Multi-polar t-conorms and uninorms

Information Sciences 301 (2015) 227–240

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Multi-polar t-conorms and uninorms Andrea Mesiarová-Zemánková Mathematical Institute, Slovak Academy of Sciences, Bratislava, Slovakia

a r t i c l e

i n f o

Article history: Received 26 March 2013 Received in revised form 27 May 2014 Accepted 30 December 2014 Available online 9 January 2015 Keywords: Multi-polar aggregation t-conorm Uninorm Nullnorm t-norm

a b s t r a c t Transformation of uninorms defined on the unit interval to the bipolar scale yields bipolar t-conorms and transformation of nullnorms on the unit interval yields bipolar t-norms. We use this relation in the investigation of the structure of bipolar and multi-polar t-norms and t-conorms. We use the same transformation of the unit interval into the bipolar scale in order to obtain bipolar aggregation operators from ordinal sum t-norms and t-conorms. The extensions of these special bipolar aggregation operators generalize both multi-polar t-norms and t-conorms and therefore we call them multi-polar uninorms. Several examples of multi-polar t-conorms and uninorms are also presented. Ó 2015 Elsevier Inc. All rights reserved.

1. Introduction Aggregation operators appear as an important tool in many domains where group of values should be replaced by one value that represents the original group of data. An aggregation operator [2,4,9] is a bounded monotone function working S on an arbitrary number of inputs from the unit interval ½0; 1, i.e., A : n2N ½0; 1n !½0; 1. Among several important classes of aggregation operators recall those of t-norms and t-conorms [1,10]. Extensions of aggregation operators to the bipolar scale started with extensions of the Choquet integral defined on non-negative numbers into the real scale, i.e., with the symmetric and the asymmetric Choquet integral [5,6,21,23] and their generalization used in the cumulative prospect theory [24]. However, the definition of a bipolar aggregation operator was introduced for the first time in [13]. Further extensions of (bipolar) aggregation operators into a multi-polar scale originate in fuzzy rule-based classification systems, where fuzzy rule outputs – represented by an absolute value and a category (class) – should be aggregated into the final system output (see [16–18] and references there in). Associativity of such an aggregation means that rule outputs can be aggregated by parts (i.e., first subgroups can be aggregated into partial results and then these are aggregated) and thus reduces the complexity. Multi-polar aggregation operators are also related to game theory and multi-criteria decision making [20]. Our aim is to investigate associative multi-polar aggregation operators which include multi-polar t-norms, t-conorms and uninorms. In the bipolar case bipolar t-norms and t-conorms play a role of AND and OR operators in bipolar fuzzy rule-bases, i.e., in the case when a truth value of a statement is from ½1; 1 instead of ½0; 1. Study of multi-polarity included introduction of multi-polar t-norms and t-conorms [14,15]. While the structure of multi-polar t-norms is completely known, multipolar t-conorms, up to the one example, were not yet studied. In this paper we investigate multi-polar t-conorms and their related multi-polar operations. We begin in this section by recalling basic definitions and results. In Section 2 we study multi-polar t-conorms and extensions of some classes of uninorms into multi-polar t-conorms. We show that in the case of multi-polarity with more

E-mail address: [email protected] http://dx.doi.org/10.1016/j.ins.2014.12.060 0020-0255/Ó 2015 Elsevier Inc. All rights reserved.

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than two polarities there is no representable multi-polar t-conorm since such an operation violates associativity. We continue with the study of generated multi-polar pre-t-conorms (Section 3). In Section 4 we define multi-polar uninorms and we study transformations of ordinal-sum t-norms and t-conorms into the bipolar scale and their extensions into a multi-polar scale. We give our conclusions in Section 5. 1.1. Basic definitions and results First we recall definition of an aggregation operator, t-norm, t-conorm, uninorm, nullnorm and an ordinal sum of t-norms and t-conorms on the unit interval ½0; 1. Definition 1. S (i) A mapping A : n2N ½0; 1n !½0; 1 is called an aggregation operator [2,4,9] if (A1) A is non-decreasing, i.e., for any n 2 N; x; y 2 ½0; 1n ; x 6 y it holds AðxÞ 6 AðyÞ; (A2) 0; 1 are idempotent elements of A, i.e., for any n 2 N; Að0; . . . ; 0Þ ¼ 0 and Að1; . . . ; 1Þ ¼ 1; |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} (A3) for n ¼ 1; AðxÞ ¼ x for all x 2 ½0; 1. ntimes ntimes 2 (ii) A triangular norm [1,10] is a binary operation T : ½0; 1 !½0; 1 which is commutative, associative, non-decreasing in both variables and 1 is its neutral element. (iii) A triangular conorm [1,10] is a binary operation C : ½0; 12 !½0; 1 which is commutative, associative, non-decreasing in both variables and 0 is its neutral element. (iv) A uninorm [7,22,25] is a binary operation U : ½0; 12 !½0; 1 which is commutative, associative, non-decreasing in both variables and e 2 0; 1½ is its neutral element. Each uninorm with a neutral element e 2 0; 1½ has the following form:

8 x y if ðx; yÞ 2 ½0; e2 ; > < e  T U ðe ; e Þ Uðx; yÞ ¼ e þ ð1  eÞ  C U ðxe ; yeÞ if ðx; yÞ 2 ½e; 12 ; 1e 1e > : Aðx; yÞ else; where minðx; yÞ 6 Aðx; yÞ 6 maxðx; yÞ and T U is a t-norm and C U is a t-conorm on ½0; 1. (v) A nullnorm [3] is a binary operation V : ½0; 12 !½0; 1 which is commutative, associative, non-decreasing, has an annihilator z 2 0; 1½ and satisfies Vð0; xÞ ¼ x for all x 6 z and Vð1; xÞ ¼ x for all x P z. Each nullnorm with an annihilator z 2 0; 1½ has the following form:

8 x y if ðx; yÞ 2 ½0; z2 ; > < zC V ðz ; zÞ Vðx; yÞ ¼ z þ ð1  zÞT V ðxz ; yzÞ if ðx; yÞ 2 ½z; 12 ; 1z 1z > : z else;

where T V is a t-norm and C V is a t-conorm on ½0; 1. (vi) Let a1 ; b1 ½ and a2 ; b2 ½ be two disjoint open subintervals of ½0; 1 and let T 1 and T 2 be two t-norms defined on ½0; 1. Then the ordinal sum of t-norms T ¼ ðha1 ; b1 ; T 1 i; ha2 ; b2 ; T 2 iÞ given for ðx; yÞ 2 ½0; 12 by

8 2 1 > a þ ðb1  a1 ÞT 1 ðbxa ; ya1 Þ if ðx; yÞ 2 a1 ; b1  ; > 1 a1 b1 a1 < 1 Tðx; yÞ ¼ a2 þ ðb2  a2 ÞT 2 ð xa2 ; ya2 Þ if ðx; yÞ 2 a2 ; b2 2 ; b2 a2 b2 a2 > > : minðx; yÞ else

is a t-norm. (vii) Let a1 ; b1 ½ and a2 ; b2 ½ be two disjoint open subintervals of ½0; 1 and let C 1 and C 2 be two t-conorms defined on ½0; 1. Then the ordinal sum of t-conorms

C ¼ ðha1 ; b1 ; C 1 i; ha2 ; b2 ; C 2 iÞ given for ðx; yÞ 2 ½0; 12 by

8 2 1 > a þ ðb1  a1 ÞC 1 ðbxa ; ya1 Þ if ðx; yÞ 2 ½a1 ; b1 ½ ; > 1 a1 b1 a1 < 1 Cðx; yÞ ¼ a2 þ ðb2  a2 ÞC 2 ð xa2 ; ya2 Þ if ðx; yÞ 2 ½a2 ; b2 ½2 ; b2 a2 b2 a2 > > : maxðx; yÞ else is a t-conorm. T-norms, t-conorms, uninorms and nullnorms are special aggregation operators. Due to the associativity all four operators can be uniquely extended to their n-ary forms (where they work on n inputs from the unit interval). Ordinal sums of t-norms (t-conorms) are continuous if and only if all summands in the ordinal sum construction are continuous. Moreover, t-norms are dual to t-conorms. Indeed, for every t-norm T the operator

Cðx; yÞ ¼ 1  Tð1  x; 1  yÞ;

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229

i.e., Cðx; yÞ ¼ T d ðx; yÞ is a t-conorm. Classes of uninorms and nullnorms are self dual in that sense that U d ðx; yÞ ¼ 1  Uð1  x; 1  yÞ is a (different) uninorm with neutral element 1  e if U has a neutral element e and V d ðx; yÞ ¼ 1  Vð1  x; 1  yÞ is a nullnorm with an annihilator 1  z if V has an annihilator z. As a uninorm has to be either conjunctive (when Uð0; 1Þ ¼ 0) or disjunctive (when Uð0; 1Þ ¼ 1) there is no self-dual uninorm. We continue with the definition of a multi-polar space and multi-polar aggregation operators. For motivation and more results on multi-polarity we recommend [15,16]. Definition 2. Let m 2 N and let K m ¼ f1; . . . ; mg be the set of m categories. Assume the input pairs of the form ðk; xÞ, with k 2 K m ; x 2 0; 1 and a neutral input 0 2 R (which belongs to each category). Then K m  ½0; 1 with the convention 0 ¼ ðk; 0Þ S n for all k 2 K m will be called a multi-polar space. A mapping M : n2N ðK m  ½0; 1Þ !K m  ½0; 1 is called an m-polar aggregation operator if (M1) M is non-decreasing, i.e., if xi 6 yi then for

Mððk1 ; x1 Þ; . . . ; ðki1 ; xi1 Þ; ðki ; xi Þ; ðkiþ1 ; xiþ1 Þ; . . . ; ðkn ; xn ÞÞ ¼ ðk; xÞ Mððk1 ; x1 Þ; . . . ; ðki1 ; xi1 Þ; ðki ; yi Þ; ðkiþ1 ; xiþ1 Þ; . . . ; ðkn ; xn ÞÞ ¼ ðp; yÞ we have one of the following cases: ðk ¼ p ¼ ki Þ ^ ðx 6 yÞ, or ðk ¼ p – ki Þ ^ ðx P yÞ, or ðk – p ¼ ki Þ. Note that in the case when minðx; yÞ ¼ 0 we assume the representation where k ¼ p. (M2) Mð0; . . . ; 0Þ ¼ 0 and Mððk; 1Þ; . . . ; ðk; 1ÞÞ ¼ ðk; 1Þ for all k 2 K m ; |fflfflfflffl{zfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} ntimes

ntimes

(M3) for n ¼ 1; Mððk; xÞÞ ¼ ðk; xÞ for all ðk; xÞ 2 K m  ½0; 1. The idea hidden in the property ðM1Þ tells us that inputs are ‘dragging’ output of the aggregation in the direction of their respective category and the bigger an input in category ki is, the more belongs the output of the aggregation to this category ki , or the less it belongs to any other category. For better understanding of this property we recommend [16,17]. In the case when m ¼ 1 (m ¼ 2) a multi-polar aggregation operator reduces to a (bipolar) aggregation operator on ½0; 1 (½1; 1). Examples of multi-polar aggregation operators include the oriented maximum operator omax (which gives the input with the maximal absolute value, and gives 0 if two inputs with maximal absolute values have different classes), the ordered category projection operator (which gives the standard aggregation of values from the most important class that is present), and the union of the projections to a single coordinate is also an m-polar aggregation operator. Generalizing the bipolar case [12,13], an ordinal sum construction for m-polar aggregation operators can be introduced as follows. Definition 3. For x ¼ ððk1 ; x1 Þ; . . . ; ðkn ; xn ÞÞ with ki 2 K m ; xi 2 ½0; 1 let

xi ¼ ðði; d1 Þ; . . . ; ði; dn ÞÞ for i ¼ 1; . . . ; m, where dj ¼ xj if j-th input belongs to i-th category, i.e., kj ¼ i, and dj ¼ 0 otherwise. Then the above is a decomposition of x to m parts according to categories (which in the bipolar case corresponds to the positive and the negative part of the input xþ ; x used for example for the symmetric and the asymmetric Choquet integral). Let Ak be an aggregation S n operator for all k 2 K m and let  : n2N ðK m  ½0; 1Þ !K m  ½0; 1 be an m-polar aggregation operator such that S n m ð0; . . . ; 0; ðk; xÞ; 0; . . . ; 0Þ ¼ ðk; xÞ for all ðk; xÞ 2 K  ½0; 1. Then the operator M  : n2N ðK m  ½0; 1Þ !K m  ½0; 1, given by

(



M ððk1 ; x1 Þ; . . . ; ðkn ; xn ÞÞ ¼

ðk; Ak ðxk ÞÞ 1

1

if ki ¼ k for all i; m

m

ðA ðx Þ; . . . ; A ðx ÞÞ else;

is an m-polar aggregation operator, which will be called an m-polar ⁄-ordinal sum of aggregation operators. We continue with the basic properties of multi-polar aggregation operators that are necessary for definition of multipolar t-norms and t-conorms. The distance on the multi-polar space K m  ½0; 1 is given by

dððk1 ; xÞ; ðk2 ; yÞÞ ¼

Definition 4. Let M :

S



j x  y j if k1 ¼ k2 ; xþy

else:

m

n

n2N ðK

 ½0; 1Þ !K m  ½0; 1 be an m-polar aggregation operator. Then

(i) M is associative if for all i 2 f1; . . . ; ng there is

Mððk1 ; x1 Þ; . . . ; ðkn ; xn ÞÞ ¼ Mððk; yÞ; ðq; zÞÞ; where ðk; yÞ ¼ Mððk1 ; x1 Þ; . . . ; ðki ; xi ÞÞ and ðq; zÞ ¼ Mððkiþ1 ; xiþ1 Þ; . . . ; ðkn ; xn ÞÞ. (ii) M is commutative if the value of Mððk1 ; x1 Þ; . . . ; ðkn ; xn ÞÞ does not depend on the order of inputs.

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(iii) M is continuous if for all i 2 f1; . . . ; ng and for all for

e > 0 there exists a d > 0 such that if dððki ; xi Þ; ðqi ; yi ÞÞ < d then

x ¼ ððk1 ; x1 Þ; . . . ; ðki1 ; xi1 Þ; ðki ; xi Þ; ðkiþ1 ; xiþ1 Þ . . . ; ðkn ; xn ÞÞ and

y ¼ ððk1 ; x1 Þ; . . . ; ðki1 ; xi1 Þ; ðqi ; yi Þ; ðkiþ1 ; xiþ1 Þ . . . ; ðkn ; xn ÞÞ we have

dðMðxÞ; MðyÞÞ < e: S n Definition 5. Let M : n2N ðK m  ½0; 1Þ !K m  ½0; 1 m ðk; xÞ 2 K  ½0; 1 is called

be

an

m-polar

aggregation

operator.

Then

the

element

(i) a k-category neutral element of M if x is a neutral element of an aggregation operator A given by

Aðx1 ; . . . ; xn Þ ¼ Mððk; x1 Þ; . . . ; ðk; xn ÞÞ: (ii) a neutral element of M if

Mððk1 ; x1 Þ; . . . ; ðk; xÞ; . . . ; ðkn ; xn ÞÞ ¼ Mððk1 ; x1 Þ; . . . ; ðkn ; xn ÞÞ for all ki 2 K m ; xi 2 ½0; 1; i ¼ 1; . . . ; n. Now we are ready to define a multi-polar t-norm and t-conorm: Definition 6. For m 2 N (i) an m-polar t-norm is a commutative, associative m-polar aggregation operator with k-category neutral elements ðk; 1Þ for all k 2 K m . (ii) an m-polar t-conorm is a commutative, associative m-polar aggregation operator with neutral element 0. The monotonicity of multi-polar aggregation operators gives us the following [15]: Proposition 1. A mapping T :

S

Tððk1 ; x1 Þ; . . . ; ðkn ; xn ÞÞ ¼

n2N ðK



m

n

 ½0; 1Þ !K m  ½0; 1 is an n-ary m-polar t-norm if it is given by

ðk; T k ðx1 ; . . . ; xn ÞÞ if ki ¼ k for all i; 0

else;

where T k is a t-norm on ½0; 1 for all k 2 K m . The above construction uses the fact that 0 is the annihilator of every t-norm T k on ½0; 1. Therefore a similar construction cannot work for multi-polar t-conorms since 0 is the neutral element of every t-conorm C k on ½0; 1 and thus CðCðð1; xÞ; ð2; yÞÞ; ð3; zÞÞ ¼ ð3; C 3 ð0; zÞÞ ¼ ð3; zÞ and Cðð1; xÞ; Cðð2; yÞ; ð3; zÞÞÞ ¼ ð1; C 1 ðx; 0ÞÞ ¼ ð1; xÞ violates the associativity. Because of the associativity we will focus mainly on the binary form of m-polar t-conorms. In the case that m ¼ 2 the bipolar t-norms are isomorphic with nullnorms on the unit interval and bipolar t-conorms are isomorphic with uninorms defined on the unit interval [14,15]. It can be seen in the following proposition which is easy to show and therefore we leave it without proof. Proposition 2. Assume an isomorphism f : ½0; 1!½1; 1. Then for any (i) Bipolar

t-norm 2

(t-conorm)

T : ½1; 12 !½1; 1 1

(C : ½1; 12 !½1; 1)

the

function

V T : ½0; 12 !½0; 1

1

(U C : ½0; 1 !½0; 1) given by V T ðx; yÞ ¼ f ðTðf ðxÞ; f ðyÞÞÞ (U C ðx; yÞ ¼ f ðCðf ðxÞ; f ðyÞÞÞ) is a nullnorm (uninorm). 1 1 (ii) Nullnorm V : ½0; 12 !½0; 1 (uninorm U : ½0; 12 !½0; 1) with annihilator z ¼ f ð0Þ (neutral element e ¼ f ð0Þ) the function T V : ½1; 12 !½1; 1 (C U : ½1; 12 !½1; 1) given by T V ðx; yÞ ¼ f ðVðf is a bipolar t-norm (t-conorm).

1

ðxÞ; f

1

ðyÞÞÞ (C U ðx; yÞ ¼ f ðUðf

1

ðxÞ; f

1

ðyÞÞÞ)

Further, it is just a question of a suitable isomorphism to transform a uninorm (nullnorm) with neutral element e 2 0; 1½ 1 1 (annihilator z 2 0; 1½) to a uninorm (nullnorm) with neutral element e ¼ f ð0Þ (annihilator z ¼ f ð0Þ). Therefore all results for uninorms and nullnorms can be transferred directly to the bipolar t-conorms and t-norms.

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2. Multi-polar t-conorms In this section we will discuss m-polar t-conorms for m P 2. One of the important classes of multi-polar t-conorms are extremal multi-polar t-conorms. First we introduce a key example of a multi-polar aggregation operator called ordered category projection. Example 1. In the case that the set K m is ordered, the m-polar aggregation operator called the ordered category projection can be assumed. In such a case let r be a permutation on K m such that rð1Þ is the most important category, rð2Þ is the second most important category and so on. Let Ak be an aggregation operator with neutral element 0 for all k 2 K m . Then the m-polar S n aggregation operator OP : n2N ðK m  ½0; 1Þ !K m  ½0; 1 given by

OPððk1 ; x1 Þ; . . . ; ðkn ; xn ÞÞ ¼ ðk; Ak ðxk ÞÞ; where k ¼ rðminðr1 ðk1 Þ; . . . ; r1 ðkn ÞÞÞ, with convention 0 ¼ ðrðmÞ; 0Þ, will be called an ordered category projection. Assume a ⁄-ordinal sum from Definition 3, where ⁄ is an ordered category projection from Example 1, and Ak is a t-conorm for all k 2 K m . In the bipolar case f1; 2g- and f2; 1g-ordered category projections correspond to the maximal and the minimal uninorms, i.e. such that are in the binary form given by Uðx; yÞ ¼ maxðx; yÞ (Uðx; yÞ ¼ minðx; yÞ) for all x; y 2 ½0; 1; x 6 e 6 y, where e is the neutral element of the uninorm U. Here f1; 2g (f2; 1g) denotes the order of importance of categories. Similarly, in an m-polar case such an ordinal sum yields an m-polar t-conorm. We will call such multi-polar tconorms extremal. Further we introduce several useful results on multi-polar t-conorms. The following result is easy to see and therefore we omit the proof. 2

Lemma 1. Let C : ðK m  ½0; 1Þ !K m  ½0; 1 be an m-polar t-conorm and assume k1 ; k2 2 K m ; k1 –k2 ; x; y 2 ½0; 1. Then if Cððk1 ; xÞ; ðk2 ; yÞÞ ¼ ðq; zÞ the monotonicity implies q 2 fk1 ; k2 g, or z ¼ 0, and if q ¼ k1 (q ¼ k2 ) then z 6 x (z 6 y). 2

Lemma 2. Let C : ðK m  ½0; 1Þ !K m  ½0; 1 be an m-polar t-conorm. Then for k1 ; . . . ; kp 2 K m mutually different and for arbitrary x1 ; . . . ; xp 2 0; 1½ the equality Cððk1 ; x1 Þ; . . . ; ðkp ; xp ÞÞ ¼ 0 implies p ¼ 2. Proof. We will provide the proof by mathematical induction. First assume that p ¼ 3. For any mutually different k1 ; k2 ; k3 2 K m assume x; y; z 2 0; 1½ such that

Cððk1 ; xÞ; ðk2 ; yÞ; ðk3 ; zÞÞ ¼ 0: In the case when Cððk1 ; xÞ; ðk2 ; yÞÞ ¼ 0 then by associativity Cððk1 ; xÞ; ðk2 ; yÞ; ðk3 ; zÞÞ ¼ ðk3 ; zÞ what is a contradiction. Similarly we can show that Cððk2 ; yÞ; ðk3 ; zÞÞ – 0 and Cððk1 ; xÞ; ðk3 ; zÞÞ – 0. Now Cððk1 ; xÞ; ðk2 ; yÞÞ belongs either to the category k1 or k2 . Without any loss of generality assume Cððk1 ; xÞ; ðk2 ; yÞÞ ¼ ðk1 ; qÞ for some q 6 x. Then there is Cððk1 ; qÞ; ðk3 ; zÞÞ ¼ 0 and for any a 2 ½0; 1 associativity and commutativity imply Cððk1 ; qÞ; ðk2 ; aÞ; ðk3 ; zÞÞ ¼ ðk2 ; aÞ. The monotonicity of C further implies Cððk1 ; qÞ; ðk2 ; aÞÞ ¼ ðk2 ; aÞ and Cððk2 ; aÞ; ðk3 ; zÞÞ ¼ ðk2 ; aÞ. If we take a ¼ y we get Cððk2 ; yÞ; ðk3 ; zÞÞ ¼ ðk2 ; yÞ and therefore Cððk1 ; xÞ; ðk2 ; yÞ; ðk3 ; zÞÞ ¼ Cððk1 ; xÞ; ðk2 ; yÞÞ – 0 what is a contradiction. Next we assume that Cððk1 ; x1 Þ; . . . ; ðkp ; xp ÞÞ – 0 for all p with 2 < p 6 P  1 for some P 2 N. We will show the result for p ¼ P. Assume Cððk1 ; x1 Þ; . . . ; ðkP ; xP ÞÞ ¼ 0 for some k1 ; . . . ; kP 2 K m mutually different and some x1 ; . . . ; xP 2 0; 1½. Then since P  1 > 2 we have Cððk1 ; x1 Þ; . . . ; ðkP1 ; xP1 ÞÞ – 0 and thus Cððk1 ; x1 Þ; . . . ; ðkP1 ; xP1 ÞÞ ¼ ðk; qÞ for some k 2 fk1 ; . . . ; kP1 g and q 2 0; 1½. Without any loss of generality assume k ¼ k1 . Then Cððk1 ; qÞ; ðkP ; xP ÞÞ ¼ 0 and associativity implies Cððk1 ; qÞ; ðk2 ; x2 Þ; ðkP ; xP ÞÞ ¼ ðk2 ; x2 Þ. The monotonicity further implies Cððk2 ; x2 Þ; ðkP ; xP ÞÞ ¼ ðk2 ; x2 Þ and thus 0 ¼ Cððk1 ; x1 Þ; . . . ; ðkP ; xP ÞÞ ¼ Cððk1 ; x1 Þ; . . . ; ðkP1 ; xP1 ÞÞ what is a contradiction. h 2

Remark 1. Let C : ðK m  ½0; 1Þ !K m  ½0; 1 be an m-polar t-conorm. Let k1 ; . . . ; kp 2 K m be mutually different and assume arbitrary x1 ; . . . ; xp 2 0; 1½. For p > 2 we know that Cððk1 ; x1 Þ; . . . ; ðkp ; xp ÞÞ – 0, i.e., Cððk1 ; x1 Þ; . . . ; ðkp ; xp ÞÞ ¼ ðk; qÞ for some q 2 0; 1½ and monotonicity implies k 2 fk1 ; . . . ; kp g. Without any loss of generality we can assume that Cððk1 ; x1 Þ; . . . ; ðkp ; xp ÞÞ ¼ ðk1 ; qÞ. By monotonicity Cððk1 ; y1 Þ; ðk2 ; y2 Þ . . . ; ðkp ; yp ÞÞ ¼ ðk1 ; sÞ for some s 2 0; 1½ for all yi 2 ½0; 1; yi 6 xi for i ¼ 2; . . . ; p and y1 2 ½0; 1; y1 P x1 . 2

Lemma 3. Let C : ðK m  ½0; 1Þ !K m  ½0; 1 be an m-polar t-conorm. If Cððk1 ; x1 Þ; ðk2 ; x2 ÞÞ ¼ 0 and Cððk3 ; x3 Þ; ðk4 ; x4 ÞÞ ¼ 0 for some x1 ; . . . ; x4 2 0; 1½ then fk1 ; k2 g ¼ fk3 ; k4 g. Proof. Assume Cððk1 ; x1 Þ; ðk2 ; x2 ÞÞ ¼ 0 and Cððk3 ; x3 Þ; ðk4 ; x4 ÞÞ ¼ 0 for some x1 ; . . . ; x4 2 0; 1½. If fk1 ; k2 g – fk3 ; k4 g then for K ¼ fk1 ; k2 g \ fk3 ; k4 g there is either CardðKÞ ¼ 0 or CardðKÞ ¼ 1. Therefore there exist k 2 fk1 ; k2 g n K and q 2 fk3 ; k4 g n K and k – q. For simplicity assume k ¼ k1 ; q ¼ k4 . Then due to the associativity there is

Cððk1 ; x1 Þ; ðk2 ; x2 Þ; ðk4 ; x4 ÞÞ ¼ ðk4 ; x4 Þ

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and monotonicity implies Cððk1 ; x1 Þ; ðk4 ; x4 ÞÞ ¼ ðk4 ; x4 Þ. On the other hand,

Cððk1 ; x1 Þ; ðk3 ; x3 Þ; ðk4 ; x4 ÞÞ ¼ ðk1 ; x1 Þ and monotonicity implies Cððk1 ; x1 Þ; ðk4 ; x4 ÞÞ ¼ ðk1 ; x1 Þ what is a contradiction since k1 – k4 . h The previous lemma shows that for an m-polar t-conorm C either Cððk1 ; x1 Þ; ðk2 ; x2 ÞÞ – 0 for all k1 ; k2 2 K m and all x1 ; x2 2 0; 1½, or there exist exactly two categories k; q from K m such that if Cððk1 ; x1 Þ; ðk2 ; x2 ÞÞ ¼ 0 for some x1 ; x2 2 0; 1½ then k1 ; k2 2 fk; qg. Remark 2. It is very easy to see that a restriction of an m-polar t-conorm (t-norm) to p categories with p < m is a p-polar tconorm (t-norm). For K  K m with CardðKÞ ¼ p we will call a restriction of an m-polar t-conorm C (t-norm T) from K m  ½0; 1 to K  ½0; 1 a partial p-polar t-conorm C K (t-norm T K ). Let C be an m-polar t-conorm, let K ¼ fk1 ; k2 g for k1 – k2 and let f : ½0; 1!½1; 1 be a linear function. Then according to Proposition 2 a partial bipolar t-conorm C fk1 ;k2 g is isomorphic to a uninorm which will be called a partial uninorm of a multi-polar t-conorm C and will be denoted by U kC1 ;k2 . Note that instead of f we can use any isomorphism, however, in order to keep original ratios of distances we prefer to work with a linear function. Further, will call the restriction of C to any category k 2 K a partial k-category t-conorm. Thus in construction of multi-polar t-conorms we can start with partial uninorms and study when are these uninorms compatible in such a way that they will not violate any of the properties of an m-polar t-conorm. In the case that associativity holds for the combination of the partial uninorms the other properties of m-polar t-conorms follow from the standard properties of these partial uninorms. Therefore associativity is the property that is the most likely to be violated. For each m 2 N, if we fix partial k-category t-conorms, we have exactly m! extremal m-polar t-conorms given by linear order of the categories. From [7, Proposition 3] it follows that if for a bipolar t-conorm (uninorm) C the partial mappings Cð; ð1; 1ÞÞ and Cð; ð2; 1ÞÞ are continuous everywhere except in the point 0 then C is an extremal bipolar t-conorm (see Example 1 and the text below). By composition of partial uninorms we can get the same result for an m-polar t-conorm for any m 2 N. 2

Proposition 3. Let C : ðK m  ½0; 1Þ !K m  ½0; 1 be an m-polar t-conorm for which partial mappings Cð; ðk; 1ÞÞ are continuous everywhere except in point 0, for all k 2 K m . Then C is an extremal m-polar t-conorm. Remark 3. On the unit interval duality is expressed by relation y ¼ 1  x, or more generally by y ¼ NðxÞ, where N is a negation on ½0; 1. By linear transformation applied in Proposition 2 the relation y ¼ 1  x is in the bipolar case transformed into  if for y  ¼ ðq; yÞ y ¼ x and extending the relation further to an m-polar case we get a category permutation, i.e.,  x is dual to y and  x ¼ ðk; xÞ there is q ¼ rðkÞ; y ¼ x. Here r : K m !K m is a permutation on the set of categories K m . Using this duality relad tion it is easy to see that U kC1 ;k2 and U kC2 ;k1 are dual uninorms, i.e., ðU kC1 ;k2 Þ ðx; yÞ ¼ 1  U kC1 ;k2 ð1  x; 1  yÞ ¼ U kC2 ;k1 ðx; yÞ. 2

Definition 7. A multipolar aggregation operator M : ðK m  ½0; 1Þ !K m  ½0; 1 will be called category commutative if

Mððk1 ; x1 Þ; . . . ; ðkn ; xn ÞÞ ¼ ðk; xÞ imply

Mððrðk1 Þ; x1 Þ; . . . ; ðrðkn Þ; xn ÞÞ ¼ ðrðkÞ; xÞ for any permutation r : K m !K m . Due to the fact that there is no self-dual uninorm there is also no category commutative m-polar t-conorm. Indeed, if we select a category permutation r such that rðk1 Þ ¼ k2 and rðk2 Þ ¼ k1 and if for example Cððk1 ; 1Þ; ðk2 ; 1ÞÞ ¼ ðk2 ; 1Þ, the category commutativity would mean that Cððrðk1 Þ; 1Þ; ðrðk2 Þ; 1ÞÞ ¼ ðrðk2 Þ; 1Þ, i.e., that Cððk2 ; 1Þ; ðk1 ; 1ÞÞ ¼ ðk1 ; 1Þ what is a contradiction. Remark 4. For every unipolar t-conorm the point 1 is an annihilator. Further, every uninorm U has an annihilator a ¼ Uð0; 1Þ 2 f0; 1g (see [7]). Therefore if C is a multi-polar t-conorm then for all k1 ; k2 2 K m there is Cððk1 ; 1Þ; ðk2 ; 1ÞÞ 2 fðk1 ; 1Þ; ðk2 ; 1Þg. This shows us that C can be restricted to the category maximal points fð1; 1Þ; . . . ; ðm; 1Þg. Then commutativity, associativity and idempotency of this restriction implies anti-symmetry, transitivity and reflexivity of an order on the set of categories given by: k1 6 k2 if and only if Cððk1 ; 1Þ; ðk2 ; 1ÞÞ ¼ ðk2 ; 1Þ (compare [19]). As any two categories are comparable in this order, we obtain a linear order on the set of categories. Assume that k is the top category of this order. Then ðk; 1Þ is an annihilator of C. Moreover, if K – ;; K # K m and q is the top category in the respective linear order on K then ðq; 1Þ is an annihilator of the restriction of C onto K  ½0; 1. From the previous remark we see that every multi-polar t-conorm possesses an annihilator. Trivially, every multi-polar t-norm has 0 as the annihilator. In [11, Theorem 6] the following result was shown. S Theorem 1. A commutative associative aggregation operator A : n2N ½0; 1n !½0; 1 with an annihilator a 2 ½0; 1 is a nullnorm if and only if its A0 and A1 sections are continuous, where A0 : ½0; 1!½0; 1 (A1 : ½0; 1!½0; 1) is a function given by A0 ðxÞ ¼ Að0; xÞ (A1 ðxÞ ¼ Að1; xÞ).

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233

This result shows us that if a restriction of a commutative associative multi-polar aggregation operator into any two categories is continuous on the boundary and possesses an annihilator then it is a multi-polar t-norm. Moreover, restriction of a multi-polar t-conorm to any two categories is non-continuous. This fact follows also from non-continuity of any uninorm. 2.1. Extensions of partial uninorms Among several classes of uninorms two perhaps most important classes are representable and almost continuous uninorms. In [22] the following result was shown: Proposition 4. Let U : ½0; 12 !½0; 1 be a uninorm continuous everywhere on the unit square expect of the two points ð0; 1Þ and ð1; 0Þ. Then U is representable, i.e., there exists such a function u : ½0; 1!½1; 1 with uðeÞ ¼ 0; uð0Þ ¼ 1; uð1Þ ¼ 1 that Uðx; yÞ ¼ u1 ðuðxÞ þ uðyÞÞ. We introduce an alternative proof of the previous proposition in Appendix. Since a representable uninorm is continuous everywhere on the unit square expect of the two points ð0; 1Þ and ð1; 0Þ we see that the above relation is if and only if. Moreover, the following result shows when is the continuity of the underlying t-norm and t-conorm enough to ensure that a uninorm is representable (compare also [8]). Proposition 5. A uninorm U : ½0; 12 !½0; 1 with the neutral element e 2 0; 1½ is representable if and only if Uj½0;e2 is a strict (i.e., continuous with no non-trivial idempotent elements) t-norm, Uj½e;12 is a strict t-conorm and there exist x; y 2 ½0; 1; x – e such that Uðx; yÞ ¼ e. Proof. The necessity is obvious. For the sufficiency assume a uninorm U such that Uj½0;e2 is a strict t-norm T; Uj½e;12 is a strict t-conorm C and there exist x; y 2 ½0; 1; x – e such that Uðx; yÞ ¼ e. Similarly as in the proof of Proposition 4 (Step 4) we can show that there exists a representable uninorm V such that Vðx; yÞ ¼ e and Uj½0;e2 ¼ Vj½0;e2 and Uj½e;12 ¼ Vj½e;12 . We will now show that U is uniquely determined and thus representable. If Uðx; yÞ ¼ e and x – e then also y – e and as 0; 1 are idempotent elements and Uð0; 1Þ is an annihilator of U we have x; y R f0; 1g. Monotonicity implies either x < e < y or y < e < x and due to commutativity we can assume x < e < y. As T and C are strict we can define, similarly as in the proof of Proposition 4 (Step ðmÞ ðmÞ 5), t 1 : 0; 1½ \ Q!½0; e with t1 ðsÞ ¼ xTn for s ¼ mn and c1 : 0; 1½ \ Q!½e; 1 with c1 ðsÞ ¼ yCn for s ¼ mn. These two functions are well defined since T and C are cancellative. By continuity of T and C we can extend these two functions to functions t 2 ; c2 with t 2 : ½0; 1!½0; e and c2 : ½0; 1!½e; 1 which are strictly monotone and continuous since T and C are strict. We will show that for every p 2 0; 1½ there exists a unique q 2 0; 1½ such that Uðp; qÞ ¼ e. First we show uniqueness. If Uðp; qÞ ¼ Uðp; q1 Þ ¼ e for some q < q1 then monotonicity implies q; q1 2 e; 1½ and since C is strict there exists an s 2 e; 1½ such that Cðq; sÞ ¼ q1 . However, then Uðp; q1 Þ ¼ Uðp; q; sÞ ¼ Uðe; sÞ ¼ s – e. Now we show the existence. Define a set E ¼ fp 2 0; 1½jthere exists q 2 0; 1½such thatUðp; qÞ ¼ eg and N ¼ 0; 1½ n E. Due to associativity we get t 2 ð0; 1½ \ QÞ 2 E and c2 ð0; 1½ \ QÞ 2 E. Further, for p 2 N there is Uðp; qÞ 2 N for all q 2 0; 1½ since if UðUðp; qÞ; zÞ ¼ e then for u ¼ Uðq; zÞ there would be Uðp; uÞ ¼ e. Assume that N is non-empty, i.e., that there exists p 2 0; 1½ such that Uðp; qÞ – e for all q 2 0; 1½. Without any loss of generality we can assume p 2 0; e½. If we take t ¼ t 1 2 ðpÞ then t 2 0; 1½ n Q. Let q ¼ c 2 ðtÞ. Since Uðp; qÞ – e we assume Uðp; qÞ > e (the case when Uðp; qÞ < e can be shown similarly). Denote s ¼ c1 2 ðUðp; qÞÞ > 0. Now take any rational number r such that r < t < r þ 2s . Due to the properties of t2 ; c2 we have Uðt 2 ðrÞ; c2 ðr þ 2s ÞÞ ¼ c2 ð2s Þ < c2 ðsÞ ¼ Uðp; qÞ. However, r < t implies t 2 ðrÞ > t2 ðtÞ ¼ p and t < r þ 2s implies q ¼ c2 ðtÞ < c2 ðr þ 2s Þ and thus monotonicity of U is violated. Summarising, have shown that for every p 2 0; 1½ there exists a unique q 2 0; 1½ such that Uðp; qÞ ¼ e. Similarly as in Proposition 4 (Step 5) we can show that inside the unit square U is uniquely determined. Now assume without loss of generality that Uð0; 1Þ ¼ 1. Then monotonicity gives us Uðp; 1Þ ¼ 1 for all p 2 ½0; 1 and Uðp; 0Þ ¼ 0 for all p 2 ½0; e. What remains is to investigate values of Uðp; 0Þ for e < p < 1. For every p with e < p < 1 there exists a unique q 2 ½0; 1 such that Uðp; qÞ ¼ e. Thus the monotonicity implies Uðp; 0Þ 6 e. Assume that there exists p such that e < p < 1 and e P Uðp; 0Þ ¼ s > 0. Then 0 < s ¼ Uðp; 0Þ ¼ Uðp; Uð0; 0ÞÞ ¼ Uðs; 0Þ, however, s ¼ Uðs; 0Þ ¼ 0 as s 6 e what is a contradiction. Therefore Uð0; pÞ ¼ 0 for all p 2 ½0; 1½ and thus U is uniquely determined on the whole unit square. h From the previous results and Lemma 3 we see that for a multi-polar t-conorm at most one of the partial uninorms can be representable. We will investigate such a case, i.e., our question is how can be a representable uninorm (a representable bipolar t-conorm) extended to three or more categories. 2

Lemma 4. Let C : ðK m  ½0; 1Þ !K m  ½0; 1 be a multi-polar t-conorm and let the partial uninorm U kC1 ;k2 be representable for some k1 ; k2 2 K m . Then Cððk1 ; xÞ; ðk; zÞÞ ¼ ðk; zÞ and Cððk2 ; xÞ; ðk; zÞÞ ¼ ðk; zÞ for all x; z 2 0; 1½ and all k 2 K m n fk1 ; k2 g. Proof. Assume that U kC1 ;k2 is representable, i.e., for all x 2 0; 1½ there exists a y 2 0; 1½ such that Cððk1 ; xÞ; ðk2 ; yÞÞ ¼ 0. Then for any z 2 0; 1½ and k R fk1 ; k2 g there is

CðCððk1 ; xÞ; ðk2 ; yÞÞ; ðk; zÞÞ ¼ Cð0; ðk; zÞÞ ¼ ðk; zÞ

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 we have Cððk2 ; yÞ; wÞ  ¼ ðk; zÞ. Comparing Cð0; wÞ  ¼w  and Cððk2 ; yÞ; wÞ  ¼ ðk; zÞ the monoand therefore if Cððk1 ; xÞ; ðk; zÞÞ ¼ w  ¼ ðk; sÞ with z 6 s. On the other hand, comparing Cð0; ðk; zÞÞ ¼ ðk; zÞ and Cððk1 ; xÞ; ðk; zÞÞ ¼ w  the monototonicity gives us w nicity gives s 6 z. Thus for all x 2 0; 1½ there is Cððk1 ; xÞ; ðk; zÞÞ ¼ ðk; zÞ. Similarly we can show that for all x 2 0; 1½ there is Cððk2 ; xÞ; ðk; zÞÞ ¼ ðk; zÞ for any z 2 0; 1½. h If we focus on three categories, this result shows us that a representable uninorm can be extended to three categories only by ordered category projection to the additional category in the open unit cube. The question remains what can be defined on the border of the unit cube. In the 3-polar case we have K 3 ¼ f1; 2; 3g. Let U 1;2 C be representable. Without loss of generality assume that Cðð1; 1Þ; ð2; 1ÞÞ ¼ ð2; 1Þ. Further, monotonicity implies Cðð2; 1Þ; ð3; zÞÞ ¼ ðk; qÞ, for k 2 f2; 3g. First assume that Cðð2; 1Þ; ð3; zÞÞ ¼ 0. Then

Cðð2; 1Þ; Cðð2; 1Þ; ð3; zÞÞÞ ¼ ð2; 1Þ but on the other hand CðCðð2; 1Þ; ð2; 1ÞÞ; ð3; zÞÞ ¼ Cðð2; 1Þ; ð3; zÞÞ ¼ 0 what is a contradiction. If Cðð2; 1Þ; ð3; zÞÞ ¼ ð2; qÞ for some q 2 0; 1½ then

Cðð2; 1Þ; Cðð2; 1Þ; ð3; zÞÞÞ ¼ Cðð2; 1Þ; ð2; qÞÞ ¼ ð2; 1Þ but CðCðð2; 1Þ; ð2; 1ÞÞ; ð3; zÞÞ ¼ Cðð2; 1Þ; ð3; zÞÞ ¼ ð2; qÞ what is a contradiction. We summarize these observations in the following lemma. 2

Lemma 5. Let C : ðK 3  ½0; 1Þ !K 3  ½0; 1 be a 3-polar t-conorm and let U 1;2 be representable with Cðð1; 1Þ; ð2; 1ÞÞ ¼ ð2; 1Þ. C Then for all z 2 0; 1½ we have one of the following options: (a) Cðð2; 1Þ; ð3; zÞÞ ¼ ð3; zÞ, (b) Cðð2; 1Þ; ð3; zÞÞ ¼ ð3; qÞ for some 0 < q < z, (c) Cðð2; 1Þ; ð3; zÞÞ ¼ ð2; 1Þ.

Remark 5. (i) If for all z 2 0; 1 we have Cðð2; 1Þ; ð3; zÞÞ ¼ ð3; zÞ (Case a) then Cðð1; 1Þ; ð3; zÞÞ ¼ Cðð1; 1Þ; Cðð2; 1Þ; ð3; zÞÞÞ ¼ CðCðð2; 1Þ; ð1; 1ÞÞ; ð3; zÞÞ ¼ Cðð2; 1Þ; ð3; zÞÞ ¼ ð3; zÞ for all z 2 0; 1. Thus on the border of the unit cube C acts as an ordered category projection and C is evidently a 3-polar t-conorm. (ii) If Cðð2; 1Þ; ð3; 1ÞÞ ¼ ð2; 1Þ then Cðð2; 1Þ; ð3; zÞÞ ¼ ð2; 1Þ for all z 2 ½0; 1. Assume Cðð2; 1Þ; ð3; 1ÞÞ ¼ ð3; 1Þ. As there is Cðð2; 1Þ; 0Þ ¼ ð2; 1Þ the monotonicity implies that there exists a v 2 ½0; 1 such that Cðð2; 1Þ; ð3; zÞÞ ¼ ð2; 1Þ for all z < v and Cðð2; 1Þ; ð3; zÞÞ ¼ ð3; qÞ for all z > v , where q 6 z. Further, if there exist a; b 2 0; v ½ such that Cðð3; aÞ; ð3; bÞÞ ¼ ð3; cÞ with c > v we get CðCðð3; aÞ; ð3; bÞÞ; ð2; 1ÞÞ ¼ Cðð3; cÞ; ð2; 1ÞÞ ¼ ð3; qÞ for some q 6 c, however,

Cðð3; aÞ; Cðð3; bÞ; ð2; 1ÞÞÞ ¼ Cðð3; aÞ; ð2; 1ÞÞ ¼ ð2; 1Þ and associativity would imply ð2; 1Þ ¼ ð3; qÞ what is not possible. Thus for all a; b 2 0; v ½ we have Cðð3; aÞ; ð3; bÞÞ ¼ ð3; cÞ, where c 6 v . Now due to the associativity we have either Cðð2; 1Þ; ð3; v ÞÞ ¼ ð2; 1Þ or Cðð2; 1Þ; ð3; v ÞÞ ¼ ð3; v Þ. Moreover, if there is a z 2 ½0; 1; v < z such that Cðð2; 1Þ; ð3; zÞÞ ¼ ð3; v Þ then associativity implies Cðð2; 1Þ; ð3; v ÞÞ ¼ ð3; v Þ. If Cðð2; 1Þ; ð3; v ÞÞ ¼ ð2; 1Þ then Cðð2; 1Þ; Cðð3; v Þ; ð3; v ÞÞÞ ¼ ð2; 1Þ and thus v is an idempotent element of the partial 3-category t-conorm C 3 . Note that if C 3 is continuous then v is an idempotent element of C 3 independently of the value Cðð2; 1Þ; ð3; v ÞÞ. (iii) We will show that equality Cðð2; 1Þ; ð3; zÞÞ ¼ ð3; qÞ for some 0 < q < z is not possible if the partial 3-category t-conorm C 3 is continuous. Assume that for some z 2 0; 1½ we have Cðð2; 1Þ; ð3; zÞÞ ¼ ð3; qÞ for some 0 < q < z. Then associativity, monotonicity and idempotency of ð2; 1Þ imply ð3; qÞ ¼ Cðð2; 1Þ; ð3; qÞÞ ¼ Cðð2; 1Þ; ð3; xÞÞ for all x 2 ½q; z. Assume any p 2 ½0; 1. If C 3 ðp; qÞ ¼ r 6 z associativity implies ð3; rÞ ¼ Cðð3; rÞ; ð2; 1ÞÞ and since q 6 C 3 ðp; qÞ ¼ r 6 z we get r ¼ q. Therefore for all p 2 ½0; 1 either C 3 ðp; qÞ ¼ q or C 3 ðp; qÞ > z. We have C 3 ð0; qÞ ¼ q and associativity and idempotency of ð2; 1Þ implies C 3 ðq; qÞ > z since C 3 ðz; qÞ ¼ C 3 ðq; qÞ P z and C 3 ðq; qÞ – z. Thus in the area ½0; q  fqg there exists a point of discontinuity of C 3 . Assume a 3-polar t-conorm C such that its partial uninorm U 1;2 is representable and Cðð2; 1Þ; ð1; 1ÞÞ ¼ ð2; 1Þ. We denote C A2 ¼ fx 2 ½0; 1jCðð2; 1Þ; ð3; xÞÞ ¼ ð2; 1Þg, A1 ¼ fx 2 ½0; 1jCðð1; 1Þ; ð3; xÞÞ ¼ ð1; 1Þg; S2 ¼ fx 2 ½0; 1jCðð2; 1Þ; ð3; xÞÞ ¼ ð3; xÞg; S1 ¼ fx 2 ½0; 1jCðð1; 1Þ; ð3; xÞÞ ¼ ð3; xÞg, and N 2 ¼ fx 2 ½0; 1jCðð2; 1Þ; ð3; xÞÞ ¼ ð3; yÞ; 0 < y < xg; N 1 ¼ fx 2 ½0; 1jCðð1; 1Þ; ð3; xÞÞ ¼ ð3; yÞ; 0 < y < xg. 2

Lemma 6. Let C : ðK 3  ½0; 1Þ !K 3  ½0; 1 be a 3-polar t-conorm and let the partial uninorm U 1;2 be representable with C Cðð2; 1Þ; ð1; 1ÞÞ ¼ ð2; 1Þ. Then S2 # S1 and A1 # A2 . Further, for x 2 Ai and q 2 Si for i 2 f1; 2g there is C 3 ðx; qÞ ¼ q.

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Proof. If q 2 S2 we have Cðð1; 1Þ; ð3; qÞÞ ¼ Cðð1; 1Þ; Cðð2; 1Þ; ð3; qÞÞÞ ¼ Cðð2; 1Þ; ð3; qÞÞ ¼ ð3; qÞ, i.e., q 2 S1 which shows that S2 # S1 . If q 2 A1 we have Cðð2; 1Þ; ð3; qÞÞ ¼ CðCðð2; 1Þ; ð1; 1ÞÞ; ð3; qÞÞ ¼ Cðð2; 1Þ; ð1; 1ÞÞ ¼ ð2; 1Þ, i.e., q 2 A2 which shows that A1 # A2 . Assume x 2 A2 and q 2 S2 (or similarly x 2 A1 and q 2 S1 ). Then associativity implies

ð3; qÞ ¼ Cðð2; 1Þ; ð3; qÞÞ ¼ CðCðð3; xÞ; ð2; 1ÞÞ; ð3; qÞÞ ¼ Cðð3; xÞ; ð3; qÞÞ; i.e. C 3 ðx; qÞ ¼ q for all x 2 A2 and q 2 S2 . The above inclusions give us the same result for the case when x 2 A1 and q 2 S2 . h Example 2. Assume a commutative 3-polar operation C such that the restriction of C onto first two categories is isomorphic 2 to a representable uninorm U 1;2 C and its restriction to the third category is a t-conorm C 3 given for all ðx; yÞ 2 ½0; 1 by

(

C 3 ðx; yÞ ¼

1

if ðx; yÞ 2

1 2

2 ;1 ;

maxðx; yÞ otherwise:

Then Cððk1 ; xÞ; ðk2 ; yÞÞ for ðk1 ; xÞ; ðk2 ; yÞ 2 K 3  ½0; 1 is given by U 1;2 if k1 ; k2 2 f1; 2g. Let Cðð3; xÞ; ð2; yÞÞ ¼ ð3; xÞ for C x; y 2 0; 1½; y < 1 and let Cðð3; xÞ; ð1; yÞÞ ¼ ð3; xÞ for x; y 2 0; 1½. We will define values Cðð2; 1Þ; ð3; xÞÞ for x 2 ½0; 1 in such a way that all three cases a), b) and c) will be present: let w 2 0; 1½ and Cðð2; 1Þ; ð3; xÞÞ ¼ ð2; 1Þ for all x < w. Now assume a S countable set of pairwise disjoint open intervals a; bi ½  ½w; 1 for i 2 I. Let S ¼ ½w; 1 n i2I ai ; bi ½. We define Cðð2; 1Þ; ð3; xÞÞ ¼ ð3; xÞ for x 2 S and Cðð2; 1Þ; ð3; xÞÞ ¼ ð3; ai Þ for x 2 ai ; bi ½. An example of the function f : f2; 3g  ½0; 1!f2; 3g  ½0; 1 given by f ððk; xÞÞ ¼ Cðð2; 1Þ; ðk; xÞÞ for ðk; xÞ 2 f2; 3g  ½0; 1 can be found on Fig. 1. In this example

f ððk; xÞÞ ¼

8 ð2; 1Þ > > > > > > < ð2; 1Þ

if k ¼ 2; if k ¼ 3 and x < 0:2;

ð3; 0:2Þ if k ¼ 3 and x 2 ½0:2; 0:4½; > > > ð3; xÞ if k ¼ 3 and x 2 ½0:4; 0:5½ [ f1g; > > > : ð3; 0:5Þ if k ¼ 3 and x 2 ½0:5; 1½:

Note that in this example the function f is right-continuous, however, this is not a necessary condition. The operation C given by the above equations is a 3-polar t-conorm. Here monotonicity, commutativity and neutral element are evident. The easy verification of associativity is left for the reader. Remark 6. Let C be a 3-polar t-conorm such that all three partial category t-conorms are continuous and Archimedean, i.e., with no non-trivial idempotent elements, and let U 1;2 C be representable with Cðð1; 1Þ; ð2; 1ÞÞ ¼ ð2; 1Þ. Then the possible cases on the border of the unit cube reduce to the following: (a) Cððk; 1Þ; ð3; zÞÞ ¼ ð3; zÞ for all z 2 ½0; 1; k 2 f1; 2g, (b) Cðð3; zÞ; ðk; 1ÞÞ ¼ ðk; 1Þ for all z 2 ½0; 1; k 2 f1; 2g, (c) Cðð3; zÞ; ð2; 1ÞÞ ¼ ð2; 1Þ and Cðð1; 1Þ; ð3; zÞÞ ¼ ð3; zÞ for all z 2 ½0; 1. Similarly as it was in the case of representable uninorms also in a general case the points x; y 2 0; 1½ such that Cððk1 ; xÞ; ðk2 ; yÞÞ ¼ 0 play an interesting role. Definition of a multi-polar t-conorm C ensures that k1 – k2 . Without any loss of generality assume that k1 ¼ 1 and k2 ¼ 2. Then Cðð1; xÞ; ð2; yÞÞ ¼ 0 and for any z 2 ½0; 1 the monotonicity (similarly as r

rrr rrrr r 

(2 , x)

(2, 1)

r

0

b

b

(3, 1)

-

(3, x)

b

Fig. 1. Function f : f2; 3g  ½0; 1!f2; 3g  ½0; 1 given by f ððk; xÞÞ ¼ Cðð2; 1Þ; ðk; xÞÞ for ðk; xÞ 2 f2; 3g  ½0; 1 from Example 2.

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in the case of representable partial uninorms) implies Cðð1; xÞ; ð3; zÞÞ ¼ ð3; zÞ and Cðð2; yÞ; ð3; zÞÞ ¼ ð3; zÞ. Moreover, the fact that Cð0; ð3; zÞÞ ¼ ð3; zÞ implies that Cðð1; qÞ; ð3; zÞÞ ¼ ð3; zÞ and Cðð2; v Þ; ð3; zÞÞ ¼ ð3; zÞ for all q 2 ½0; x; v 2 ½0; y. 3. Generated multipolar pre-t-conorms As we have seen in the previous section, there is no representable m-polar t-conorm for m > 2. This is confirmed by the result from [15, Remark 2(iv)] which states that for m > 2 there is no continuous associative m-polar aggregation operator. The same result can be shown also for m-polar aggregation operators that are continuous on the open unit hyper-cube. In the case of uninorms, representability means isomorphism with addition. The problem for m > 2 is in the fact that addition is not associative. Therefore there are two possibilities: define multi-polar t-conorms as commutative, associative, m-polar aggregation operators with neutral element 0; or define m-polar aggregation operators isomorphic with m-polar addition. We have decided to call the later generated multi-polar pre-t-conorms. Before we continue, we recall the definition of addition on a multi-polar space. Here clxðxÞ denotes the class of the input with the maximal absolute value. If it is not unique we take clxðxÞ ¼ 1. P Definition 8. Let AðdÞ ¼ ni¼1 di for d ¼ ðd1 ; . . . ; dn Þ and for input ððk1 ; x1 Þ; . . . ; ðkn ; xn ÞÞ let r be a permutation such that xrð1Þ P    P xrðmÞ . Further assume 2 6 q 6 m and

U q ððk1 ; x1 Þ; . . . ; ðkn ; xn ÞÞ ¼ ðclxðxÞ; maxð0; xrð1Þ  xrð2Þ      xrðqÞ ÞÞ: Then the operation Lq :

S

n2N ðK

m

Lq ððk1 ; x1 Þ; . . . ; ðkn ; xn ÞÞ ¼

n

 ½0; 1½Þ !K m  ½0; 1½ given by



if ki ¼ k for all i;

ðk; AðxÞÞ

U q ðð1; Aðx1 ÞÞ; . . . ; ðm; Aðxm ÞÞÞ else;

will be called a q-summation on K m  ½0; 1½. Additionally, for q ¼ 1 the 1-summation is given by L1 ððk1 ; x1 Þ; . . . ; ðkn ; xn ÞÞ ¼ U 1 ðð1; Aðx1 ÞÞ; . . . ; ðm; Aðxm ÞÞÞ, where U 1 ¼ omax. The oriented maximum operator omax (see Section 1.1) in the previous definition is one of the possible extensions of the maximum operator to the multi-polar scale. For q > 1 a q-summation extends bipolar addition. An example from [16] shows us that m-summation corresponds to a real-world situation. However, in some cases other q-summations can be preferable. The addition was so far defined only for finite absolute values. We would like to extend it also to infinite points. The bipolar case shows us that some convention is necessary: here either 1 þ ð1Þ ¼ 1 or 1 þ ð1Þ ¼ 1. If all infinities are in one category the definition is evident. Thus if we assume an input vector x ¼ ððk1 ; x1 Þ; . . . ; ðkn ; xn ÞÞ and a set IK ¼ fki jxi ¼ 1g then IK ¼ fkg implies Lm ðxÞ ¼ ðk; 1Þ. Associativity of the bipolar addition further implies that if IK ¼ fk1 ; k2 g for k1 – k2 then either Lm ðxÞ ¼ ðk1 ; 1Þ or Lm ðxÞ ¼ ðk2 ; 1Þ. In general, Lm ðxÞ ¼ Lm ðyÞ, where y ¼ ððki ; xi Þjxi ¼ 1Þi¼1;...;n . For m > 2 addition is no longer associative, however, we will require associativity of the restriction of this addition to infinite points. In such a case, similarly as in Remark 4, we can define an order of categories by k1 6 k2 if Lm ððk1 ; 1Þ; ðk2 ; 1ÞÞ ¼ ðk2 ; 1Þ. Further generalization is possible by dropping the associativity requirement. Definition 9. Assume q 6 m. Let Lq : S n P : n2N ðK m  ½0; 1Þ !K m  ½0; 1 be given by

Pððk1 ; x1 Þ; . . . ; ðkn ; xn ÞÞ ¼ f

1

S

n2N ðK

m

n

 ½0; 1½Þ !K m  ½0; 1½

be

the

q-summation

and

let

ðLq ðf ððk1 ; x1 ÞÞ; . . . ; f ððkn ; xn ÞÞÞÞ;

where f : K m  ½0; 1!K m  ½0; 1 is given by f ððk; xÞÞ ¼ ðk; ðf k ðxÞÞ and f k : ½0; 1!½0; 1 are continuous increasing functions, f k ð0Þ ¼ 0; f k ð1Þ ¼ 1 for k ¼ 1; . . . ; m. Then P is called a generated m-polar q-pre-t-conorm. Depending on the convention concerning addition of infinities we obtain a different behavior on the boundaries of the multi-polar space. Example 3. Let f i : ½0; 1!½0; 1 be given by f i ðxÞ ¼  lnð1  xÞ for i ¼ 1; . . . ; m and let f : K m  ½0; 1!K m  ½0; 1 be given by f ððk; xÞÞ ¼ ðk; ðf k ðxÞÞ. Assume addition of infinities given by the ordered category projection with order of importance of categories f1; 2; . . . ; mg (i.e., 1 is the most important category). Then the generated m-pre-t-conorm P generated by f is for maxðx1 ; . . . ; xn Þ < 1 given by

( Pððk1 ; x1 Þ; . . . ; ðkn ; xn ÞÞ ¼

ðk; 1  ð1  x1 Þ    ð1  xm ÞÞ if ki ¼ k for all i; k 2

Þ ðk; 1  AðxAðx 1 ÞAðxm ÞÞ

if Aðxk Þ ¼ minðAðx1 Þ; . . . ; Aðxm ÞÞ;

ð1Þ

where Aðd1 ; . . . ; dn Þ ¼ ð1  d1 Þ    ð1  dn Þ. In the case that maxðx1 ; . . . ; xn Þ ¼ 1 there is Pððk1 ; x1 Þ; . . . ; ðkn ; xn ÞÞ ¼ OPððq1 ; 1Þ; . . . ; ðqs ; 1ÞÞ, where OP is an ordered category projection with order of importance of categories f1; 2; . . . ; mg and qi 2 fkj jxj ¼ 1g for all i ¼ 1; . . . ; s. If a uninorm is representable then its generator can be constructed from generators of its underlying t-norm and t-conorm, knowing the value Uð0; 1Þ (i.e., the order of the two categories corresponding to the bipolar t-conorm) and a value in at least one point from 0; e½  e; 1½. This point will fix the ratio of multiplicative constants applied to additive generators t and c of underlying t-norm and t-conorm, respectively. The same holds for m-polar pre-t-conorms. Here for a generator f of a

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generated m-polar pre-t-conorm, with f ððk; xÞÞ ¼ ðk; ðf k ðxÞÞ, there exists constants q1 ; . . . ; qm 2 Rþ such that f k ¼ qk  ck , where ck is an additive generator of the partial k-category t-conorm C k . Thus generator of a generated m-polar pre-t-conorm can be constructed from generators ci of partial category t-conorms C i for i ¼ 1; . . . ; m, knowing the order of the categories (i.e., when addition of infinities is given by some convention) and the value in m  1 points of the form x ¼ ðð1; yÞ; ðk; zÞÞ; y; z 2 0; 1½; k ¼ 2; . . . ; m. These points will fix the ratio of multiplicative constants applied to additive generators ci . 4. Multi-polar uninorms In the case of bipolar t-norms and t-conorms we have seen that they are nothing else than isomorphic transformations of nullnorms and uninorms. On the other hand, bipolar aggregation operators are noting else then isomorphic transformations of aggregation operators with a non-trivial idempotent element. Therefore it is interesting to study known aggregation operators with non-trivial idempotent elements in order to see what bipolar aggregation operators they are isomorphic to and study whether it is possible to extend these operators to m-polar case for m > 2. The simplest examples of aggregation operators with a non-trivial idempotent element are ordinal sums of t-norms and t-conorms. Let T ¼ ðh0; 12 ; T 1 i; h12 ; 1; T 2 iÞ and C ¼ ðh0; 12 ; C 1 i; h12 ; 1; C 2 iÞ. If C 1 (T 1 ) is dual to T 1 (C 1 ) then taking the linear isomorphic transformation the bipolar aggregation operator MT isomorphic to T is given by

8 ð2; T 2 ðx; yÞÞ > > > < ð1; C 1 ðx; yÞÞ MTððk1 ; xÞ; ðk2 ; yÞÞ ¼ > ð1; xÞ > > : ð1; yÞ

if k1 ¼ k2 ¼ 2; if k1 ¼ k2 ¼ 1; if 1 ¼ k1 – k2 ; if k1 – k2 ¼ 1;

and the bipolar aggregation operator MC isomorphic to C is given by

8 ð2; C 2 ðx; yÞÞ > > > < ð1; T 1 ðx; yÞÞ MCððk1 ; xÞ; ðk2 ; yÞÞ ¼ > ð2; yÞ > > : ð2; xÞ

if k1 ¼ k2 ¼ 2; if k1 ¼ k2 ¼ 1; if 1 ¼ k1 – k2 ; if k1 – k2 ¼ 1:

This shows us that both operators work as ordered category projection, where the more important is always the category where the underlying partial k-category aggregation operator is a t-conorm. Moreover, both operators are dual with respect to the category permutation if and only if the original ordinal sums are dual to each other. Similar results can be obtained for any isomorphic transformation. This implies that both ordinal sum of t-norms and ordinal sum of t-conorms yield the same kind of bipolar aggregation operators. It also induces its associative m-polar extension. Since the above mentioned bipolar aggregation operators have as their underlying partial k-category aggregation operators a t-norm and a t-conorm, in more polar cases we can put together more t-norms and t-conorms. As such a multi-polar aggregation operator restricted to categories corresponding to underlying partial k-category t-norms (t-conorms) is a multi-polar t-norm (t-conorm), this operator is in fact combination of a multi-polar t-norm and a multi-polar t-conorm where inter category aggregation works as a category projection into categories corresponding to the multi-polar t-conorm. The question is how we will call this operator. Tnorms and t-conorms are generalized by both uninorms and nullnorms. However, the name uninorm was selected exactly because it unifies t-norms and t-conorms and therefore we think that the name ‘multi-polar uninorm’ is suitable in this case. It is easy to see that a multi-polar t-norm (t-conorm) can be described as an associative, commutative multi-polar aggregation operator such that its restriction to each category is a unipolar t-norm (t-conorm). Therefore we can give the following definition. Definition 10. An associative, commutative multi-polar aggregation operator such that its restriction to each category is either a (unipolar) t-norm or a (unipolar) t-conorm will be called a multi-polar uninorm. We immediately obtain the following result. 2

2

Proposition 6. Let C : ðK m  ½0; 1Þ !K m  ½0; 1 be an m-polar t-conorm and T : ðK s  ½0; 1Þ !K s  ½0; 1 be an s-polar tnorm. Assume any two disjoint subsets S1 and S2 of K mþs ¼ f1; . . . ; m þ sg where cardðS1 Þ ¼ m and cardðS2 Þ ¼ s, and two 2 bijections f : S1 !f1; . . . ; mg and g : S2 !f1; . . . ; sg. Then U C;T : ðK mþs  ½0; 1Þ !K mþs  ½0; 1 given by

8 1 > > > f ðCððf ðk1 Þ; x1 Þ; ðf ðk2 Þ; x2 ÞÞÞ > < 1 g ðTððgðk1 Þ; x1 Þ; ðgðk2 Þ; x2 ÞÞÞ U C;T ððk1 ; x1 Þ; ðk2 ; x2 ÞÞ ¼ > ðk2 ; x2 Þ > > > : ðk1 ; x1 Þ is an m þ s-polar uninorm.

if k2 ; k1 2 S1 ; if k2 ; k1 2 S2 ; if k2 2 S1 ; k1 2 S2 ; if k1 2 S1 ; k2 2 S2

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For any multi-polar uninorm if we denote by S1 categories corresponding to the multi-polar t-conorm part and by S2 categories corresponding to the multi-polar t-norm part we see that 0 is a k-category neutral element for categories k 2 S1 and ðk; 1Þ is a k-category neutral element for categories k 2 S2 . Due to the associativity the n-ary extension of an m-polar uninorm is uniquely given. If m ¼ 0 (s ¼ 0) we obtain the original s-polar (m-polar) t-norm (t-conorm) and therefore multi-polar uninorms generalize both multi-polar t-norms and t-conorms. The following lemma shows us that a multi-polar uninorm with given underlying multi-polar t-norm and a multi-polar tconorm is uniquely given. Lemma 7. Assume a commutative associative aggregation operator A : ½0; 12 !½0; 1 such that restrictions Aj 1 2 and Aj 1 2 are ½0;2 ½2;1 linear transformations of t-norms T 1 and T 2 , respectively. Then

8  2 1 > if ðx; yÞ 2 0; 12 > < 2  T 1 ð2x; 2yÞ Aðx; yÞ ¼ 1  ðT 1 ð2x  1; 2y  1Þ þ 1Þ if ðx; yÞ 2 1 ; 12 > 2 2 > : minðx; yÞ else for all ðx; yÞ 2 ½0; 12 . Proof. Assume x < 12 and y > 12. Then Að12 ; 12Þ ¼ 12 ; Aðx; 12Þ ¼ x and Að12 ; yÞ ¼ 12. Further, let Aðx; yÞ ¼ z. Then the associativity of A implies Að12 ; zÞ ¼ Að12 ; Aðx; yÞÞ ¼ Aðx; yÞ ¼ z, however, Aðz; 12Þ ¼ AðAðx; yÞ; 12Þ ¼ Aðx; 12Þ ¼ x. The commutativity implies z ¼ x thus finishing the proof. h The previous lemma shows us that if at least one input ðk; xÞ from the category related to the underlying multi-polar tconorm is present then we can ‘absorb’ all inputs from categories related to the underlying multi-polar t-norm into ðk; xÞ and therefore is the output of the aggregation equal to the aggregation of inputs from categories related to the underlying multipolar t-conorm only. Example 4. Assume an extremal bipolar t-conorm C with order of categories f2; 1g and partial k-category t-conorms equal to max, and the minimum t-norm T M ; T M ðx; yÞ ¼ minðx; yÞ for all ðx; yÞ 2 ½0; 12 . The combination of the bipolar t-conorm C and the unipolar t-norm T M yields a 3-polar uninorm MU given by

8 ð1; maxðx; yÞÞ > > > > > ð2; maxðx; yÞÞ > > > > > > < ð3; minðx1 ; x2 ÞÞ MUððk1 ; x1 Þ; ðk2 ; x2 ÞÞ ¼ ð2; x1 Þ > > > ð2; x2 Þ > > > > > ð1; x1 Þ > > > : ð1; x2 Þ

if k1 ¼ k2 ¼ 1; if k1 ¼ k2 ¼ 2; if k1 ¼ k2 ¼ 3; if k1 ¼ 2; k2 – 2; if k1 – 2; k2 ¼ 2; if k1 ¼ 1; k2 ¼ 3; if k1 ¼ 3; k2 ¼ 1:

5. Conclusions We have studied the structure of multi-polar t-conorms as extensions of uninorms to more than 2-polar case. Although the structure of multi-polar t-norms is very simple, the structure of multi-polar t-conorms, similarly as the structure of multi-polar uninorms is much more complicated. We have shown several properties of multi-polar t-conorms and discussed several special cases of partial uninorms of multi-polar t-conorms. We have introduced generated multi-polar pre-t-conorms as isomorphic transformations of the multi-polar addition. Generalization of both multi-polar t-norms and t-conorms are multi-polar uninorms. We have shown that these special operators are multi-polar extensions of transformations of ordinal sums of t-norms and t-conorms on the unit interval. We have included several examples. We expect applications of our results in all domains where use of a multi-polar setting is beneficial, where multi-polar t-conorms and uninorms can serve as basic operations. Multi-polar uninorms can distinguish between categories, where some categories can be chosen as conjunctive and the others with disjunctive behavior. Our results bring also new views on the connection between t-norms, tconorms and uninorms. Acknowledgements This work was supported by grants VEGA 2/0049/14, APVV-0073-10 and Program Fellowship of SAS.

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Appendix A

Proposition 4. Let U : ½0; 12 !½0; 1 be a uninorm continuous everywhere on the unit square expect of the two points ð0; 1Þ and ð1; 0Þ. Then U is representable, i.e., there exists such a function u : ½0; 1!½1; 1 with uðeÞ ¼ 0; uð0Þ ¼ 1; uð1Þ ¼ 1 that Uðx; yÞ ¼ u1 ðuðxÞ þ uðyÞÞ. Proof. From [7] we know that either Uð1; 0Þ ¼ 1 or Uð1; 0Þ ¼ 0. Without any loss of generality assume that Uð1; 0Þ ¼ 1 (the case when Uð1; 0Þ ¼ 0 can be proved analogically). We will provide the proof in 5 steps: 1. Show that Uð1; xÞ ¼ Uðx; 1Þ ¼ 1 for all x 2 ½0; 1 and Uð0; xÞ ¼ Uðx; 0Þ ¼ 0 for all x 2 ½0; 1½. 2. Show that there are no other idempotent elements than 0; 1; e. 3. Show that the underlying t-norm T and t-conorm C related to U are strict, i.e., generated by continuous additive generators t and c with tð0Þ ¼ cð1Þ ¼ 1. 4. Show that for any point p 2 0; e½ there is a q 2 e; 1½ such that Uðp; qÞ ¼ e and there exists a representable unimorm U f such that U f ðp; qÞ ¼ e; U f j½0;e2 ¼ T; U f j½e;12 ¼ C. 5. Show that there is only one uninorm U such that Uðp; qÞ ¼ e and Uj½0;e2 ¼ T; Uj½e;12 ¼ C for a fixed p; q 2 ½0; 1 and thus the given uninorm has to be equal to the representable uninorm U f . 1:: If Uð1; 0Þ ¼ 1 then the monotonicity gives us 1 ¼ Uð1; 0Þ 6 Uð1; xÞ 6 Uð1; 1Þ ¼ 1, i.e., commutativity gives us Uð1; xÞ ¼ Uðx; 1Þ ¼ 1 for all x 2 ½0; 1. Further, 0 6 Uð0; xÞ 6 Uð0; eÞ ¼ 0 gives us Uð0; xÞ ¼ 0 for all x 2 ½0; e. Assume that Uð0; xÞ ¼ e for some x 2 e; 1½. Then e ¼ Uð0; xÞ ¼ UðUð0; 0Þ; xÞ ¼ Uð0; Uð0; xÞÞ ¼ Uð0; eÞ ¼ 0 what is not possible. Thus continuity gives us Uð0; xÞ < e for all x 2 ½0; 1½. Now assume that Uð0; xÞ ¼ y > 0 for some x 2 e; 1½. Then y < e and 0 < y ¼ Uð0; xÞ ¼ UðUð0; 0Þ; xÞ ¼ Uð0; Uð0; xÞÞ ¼ Uð0; yÞ ¼ 0 what is a contradiction. Thus Uð0; xÞ ¼ Uðx; 0Þ ¼ 0 for all x 2 ½0; 1½. 2:: Assume that there is an idempotent element a – f0; e; 1g. Without any loss of generality assume that a > e (the case when a < e can be proved analogically). Now we have Uðe; aÞ ¼ a > e and Uð0; aÞ ¼ 0. The continuity implies that there exists a point p < e such that Uðp; aÞ ¼ e. Then we have e ¼ Uða; pÞ ¼ UðUða; aÞ; pÞ ¼ Uða; Uða; pÞÞ ¼ Uða; eÞ ¼ a what is a contradiction. 3:: Let T be a t-norm on ½0; e and C a t-conorm on ½e; 1 such that Uj½0;e2 ¼ T and Uj½e;12 ¼ C. Then T and C are continuous and since they have no idempotent elements different to 0; e; 1 we know that T and C are either nilpotent or strict. We will show that C is strict (the proof that T is strict is analogical). Suppose that C is nilpotent. Then there exist such two points x; y 2 e; 1½ that Uðx; yÞ ¼ Cðx; yÞ ¼ 1. Then Uðe; xÞ ¼ x > e and Uð0; xÞ ¼ 0 and therefore the continuity implies that there exists a p 2 ½0; e½ such that Uðp; xÞ ¼ e. Then the associativity implies: y ¼ Uðe; yÞ ¼ UðUðp; xÞ; yÞ ¼ Uðp; Uðx; yÞÞ ¼ Uðp; 1Þ ¼ 1 what is a contradiction. 4:: Since t-norm T (t-conorm C) is strict there exists an additive generator t : ½0; e!½0; 1 (c : ½e; 1!½0; 1) unique up to a positive multiplicative constant such that tð0Þ ¼ 1 and tðeÞ ¼ 0 (cð1Þ ¼ 1 and cðeÞ ¼ 0) and Tðx; yÞ ¼ t 1 ðtðxÞ þ tðyÞÞ for ðx; yÞ 2 ½0; e2 (Cðx; yÞ ¼ c1 ðcðxÞ þ cðyÞÞ for ðx; yÞ 2 ½e; 12 ). Now assume any point p 2 0; e½. Since Uðe; pÞ ¼ p < e and Uð1; pÞ ¼ 1 from continuity there exist a point q 2 e; 1½ such that Uðp; qÞ ¼ e. We will construct a function f : ½0; 1!½1; 1 as follows:

( f ðxÞ ¼

tðxÞ

if x 2 ½0; e;

tðpÞ cðxÞ cðqÞ

otherwise: 1

Then f ðeÞ ¼ 0; f ð0Þ ¼ 1; f ð1Þ ¼ 1 and for the uninorm U f given by U f ðx; yÞ ¼ f ðf ðxÞ þ f ðyÞÞ we have U f ðp; qÞ ¼ e; U f j½0;e2 ¼ T and U f j½e;12 ¼ C. 5:: Assume any uninorm U with Uðp; qÞ ¼ e for some fixed ðp; qÞ 2 0; e½  e; 1½ and Uj½0;e2 ¼ T and Uj½e;12 ¼ C. We will show that all other values of U on ½0; 12 are uniquely given. If Uðp; qÞ ¼ e then also ðnÞ ðnÞ UðUðp; pÞ; Uðq; qÞÞ ¼ UðUðp; qÞ; Uðp; qÞÞ ¼ Uðe; eÞ ¼ e and similarly UðpT ; qC Þ ¼ e where ðnÞ

ðnÞ

pT ¼ pU ¼ Uðp; Uðp; . . .ÞÞ ¼ Tðp; Tðp; . . .ÞÞ; |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} ntimes

ðnÞ

ntimes

ðnÞ

qC ¼ qU ¼ Uðq; Uðq; . . .ÞÞ ¼ Cðq; Cðq; . . .ÞÞ; |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} ntimes

ntimes

ð1Þ

ð1Þ

for all n 2 N. Similarly since T and C are strict we can define pTn (qCn ) as such unique number an (bn ) for which

p ¼ Uðan ; Uðan ; . . .ÞÞ ¼ Tðan ; Tðan ; . . .ÞÞ; |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} ntimes

ntimes

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q ¼ Uðbn ; Uðbn ; . . .ÞÞ ¼ Cðbn ; Cðbn ; . . .ÞÞ: |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} ntimes

ntimes

ðnÞ

ðnÞ

Then, however, for a number zn ¼ Uðan ; bn Þ we get ðzn ÞU ¼ Uðp; qÞ ¼ e and since both T and C are strict we have aU ¼ e for ð1nÞ

ð1nÞ

some a 2 ½0; 1 if and only if a ¼ e and thus UðpT ; qC Þ ¼ zn ¼ e. Similarly if we assume any ðx; yÞ 2 0; e½  e; 1½ such that ð1Þ

ð1Þ

ðmÞ

ðmÞ

Uðx; yÞ ¼ e we get UðxTn ; yCn Þ ¼ e for all n 2 N. Combining these results we have UðpTn ; qCn Þ ¼ e for all rational numbers ð1nÞ

ð1nÞ

ðmÞ

s ¼ mn 2 0; 1½. We have also limn!1 pT ¼ e; limn!1 qC ¼ e and limm!1 pT ðmÞ pTn

ðmÞ

¼ 0; limm!1 qC

¼ 1. So far we have two funcðmÞ

for s ¼ mn and c1 : 0; 1½ \ Q!½e; 1 with c1 ðsÞ ¼ qCn for s ¼ mn. By continuity we tions t 1 : 0; 1½ \ Q!½0; e with t 1 ðsÞ ¼ can extend these two functions to functions t 2 ; c2 with t2 : ½0; 1!½0; e and c2 : ½0; 1!½e; 1. Now for any point u 2 0; e½ there is a unique point v 2 e; 1½ such that Uðu; v Þ ¼ e and it is given by v ¼ c2 ðt1 2 ðuÞÞ. Thus all points ðx; yÞ 2 0; e½  e; 1½ such that Uðx; yÞ ¼ e are uniquely determined by a single point ðp; qÞ. Finally, we will show that if Uðx; yÞ ¼ e for some ðx; yÞ 2 0; e½  e; 1½ and Uj½0;e2 ¼ T; Uj½e;12 ¼ C then all values Uðx; dÞ for d 2 ½y; 1 and Uða; yÞ for a 2 ½0; x are uniquely determined. We will show only that Uðx; dÞ for d 2 ½y; 1 is uniquely determined (the case with Uða; yÞ for a 2 ½0; x can be shown analogically). If d 2 ½y; 1 then since Cðe; yÞ ¼ y and Cð1; yÞ ¼ 1 from the continuity there exists a unique point p 2 ½e; 1 such that Cðp; yÞ ¼ d. Now Uðx; dÞ ¼ Uðx; Uðy; pÞÞ ¼ UðUðx; yÞ; pÞ ¼ Uðe; pÞ ¼ p. Thus for any d 2 ½y; 1 the value Uðx; dÞ is uniquely determined by C. Therefore if we have any point ðx; yÞ 2 ½0; 12 the value of Uðx; yÞ is: Cðx; yÞ if ðx; yÞ 2 ½e; 12 ; Tðx; yÞ if ðx; yÞ 2 ½0; e2 ; 1 if maxðx; yÞ ¼ 1; 0 if minðx; yÞ ¼ 0 and maxðx; yÞ < 1; if ðx; yÞ 2 0; e½  e; 1½: assume points p 2 e; 1½ with Uðx; pÞ ¼ e and q 2 0; e½ with Uðq; yÞ ¼ e. Now if y ¼ p we have Uðx; yÞ ¼ e, if y > p then Uðx; yÞ ¼ a where y ¼ Cðp; aÞ and finally if y < p then Uðx; yÞ ¼ b where x ¼ Tðb; qÞ; (vi) Uðx; yÞ ¼ Uðy; xÞ if ðx; yÞ 2 e; 1½  0; e½.

(i) (ii) (iii) (iv) (v)

Summarising, given the point ðp; qÞ 2 0; e½  e; 1½ such that Uðp; qÞ ¼ e and Uj½0;e2 ¼ T; Uj½e;12 ¼ C the uninorm on ½0; 12 is uniquely determined and thus it have to coincide with the uninorm U f generated by the additive generator f. h References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

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