very important social meanings. The liking partial hypergroupoid is a hypergroupoid if the following property of “liking convergence” holds: “for every pair of individuals (x, y) of the set U there exists at least an element z U that receives some liking by both x and y, i.e. x and y agree to offer at least a minimal consideration on an individual z of the social group U.” If such condition is not valid we say that the social group has a “disconnected liking agreement”, i.e. there are “opposing views in liking”. Similarly, the receiving partial hypergroupoid is a hypergroupoid if the following property of “receiving convergence” holds: “for every pair of individuals (x, y) of the set U there exists at least an element zU that gives some liking to both x and y, i.e. there exists an individual z of the social group U that have at least a minimal consideration on both x and y.” If such condition is not valid we say that the social group has a “disconnected receiving agreement” or “opposing views in receiving”. Finally, the intermediate partial hypergroupoid is a hypergroupoid if the following property of “intermediate convergence” holds: “for every pair of individuals (x, y) of the set U there exists at least an element z U that x gives some liking to z and z to y, i.e. there exists an individual z of the social group U that creates, as an intermediate, at least a minimal consideration of x to y.” If such condition is not valid we say that the social group has a “disconnected intermediate agreement”. In the sequel we assume that the social group we consider satisfy properties of liking, receiving, intermediate convergence, respectively. In this case (U, ) and (U, ) are commutative hypergroupoids.
3. 3. Geometric hypergroupoid associated social relations In the social context, let us call “geometric hypergroupoid” a hypergroupoid (U, ) commutative and such that xxx. The choice of this name is due to the fact that, under the above conditions, it is possible to give some geometric representations. Some very significant hyperstructures associated to social relations are the following. Let R be a fuzzy relation. We can introduce: the hyperstructure of “between like relation” defined (x, y)UU as: xy = {uU: zU, R(x,z) ≤ R(u,z) ≤ R(y,z) or R(y,z) ≤ R(u,z) ≤ R(x,z)}; the hyperstructure of “between receiving relation” defined (x, y)UU as x y = {uU: zU, R(z,x) ≤ R(z,u) ≤ R(z,y) or R(z,y) ≤ R(z,u) ≤ R(z,x)}. For every x, yU, {x, y}xy and {x, y}xy, then (U, ) and (U, ) are hypergroupoids. The commutative property is evident. We can prove that they are commutative hypergroups
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Example 3.3 To illustrate the concepts we give an example of “between like relation ” and “between receiving relation ” associated to a fuzzy relation R. Let us define the fuzzy relation R by the table:
R a b c
a 0.1 0.2 0.1
b 0.3 0.4 0.4
c 0.3 0.7 0.6
Than we have a b={a,b,c}, a c={a,c}, b c={b,c}. a b={a,b}, a c={a, b, c}, b c={b,c}.
4.
Some additional mathematical formalization for social groups
Let U be a set of individuals and let R be a reflexive fuzzy relation on S, i.e., for every xU, x R x = 1. Moreover we assume R represents a meaningful linguistic relation in social sciences as “friendship”, “likes”, “loves”, etc. We denote with R* the relation RR-1, called the symmetric closure of R, where R-1 is the inverse of R, i.e., for every x, y in U, x R* y = = max{x R y, y R x}. For some social problem the following concepts of topological type can be useful. They describe, in some way, the areas of influence of individuals in the social group.
4.1. Neighborhoods associated to a reflexive fuzzy relation Definition 4.1.: Let R be a reflexive fuzzy relation on U. Then we define the active R-neighborhood of order 1 of xU as the set AR(x) = A1R(x)= {zU: x R z > 0}; the active R-neighborhood of order n>1 of xU as the set AnR(z) = {zU: x1, x2, ..., xn-1U: x R x1 > 0, i< n-2, xi R xi+1 > 0, xn-1R z > 0}; the passive R-neighborhood of order 1 of xU as the set PR(x)= P1R(x) = {zU: z R x > 0}; the passive R-neighborhood of order n>1 of xU as the set PnR(x) = {zU: x1, x2, ..., xn-1S: z R x1 > 0, i< n-2, xi R xi+1> 0, xn-1R x > 0}. Definition 4.2: Let R be a reflexive fuzzy relation and let R* its symmetric closure. Then we define the global R-neighborhood of order 1 of xU as the set GR(x) = G1R(x)= {zU: z R*x > 0}; the global R-neighborhood of order n>1 of xU as the set GnR(x) = {zU: x1, x2, ..., xn-1U: z R*x1 > 0, i< n-2, xi R* xi+1> 0, xn-1R* x > 0}. Remark 4.3 From a social point of view, if R reduces to a crisp relation, an active neighborhood of order 1 of x can be seen as the set of all people that x chooses according to the rules given by the relation R, with the condition that he/she chooses also itself, while an active neighborhood of order 2 is the set of choices or by x or by an intermediary z. Moreover a passive neighborhood of order 1 of x is the set of people that choose x. A passive neighborhood of order 2 of x is the set of people who choose z either directly or through an intermediary. A global neighborhood of order 1 of z is the union of the set of all people that z chooses and the set of all people chosen by z, according to the rules. If R is a reflexive fuzzy relation, then an active neighborhood of order 1 of x can be seen as the set of all people z that x chooses with a positive agreement or a positive probability, etc. Analogous interpretations can be given for passive neighborhood or global neighborhood. We can also define the areas of influence of a coalition X, as follows. For every n N, X U, we put AnR(X) =⋃z X AnR(z) , PnR(X) = ⋃z X PnR(z), GnR(X) = ⋃z X GnR(z).
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4.2. -hyperoperations associated to fuzzy relations In addition to the hyperoperations defined in the previous section, there are many others that can be very useful to investigate the behavior or attitudes of a social group. Let us limit, by way of example, to the following, which we call -hyperoperations. Definition 4.3. Let R be a reflexive fuzzy relation on U. For x, yU we define the -hyperoperations: “rigth directed”, r1: (x, y)UU {z U: xRz zRy>0}; “left directed”, l1: (x, y)UU {z U: yRz zRx>0}; “active” a1: (x, y)UU {z U: xRz yRz>0}; “passive” p1: (x, y)UU {z U: zRx zRy>0}. These -hyperoperations are not commutative. For example r1(x, y) = Ar(x) Pr(y), in general different from r1(y, x) = Ar(y) Pr(x). We can obtain a commutative -hyperoperation if we replace R with R*. Definition 4.4.: Let R be a reflexive fuzzy relation and let R* be its symmetric closure. Then we define symmetric -hyperoperation associated to R the hyperoperation ̅1R as follows: ̅1R : (x, y)UU ={zS: x R* z z R* y>0}. We have ̅1R(x, y) = GR(x) GR(y) = ̅1R(y, x), then ̅1R is commutative and so (U, ̅1R) is a geometric (eventually partial) hypergroupoid.
5.
Conclusion and new perspective of research
We can introduce many other hyperoperations associated to a fuzzy relations. For example, by analogy with the -hyperoperations we can introduce the -hyperoperations. More generally it can be considered a t-conorm “” or a t-norm “” and define -hyperoperations or -hyperoperations associated to a relation R on the set of individuals U. If is one of these hyperoperation, the most important aspect to consider is that xy is the result of a joint action between x and y, for example, a coalition. If the relation R is symmetric then is commutative and the role of x and y is equal. If R is not symmetric one of the two is at different level than the other, for example, has a different power. The hypothesis that (U, ) is a hypergroupoid is equivalent to saying that the interaction between x and y produces at least the adhesion of an individual. The associativity corresponds to the fact that, in the agreement between three individuals x, y, z, in order, the result is not determined on the pair of individuals that accords first, and then agree with the third individual. Typically, in social relations, that is only a weak associativity. (U, ) is a quasi – hypergroup, if, for every x, y, there exist z, u, such that yxz, and yux, i.e. y belongs at least to a coalition formed by x as first element and to one in which x is the second element. If it does not, we say that x can marginalize y. In conclusion we can say that the condition that (U, ) is a commutative hypergroup is equivalent to require conditions of social equity in forming coalitions xy, i.e. x and y have an equal role (commutativity), coalitions do not depend on the order in which three individuals make agreements, no one is marginalized by another.
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