aa r X i v : . [ m a t h . L O ] F e b NEGOTIATION SETS: A GENERAL FRAMEWORK
TOMASZ WITCZAK
Abstract.
It is well-known fact that there exists 1 − Introduction
Intuitionistic sets have been introduced by C¸ oker (see [3]). In fact, theyare special (crisp) type of intuitionistic fuzzy sets (investigated earlier byAtanassov). Analogously, intuitionistic topological spaces (introduced byC¸ oker in [4]) can be considered as a crisp case of intuitionistic fuzzy topolog-ical spaces. Many authors contributed to the development of intuitionistictopologies since the beginning of XXI century. Nowadays, there are intu-itionistic analogues of the majority of well-known topological notions (likeinterior, closure, density, continuity, compactness etc.). However, in this ar-ticle we do not deal too much with topology. Rather, we are interested insets as such.One thing should be pointed out: we do not participate in a terminolog-ical debate about whether it is sensible to call these sets (be they fuzzy ornot) ”intuitionistic”. This topic has been extensively discussed by Guti´errezGarcia and Rodabaugh in [5], Cattaneo and Ciucci in [1] and also by Ciucciin [2]. In general, we may agree with these authors that the term in question
Mathematics Subject Classification.
Primary: 03B45, 03E75 ; Secondary: 68T27,68T30.
Key words and phrases.
Intuitionistic sets, double sets, formal concept analysis, dataclustering. is not adequate. However, it seems that the notion of intuitionistic set hasbeen widely accepted or at least tolerated by many mathematicians.Be as it may, the idea of intuitionistic set is simple: if we have non-empty universe X with two subsets A and A such that A ∩ A = ∅ ,then we define intuitionistic set A as [ A , A ], assuming that if A and B are both intuitionistic sets, then A ∩ B = [ A ∩ B , A ∪ B ] and A ∪ B =[ A ∪ B , A ∩ B ]. These notions are properly defined because the result ofboth operations is an intuitionistic set too. On the other hand, double sets (see [6]) are defined as ordered pairs [ A , A ] such that A ⊆ A . However,we assume that if A and B are double, then A ∩ B = [ A ∩ B , A ∩ B ]and A ∪ B = [ A ∪ B , A ∪ B ]. Again, these operations do not lead outof the class. Now it is simple to show that each double set [ A , A ] can beconsidered as an intuitionistic set [ A , − A ], where − A is just a complementof − A with respect to the universe X .But there the question arises: is it reasonable to combine intuitionisticdefinitions of intersection and union with double sets? First of all, let usexplain our motivation. Imagine that A and A are two sets of arguments,formulas, scenarios or certain other objects. We do not precise their nature:they can be items, scenarios or logical formulas. If these objects are only possible (or allowable a.k.a. admissible ), then they are somewhere in A . Ifthey are necessary (or obligatory ), then we put them into A . Hence, it asnatural that A is contained in A (because what is necessary should alsobe possible; this is natural assumption, well-known from almost all modallogics which are not extremely weak). Assume now that we have two doublesets A and B . Let us consider the following new objects (or, equivalently,operations):[ A ∩ B , A ∪ B ][( A ∪ B ) ∩ ( A ∩ B ) , A ∩ B ]These operations make sense in the light of our interpretation. Let ustreat different double sets as different points of view of certain agents (ordecision makers). Then our new objects describe two variants of compro-mise between two agents, represented by A and B . In the first case, more things are acceptable now and fewer things are necessary. The second case isnearly symmetrical: more things are necessary and fewer things are accept-able. However, we must ensure ourselves that our new range of necessity iscontained in the range of admissibility of both agents. For this reason, thiskind of compromise is slightly more complicated than the first one (we in-tersect A ∪ B with the intersection of A and B ). From the mathematicalpoint of view, it means that our new object should still be a double set.Our motivation is practical. For this reason, we are interested mostlyin finite universes (collections of objects), even if it is possible to consider Later we shall identify agents with their double sets.
EGOTIATION SETS: A GENERAL FRAMEWORK 3 arbitrary ones. Moreover, we admit the thought that some versions of ourframework may not solve each conflict but only some of them. This is typicalfor many decision systems: that some cases are left as unsolvable and somescenarios are equally good. In such situations we must use other, externalcriteria. These topics will be discussed in the last section of our paper.2.
Formal exposition
In this section we shall see formal (and more general) definitions of thefunctions and objects introduced earlier. Also, their basic algebraic proper-ties will be studied.2.1.
Basic definitions.
The first three definitions are typical for double sets(see [6] for example). However, we shall use the notion of negotiation set tospeak about double sets considered in the context of our new operations.
Definition 2.1.
Assume that X is a non-empty universe (consisting of objects ), A , A ⊆ X and A ⊆ A . An ordered pair [ A , A ] is called a negotiation set (on X ). We say that A is range of necessity (of A ), while A is its range of admissibility . Definition 2.2.
Assume that X is a non-empty universe and A, B aretwo negotiation sets. We define complement of A as − A = [ − A , − A ].Moreover, we define difference of A and B as A \ B = [ A \ B , A \ B ]. Wesay that A ⊆ B if and only if A ⊆ B , A ⊆ B . Definition 2.3.
Assume that X is a non-empty universe, J is an arbitraryset and { A j : j ∈ J } is a family of negotiation sets on X . We define thefollowing operations:(1) Generalized union : S j ∈ J A j = [ S j ∈ J A j , S j ∈ J A j ](2) Generalized intersection : T j ∈ J A j = [ T j ∈ J A j , T j ∈ J A j ]It is easy to check that both complement and difference are well-defined:they result in a new negotiation set. The same can be said about union andintersection.The next definition is new: Definition 2.4.
Assume that X is a non-empty universe, J is an arbitraryset and { A j : j ∈ J } is an indexed family of negotiation sets on X . We definethe following operations:(1) Generalized minimalization (of necessities): ⊙ j ∈ J A j = [ T j ∈ J A j , S j ∈ J A j ](2) Generalized relative maximalization (of necessities): ⊕ i ∈ J A j = [ S j ∈ J A j ∩ T j ∈ J A j , T j ∈ J A j ] TOMASZ WITCZAK
Note that for J = { , } we obtain binary instances that were presentedin the introduction. In fact, we shall deal mostly with them: A ⊙ B = [ A ∩ B , A ∪ B ], A ⊕ B = [( A ∪ B ) ∩ ( A ∩ B ) , A ∩ B ].We can prove the following lemma: Lemma 2.5.
Let X be a non-empty universe. Assume that { A j ; j ∈ J } is a family of negotiation sets and B is also negotiation set:(1) If A j ⊆ B for any j ∈ J , then ⊙ j ∈ J A j ⊆ S j ∈ J A j ⊆ B .(2) If B ⊆ A j for any j ∈ J , then B ⊆ T j ∈ J A j ⊆ ⊕ j ∈ J A j .Proof. (1) Assume that for each j ∈ J we have A j ⊆ B and A j ⊆ B .Then it is clear that T j ∈ J A j ⊆ B and S j ∈ J A j ⊆ B . Of coursewe can say more: that T j ∈ J A j ⊆ S j ∈ J A j ⊆ B . Thus we obtainour expected conclusion.(2) For any j ∈ J we have B ⊆ A j and B ⊆ A j . If x ∈ B , then x ∈ T j ∈ J A j ⊆ S j ∈ J A j . However, for any j ∈ J , A j ⊆ A j . Hence x ∈ T j ∈ J A j ⊆ S i ∈ J A j ∩ T j ∈ J A j . If x ∈ B , then x ∈ A j for any j ∈ J . Thus x ∈ T j ∈ J A j and we obtain our final conclusion. (cid:3) Let us introduce another, weaker type of inclusion:
Definition 2.6.
Let X be a non-empty universe and assume that A, B are two negotiation sets on X . We say that A ⊆ B if and only if A ⊆ B .Analogously, A ⊆ B if and only if A ⊆ B .Now we may formulate the following theorem: Theorem 2.7.
Let X be a non-empty universe. Assume that { A j ; j ∈ J } is a family of negotiation sets. Then we have:(1) − ⊙ j ∈ J A j ⊆ ⊕ j ∈ J ( − A j ) .(2) ⊕ j ∈ J ( − A j ) ⊆ − ⊙ j ∈ J A j .(3) ⊙ j ∈ J ( − A j ) ⊆ − ⊕ j ∈ J A j .(4) − ⊕ j ∈ J A j ⊆ ⊙ j ∈ J ( − A j ) .Proof. First, let us write: − ⊙ j ∈ J A j = [ − S j ∈ J A j , − T j ∈ J A j ], ⊕ j ∈ J ( − A j ) = [ S j ∈ J ( − A j ) ∩ T j ∈ J ( − A j ) , T j ∈ J ( − A j )].Second, we have: − ⊕ j ∈ J A j = [ − T j ∈ J A j , − (cid:16)S j ∈ J A j ∩ T j ∈ J A j (cid:17) ], ⊙ j ∈ J ( − A j ) = [ − S j ∈ J A j , − T j ∈ J A j ].Now we can prove all the cases: EGOTIATION SETS: A GENERAL FRAMEWORK 5 (1) Assume that x ∈ − S j ∈ J A j . This means that for any j ∈ J , x / ∈ A j .Now it is easy to show that x ∈ S j ∈ J ( − A j ). Moreover, for any j ∈ J we have A j ⊆ A j . Hence for any j ∈ J , x / ∈ A j . Thus x ∈ T i ∈ J ( − A j ).(2) It follows from the general principles of set theory that T j ∈ J ( − A j ) ⊆− T j ∈ J A j .(3) Assume that x ∈ − S j ∈ J A j . Again, from the basic principles x ∈− T j ∈ J A j .(4) Let x ∈ − (cid:16)S j ∈ J A j ∩ T j ∈ J A j (cid:17) . Hence x / ∈ (cid:16)S j ∈ J A j ∩ T j ∈ J A j (cid:17) .It means that x / ∈ S j ∈ J A j or x / ∈ T j ∈ J A j . Consider the first option.In particular, it means that x / ∈ T j ∈ J A j , i.e. x ∈ − T j ∈ J A j . Asfor the second one: there must be k ∈ J such that x / ∈ A k . Then x / ∈ A k . Again, x / ∈ T j ∈ J A j . (cid:3) We cannot replace weaker inclusions by equalities. Consider the followingcounter-example for Th. 2.7 1, 2.
Example 2.8.
Assume that X = { a, b, c, d, e, f, g } , A = [ { a, b } , { a, b, c, d } ]and B = [ { c, d } , { c, d, g } ]. Now: A ⊙ B = [ ∅ , { a, b, c, d, g } ]. Then − ( A ⊙ B ) =[ { e, f } , { a, b, c, d, e, f, g } ]. On the other hand − A = [ { e, f, g } , { c, d, e, f, g } ], − B = [ { a, b, e, f } , { a, b, e, f, g } ]. Then − A ⊕ − B = [ { e, f, g } , { e, f, g } ] * − ( A ⊙ B ). Moreover, − ( A ⊙ B ) * − A ⊕ − B .In practical applications we limit our early attention to the collection ofsome distinguished negotiation sets: the initial ones (just like A and B inthe preceding example). They represent points of view of several discus-sants. Actually, this collection can be (and probably will be ) smaller thanthe class of all imaginable negotiation sets. On the other hand, negotiatingsets resulting from the application of ⊙ or ⊕ can be considered as coalitionsof agents . Example 2.9.
Adam ( A ), Bernard ( B ) and Clara ( C ) are planning jointtrip by car. They are discussing which items should be taken to the trunk.They want to determine which things are necessary and which are onlyoptional. Let us say that X = { a = map , b = flashlight , c = shoe polish , d =first aid kit , e = fishing rod , f = ball , g = night-vision device , h = tent , i =riffle , k = guitar , l = chest expander } .Assume that their negotiation sets are: A = [ { a, d } , { a, d, f, g, h } ] B = [ { a, b, d } , { a, b, d, f, i, l } ] C = [ { a, h } , { a, d, h, k } ]. TOMASZ WITCZAK
Suppose that they are interested in minimalization of necessities. Then :( A ⊙ B ) ⊙ C = [ { a, d } , { a, b, d, f, g, h, i, l } ] ⊙ C = [ { a } , { a, b, d, f, g, h, i, k, l } ].( A ⊕ B ) ⊕ C = [ { a, d } , { a, d, f } ] ⊕ C = [ { a } , { a, d } ].We may imagine that at the first step Bernard and Clara use, say, ⊕ tofind their compromise, and then they use ⊙ to discuss their proposition withAdam. Hence they obtain:( B ⊕ C ) ⊙ A = [ { a } , { a, d } ] ⊙ A = [ { a } , { a, d, f, g, h } ].In each of these three cases they are only sure that a map should betaken. However, they have obtained three different ranges of admissibility.Of course their debate may take longer.The next example is more abstract and rather theoretical: Example 2.10.
Let X = C (the set of complex numbers). Assume that A = [ N , R ], B = [ { , , } , N ] and C = [ Z , C ]. We may calculate (amongmany other possible compositions): A ⊙ B = [ { , , } , R ], B ⊕ C = [ N , N ], ( A ⊕ B ) ⊙ C = [ N , N ] ⊙ [ Z , C ] = [ N , C ].2.2. Algebraic properties.
Undoubtedly, some fundamental algebraic prop-erties of our system should be checked. This is essentially a content of Th.2.11 and Th. 2.12.
Theorem 2.11.
Let X = ∅ , J be an arbitrary set and { A j : j ∈ J } bea family of negotiation sets on X . Then the following features of binary ⊙ and ⊕ operations hold :(1) Idempotence
Proof.
Clearly, A ⊕ A = A ⊙ A = A . (cid:3) (2) Commutativity .Proof.
This is trivial and based on the commutativity of ∪ and ∩ operations. (cid:3) (3) ⊕⊙ - Absorption law .Proof.
We may write:[ A , A ] ⊕ [ A ∩ B , A ∪ B ] = [( A ∪ ( A ∩ B )) ∩ ( A ∩ ( A ∪ B )) , A ∩ ( A ∪ B )] = [ A ∩ A , A ] = [ A , A ]. (cid:3) (4) Associativity .Proof.
First, consider ⊙ . We have: A ⊙ ( B ⊙ C ) = A ⊙ [ B ∩ C , B ∪ C ] = [ A ∩ B ∩ C , A ∪ B ∪ C ] =( A ⊙ B ) ⊙ C .Second, we write: In the next section we shall prove that both ⊙ and ⊕ are associative. EGOTIATION SETS: A GENERAL FRAMEWORK 7 A ⊕ ( B ⊕ C ) = [ A , A ] ⊕ [( B ∪ C ) ∩ ( B ∩ C ) , B ∩ C ] =[( A ∪ (( B ∪ C ) ∩ ( B ∩ C ))) ∩ ( A ∩ B ∩ C ) , A ∩ B ∩ C ] =[( A ∪ B ∪ C ) ∩ ( A ∪ ( B ∩ C )) ∩ ( A ∩ B ∩ C ) , A ∩ B ∩ C ].and, simultaneously:( A ⊕ B ) ⊕ C = [( A ∪ B ) ∩ ( A ∩ B ) , A ∩ B ] ⊕ ( C , C ) =[((( A ∪ B ) ∩ ( A ∩ B )) ∪ C ) ∩ ( A ∩ B ∩ C ) , A ∩ B ∩ C ] =[( A ∪ B ∪ C ) ∩ (( A ∩ B ) ∪ C )) ∩ ( A ∩ B ∩ C ) , A ∩ B ∩ C ].We see that ranges of admissibility of resulting sets are identical.What about their ranges of necessity? i) Let x ∈ ( A ∪ B ∪ C ) ∩ ( A ∪ ( B ∩ C )) ∩ ( A ∩ B ∩ C ).In particular, it means that x ∈ A ∩ B ∩ C . Assume that x / ∈ ( A ∪ B ∪ C ) ∩ (( A ∩ B ) ∪ C )) ∩ ( A ∩ B ∩ C ). The onlyoption worth considering is x / ∈ ( A ∩ B ) ∪ C . However, in thiscase x / ∈ A ∩ B , and this is contradiction. ii) Let x ∈ ( A ∪ B ∪ C ) ∩ (( A ∩ B ) ∪ C )) ∩ ( A ∩ B ∩ C ).In particular, it means that x ∈ A ∩ B ∩ C . Assume that x / ∈ ( A ∪ B ∪ C ) ∩ ( A ∪ ( B ∩ C )) ∩ ( A ∩ B ∩ C ). The only optionworth considering is x / ∈ A ∪ ( B ∩ C ). However, in this case x / ∈ A and x / ∈ B ∩ C . This is contradiction. (cid:3) Theorem 2.12.
Let X = ∅ , J be an arbitrary set and { A j : j ∈ J } bea family of negotiation sets on X . Then the following features of binary ⊙ and ⊕ operations do not hold:(1) Distributivity .Proof.
Counterexamples: First, consider
A, B, C where A = { x } , A = { a, x } , B = { b } , B = { b, d } , C = { c } , C = { c, x } . Now: B ⊙ C = [ { b } ∩ { c } , { b, d } ∪ { c, x } ] = [ ∅ , { b, c, d, x } ], A ⊕ ( B ⊙ C ) = [ { x } ∩ { x } , { x } ] = [ { x } , { x } ], A ⊕ B = [ { x, b } ∩ ∅ , ∅ ] = [ ∅ , ∅ ], A ⊕ C = [ { x, c } ∩ { x } , { x } ] = [ { x } , { x } ].Finally, we obtain:( A ⊕ B ) ⊙ ( A ⊕ C ) = [ ∅ ∩ { x } , ∅ ∪ { x } ] = [ ∅ , { x } ] = A ⊕ ( B ⊙ C ). Second, consider
A, B, C where A = { a } , A = { a } , B = { x } , B = { x, b } , C = { x, a } , C = { x, a, c } . Now: A ⊙ B = [ { a } ∩ { x } , { a } ∪ { x, b } ] = [ ∅ , { a, x, b } ], A ⊙ C = [ { a } ∩ { x, a } , { a } ∪ { x, a, c } ] = [ { a } , { x, a, c } ],( A ⊙ B ) ⊕ ( A ⊙ C ) = [( ∅ ∪ { a } ) ∩ { a, x } , { a, x } ] = [ { a } , { a, x } ], B ⊕ C = [( { x } ∪ { x, a } ) ∩ { x } , { x } ] = [ { x } , { x } ].Finally, we obtain: TOMASZ WITCZAK A ⊙ ( B ⊕ C ) = [ { a } ∩ { x } , { a } ∪ { x } ] = [ ∅ , { a, x } ] = ( A ⊙ B ) ⊕ ( B ⊙ C ) (cid:3) (2) ⊙⊕ - Absorption law .Proof.
Counterexample:Consider
A, B where A = { x } , A = { x } , B = { b } , B = { b } .Now: A ⊕ B = [( { x } ∪ { b } ) ∩ ( { x } ∩ { b } ) , { x } ∩ { b } ] = [ ∅ , ∅ ].Hence we get the following result: A ⊙ ( A ⊕ B ) = [ { x } ∩ ∅ , { x } ∪ ∅ ] = [ ∅ , { x } ] = [ { x } , { x } ] = A . (cid:3) The two theorems above allow us to say that the set of all negotiation setson X forms a structure which is slightly weaker than lattice (and slightlystronger than semi-lattice). As we could see, it satisfies only one absorptionlaw.In this environment we can speak about analogues of empty set and thewhole universe: Definition 2.13.
Assume that X is a non-empty universe and x ∈ X . Wepoint out two special types of negotiation sets: ∅ N = [ ∅ , ∅ ] and X N = [ X, X ]. Lemma 2.14.
Assume that X is a non-empty universe and A is negoti-ation set on X . Then we have:(1) A ⊙ X N = [ A , X ] .(2) A ⊙ ∅ N = [ ∅ , A ] .(3) A ⊕ X N = [ A , A ] .(4) A ⊕ ∅ N = ∅ N . As we can see, there is no full analogy between ∅ N (resp. X N ) and latticebottom (resp. top ). We may also check half-empty set X P = [ ∅ , X ]. Lemma 2.15.
Assume that X is a non-empty universe and A is negoti-ation set on X . Then we have:(1) A ⊙ X P = X P .(2) A ⊕ X P = A . Here we see that X P plays the role of : if ⊙ and ⊕ are treated as (resp.) ∨ and ∧ . Note that this identification allows us to say that ∅ N behaves in away like . However, this correspondence is only partial: note that A ⊕ ∅ N does not result in A . Per analogiam with intuitionistic points, we may speak about negotiationpoints : Definition 2.16.
Assume that X is a non-empty universe and x ∈ X .We point out two special types of negotiation sets: x . = [ ∅ , { x } ] and x =[ { x } , { x } ]. EGOTIATION SETS: A GENERAL FRAMEWORK 9
Lemma 2.17.
Let x, y ∈ X and x = y . Then we have the followingresults:(1) x . ⊕ y . = x . ⊕ y = x ⊕ y = [ ∅ , ∅ ] .(2) x . ⊙ y . = x . ⊙ y = x ⊙ y = [ ∅ , { x, y } ] . In the next section we shall deal not only with our negotiation sets butalso with the internal structure of universe X .3. Contradictions and relationships
We assumed, by default, that our basic objects (the elements of X ) are insome sense mutually independent and peacefully coexisting, while our pointsof view (negotiation sets) are equal: there is no any specified hierarchy be-tween them. However, these assumptions do not exhaust the whole richnessof reality. First of all, we may easily imagine that there is a kind of conflictbetween some of our objects. In short, they can be mutually contradictory.For example, if our objects are logic formulas (sentences), then it is enoughto consider pair x = ϕ and y = ¬ ϕ (we assume tacitly that our logic is notparaconsistent). It is also possible that x and y are two scenarios which are(in practice) mutually exclusive. For example, x means ”holidays on Tris-tan da Cunha”, while y means ”holidays on St Kilda” (at the same time andwith the same participants). Clearly, it is not possible to be in two differentlocations at once. Moreover, it is possible that (due to the certain reasonslike lack of cash) we cannot visit these two places even one after another.Hence, these scenarios are still incompatible.Note that these contradictions are inherent for the elements in question:they do not depend on the points of view of the agents. Of course, onecould discuss another approach: that x and y are contradictory from theperspective of agent A but agent B does not recognize any conflict betweenthem. Undoubtedly, this model should also be investigated but it is beyondthe scope of present study.In this section we shall introduce two variants of contradiction: strongerand weaker one. The idea is that our negotiation sets should be consistent (in some sense which will be defined later). If a given set is consistent in thissense, then we say that it is admitted to discussion: it belongs to the class Disc .Let us introduce some basic definitions:
Definition 3.1.
Assume that X is a non-empty universe and , ≀ aretwo binary, symmetric and anti-reflexive relations on X . We define Disc asthe following class of negotiation sets: A ∈ Disc if and only if the tworequirements below are satisfied:(1) There are no any x , y in A such that x y . In the sense of vacations. (2) There are no any x , y in A such that x ≀ y and ( x ∈ A or y ∈ A ).This definition says that reflects the idea of strong contradiction: itis not possible that two strongly contradictory objects are in the range ofadmissibility of a consistent negotiation set. It is possible that two weaklycontradictory objects (i.e. such that x ≀ y ) are in this range. However,they both should be in A \ A . It means that we leave them for a furtherdiscussion (so they are temporarily admissible) but cannot assume that evenone of them is necessary.We may prove the following theorem: Theorem 3.2.
Let X be a non-empty universe with relations and ≀ .Then the set Disc is closed under operation ⊕ .Proof. Assume that
A, B ∈ Disc and suppose that our thesis is not true:( A ⊕ B ) / ∈ Disc . There are two possible reasons:(1) There are x, y ∈ ( A ⊕ B ) such that x y . However, ( A ⊕ B ) = A ∩ B , hence x, y ∈ A and x, y ∈ B . This is contradiction,because A, B ∈ Disc .(2) There are x, y ∈ ( A ⊕ B ) such that x ≀ y and (without loss of gener-ality) x ∈ ( A ⊕ B ) , y ∈ ( A ⊕ B ) . Hence x ∈ ( A ∪ B ) ∩ ( A ∩ B ).In particular, it means that x ∈ A or x ∈ B . However, at the sametime y ∈ A and y ∈ B . This gives us contradiction because x ≀ y and A, B ∈ Disc , hence it is not possible that x ∈ A , y ∈ A or x ∈ B , y ∈ B . (cid:3) What about ⊙ operation? We have this partial result: Lemma 3.3.
Let X be a non-empty universe with relation ≀ and A, B ∈ Disc . Then there are no such x, y ∈ ( A ⊙ B ) that x ≀ y and (without lossof generality) x ∈ ( A ⊙ B ) , y ∈ ( A ⊙ B ) .Proof. Assume the contrary. Hence, x ∈ ( A ⊙ B ) , y ∈ ( A ⊙ B ) . It meansthat x ∈ A and x ∈ B . At the same time y ∈ A or y ∈ B . Clearly, weobtain contradiction (because x ≀ y ). (cid:3) However, things are more complicated if we consider relation. Let usdiscuss the following (counter)-example:
Example 3.4.
Let X be a non-empty universe with relation . Assumethat a b (in particular, it means also that a = b ) and A = [ { a } , { a } ], B =[ { b } , { b } ]. Then both A and B belong to Disc . Now consider A ⊙ B =[ ∅ , { a, b } ]. Then A ⊙ B / ∈ Disc .Several solutions to this problem can be applied:
EGOTIATION SETS: A GENERAL FRAMEWORK 11 (1) We may assume that only some actions with use of ⊙ are permissible .If there a problem occurs (similar to the one from Ex. 3.4), then wejust say that it is not possible to solve it. This round of negotiationsfails.(2) We may assume that each two strongly contradictiory objects x , y arealso connected by asymmetric, transitive and anti-reflexive relation > . The interpretation goes as follows: if A, B ∈ Disc but A ⊙ B / ∈ Disc because of some x, y ∈ ( A ⊙ B ) such that x y and (withoutloss of generality) x > y , then we leave only x in ( A ⊙ B ) . Thissolution has one advantage: each conflict can be solved. However,relation > does not depend on the point of view: it is inherent for thepairs of objects themselves. While in the case of this approach isnatural, the same cannot be said about > . Clearly, it means that allparticipants of discussion agree that, for example, holidays on Marsare always better when compared to the holidays on the Moon.(3) We may determine certain (total) order not between objects butbetween our (initial) negotiation sets. This would mean that ourpoints of view are hierarchically ranked. In other words, if agent A has priority over B and B has priority over C , then we may write: A > B > C . If
A, B ∈ Disc but A ⊙ B / ∈ Disc because of some x, y ∈ ( A ⊙ B ) such that x y and x ∈ A , then we leave only x in( A ⊙ B ) .This solution has one serious flaw: it says nothing about the hier-archy of sets obtained by means of ⊕ and ⊙ . Basically, it refers onlyto the initial, ”pure” points of view. For example, it is natural thatthe president’s opinion is more important that the opinion of vice-president; and the opinion of the latter is stronger than the opinionof secretary of state . However, if we assume arbitrarily that, say, A > B > C , then it does not give us any information about therelationship between A ⊕ B and C or A ⊙ C and B ⊕ C . One couldsay that if we have finitely many finite initial sets, then we can de-termine hierarchy of each two possible results. However, this wouldbe artificial and impractical.Of course we may invent more precise relations. For example,we can assume that A > B if A contains fewer elements than B .It would mean that priority goes to those agents (or coalitions ofagents) which have lesser expectations regarding necessity. However,such relation is not total. Hence, it does not solve all contradictions. Clearly, this description is simplified and skips some nuances. Further studies
Other problems should also be a matter of further studies. The fact thatcertain object is strongly contradictory with another one does not alwaysmean that choosing one of them is satisfying. For example, x = ”holidaysin a tropical country” is contradictory with y = ”holidays on Svalbard”.However, one could say that these scenarios belong to different classes. Beingon Svalbard is something concrete, while being ”in a tropical country” isvague. In some sense, these are philosophical and linguistic considerations.On the other hand, such subtle distinctions can be expressed in a formallanguage of functions and relations.Moreover, there are also algorithmic issues. The question is: which ne-gotiation set should be considered as a final effect of discussion; and is itpossible to determine steps leading to the achievement of this goal?Another interesting topic is topology. It is not clear which of our op-erations is ”closer” to the idea of union and which resembles intersection.Perhaps ⊙ has more to do with union because it joins together external partsof negotiation sets, hence resulting sets are ”bigger” when considered just assubsets of X . Hence, it may be valuable to consider families closed under ar-bitary minimalizations of necessities (that is, ⊙ ) and finite maximalizations(that is, ⊕ ). However, we may imagine that the roles of these two operatorsare replaced. References [1] G. Cattaneo, D. Ciucci,
Basic intuitionistic principles in fuzzy set theories and its ex-tensions (a terminological debate on Atanassov IFS) , Fuzzy Sets and Systems, Decem-ber 2006, https://boa.unimib.it/retrieve/handle/10281/1424/720/fss157.pdf .[2] D. Ciucci,
Orthopairs: a simple and widely used way tomodel uncertainty , Fundamenta Informaticae 108(3): 287-304, https://boa.unimib.it/retrieve/handle/10281/25265/32626/09-RST-SITO.pdf .[3] D. C¸ oker,
A note on intuitionistic sets and intuitionistic points , Turkish Journal ofMathematics, 20 (1996), pp. 343-351.[4] D. C¸ oker,
An introduction to intuitionistic topological spaces , BUSEFAL 81 (2000),pp. 51 - 56.[5] J. Gutierrez Garcia, S. E. Rodabaugh,
Order-theoretic, topological, categorical re-dundancies of interval-valued sets, grey sets, vague sets, interval-valued intuitionisticsets, intuitionistic fuzzy sets and topologies , Fuzzy Sets and Systems, 156 (2005), pp.445-484.[6] A. Kandil, S. A. El Sheikh, M. M. Yakout, S. A. Hazza,
Some types of compactnessin double topological spaces , Annals of Fuzzy Mathematics and Informatics 10 (10)(2015) 87 - 102.
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