NON-COMMUTATIVE NULLNORMS 1. Introduction De nition 1. A

International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems Vol. 0, No. 0 (1993) 000{000 cfWorld Scienti c Publishing Company

NON-COMMUTATIVE NULLNORMS RONALD R. YAGER Machine Intelligence Institute Iona College 715 North Avenue, New Rochelle, NY 10801-1890, USA email [email protected] VLADIK KREINOVICH Department of Computer Science University of Texas at El Paso El Paso, TX 79968, USA email [email protected] Received Revised In their recent paper, Calvo, de Baets, and Fodor introduced a new generalization of t-norms and t-conorms called a nullnorm, and describe all possible nullnorms. In this paper, we provide a further generalization of nullnorms by considering non-commutative nullnorms, and extend a description of Calvo et al. to this more general case; it turns out that while commutative nullnorms have a unique null element, non-commutative nullnorms have the whole interval of null elements. Keywords : Null-norms, t-norms, t-conorms, non-commutative.

1. Introduction

The t-norm and the t-conorm (which are de ned in the following text) provide an important class of aggregation operations (see, e.g.,6 7 10): De nition 1. A function f& : [0; 1]  [0; 1] ! [0; 1] is called a t-norm if it satis es the following four conditions for all x, y, z , and t: ; ;

   

f& (1; x) = x (identity); f& (x; y) = f& (y; x) (commutativity); f& (x; f& (y; z )) = f& (f& (x; y); z )) (associativity); if x  z and y  t, then f& (x; y)  f& (z; t) (monotonicity).

1

2 Non-Commutative Nullnorms

De nition 2. A function f& : [0; 1]  [0; 1] ! [0; 1] is called a t-conorm if it satis es

the following four conditions for all x, y, z , and t:

   

f& (0; x) = x (identity); f& (x; y) = f& (y; x) (commutativity); f& (x; f& (y; z )) = f& (f& (x; y); z )) (associativity); if x  z and y  t, then f& (x; y)  f& (z; t) (monotonicity).

(It is also often required that the t-norm and t-conorm be continuous.) Both t-norms and t-conorms are commutative associative monotonic functions. They both have an identity (unit) element, i.e., an element e for which f (e; x) = x for all x. The main dierence between them is that for a t-norm, e = 1 is a unit element, while for a t-conorm, e = 0 is a unit element. This distinction leads to f&(x; y)  min(x; y) and f_(x; y)  max(x; y). It is worth mentioning that there is a generalization of t-norms and t-conorms in which there exists a unit element which can be an arbitrary number e 2 [0; 1]; this generalization is called a uninorm (see, e.g.,2 3 5 12 13): Another dierence between a t-norm and a t-conorm is that they have dierent absorbing (null) elements. An element g is called an absorbing element of an operation f if f (g; x) = g for all x. Thus, the appearance of g as one of the arguments of an aggregation procedure always results in g being the aggregated value: ; ; ;

;

 For a t-norm, since f& (1; x) = x for all x, we have f& (1; 0) = 0. Hence, monotonicity implies that f&(x; 0) = 0, i.e., that 0 is an absorbing element.  For a t-conorm, since f_ (0; x) = x for all x, we have f& (0; 1) = 1. Hence, monotonicity implies that f&(x; 1) = 1, i.e., that 1 is an absorbing element. In1 , T. Calvo, B. de Baets, and J. Fodor propose a new generalization of t-norms and t-conorms called a nullnorm: De nition 3. Let g be a real number from the interval [0; 1]. A commutative associative monotonic function f : [0; 1]  [0; 1] ! [0; 1] is called a nullnorm with an absorbing element g if it satis es the following three conditions:

 for every x, we have f (g; x) = g;  for every x  g, we have f (x; 0) = x;  for every x  g, we have f (x; 1) = x. It is easy to see that when the absorbing (null) element g is equal to 0, the de nition of a nullnorm turns into the de nition of a t-norm, and when g = 1, the de nition of a nullnorm turns into a de nition of a t-conorm. Nullnorms corresponding to g 2 (0; 1) are new binary operations.

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The authors of1 provide the general description of nullnorms: for g = f (0; 1), a nullnorm is equal to a t-norm for x; y  g, it is equal to a t-conorm for x; y  g, and f (x; y) = g when x  g and y  g. x 2 [0; g] x 2 [g; 1] y 2 [g; 1] g t-norm y 2 [0; g] t-conorm g Let us describe this result in precise terms. De nition 4. Let g 2 (0; 1) be a real number, f&(x; y) be a t-norm, and f_(x; y) be a t-conorm. By their combination C (g; f& ; f_), we mean the following operation f (x; y):    when g  x; y  1, we have f (x; y) = g + (1 ? g)
f& x1 ?? gg ; y1 ?? gg ;    when 0  x; y  g, we have f (x; y) = g
f_ xg ; yg ;  when x < g < y or y < g < x, we have f (x; y) = g:

Theorem1.  Every t-norm f& is a nullnorm, every t-conorm f_ is a nullnorm, and every combination C (g; f& ; f_) is a nullnorm.  Every nullnorm is either a t-norm, or a t-conorm, or a combination C (g; f& ; f_ ).

The above combination f (a; b) acts as a t-conorm (i.e., as \or") for small a; b, and as a t-norm (i.e., as \and") for larger a and b. This combination is similar to the \ordinal sum" construction which describes general t-norms and t-conorms (see, e.g.,7 ), with one main dierence:  in the \ordinal sum" construction, we combine several t-norms with each other or several t-conorms with each other, while  in the above construction, we combine a t-norm with a t-conorm. Comment. The above construction of a nullnorm from a t-norm and a t-conorm is similar to a construction which generates a uninorm from a t-norm and a t-conorm, with the following main dierence: the description of a nullnorm has the t-norm \above" the t-conorm (t-norm for larger values and t-conorm for smaller values), while the uninorm has them in the opposite order.

2. Motivations for Using Nullnorms

A t-norm f& (a; b) describes the degree to which two conditions A and B are both satis ed if we know that the rst condition A is satis ed with a degree a, and the second condition B is satis ed with a degree b.

4 Non-Commutative Nullnorms

In eect, t-norms describe the situations when both conditions are absolutely necessary, so that if one of the conditions is not satis ed, we completely reject the corresponding alternative. There are many such situations, but there are also many other situations, in which, although we say that we want the rst condition to be satis ed and the second condition to be satis ed, etc., but if one of these conditions is not satis ed, we may still consider the corresponding alternative. For example, a computer science department may be looking for a person who is a brilliant researcher and a very good lecturer and is knowledgeable in all the areas of computer science, i.e., in data structures and in operating systems and in software engineering etc. Ideally, all these conditions should be met. However, if a brilliant researcher with a reputation of a good lecturer applies for a position, then, even if he does not know anything about operating systems, a department would most probably not de nitely reject him. In short, in many real-life situations, even if one of the conditions A, B is not satis ed at all, e.g., if a = 0, we may still have some non-zero degree of belief in the conjunction A&B { in direct contrast to the fact that for a t-norm, in this case, f& (0; b) = 0. This dierence between the formal notion of a t-norm and the human use of \and" was noticed several decades ago, in the experiments of H.-J. Zimmermann and P. Zysno described in14 . To get a more adequate description of human \and"-operations, the authors of14 propose to use, instead of t-norm fe&(a; b), a combination (e.g., linear combination) of a t-norm fe& (a; b) and a tconorm fe_ (a; b) (a fuzzy equivalent of \or"), e.g., to use a combination f& (a; b) = (1 ? )
fe& (a; b) + 
fe_(a; b). Such a combination satis es the desired property f&(0; 1) > 0, and it keeps some properties of the t-norm such as commutativity, but by using this operation, we lose another important property of \and"-operations: associativity. Let us see whether we can keep associativity by weakening some other less intuitive conditions from the de nition of a t-norm. Of the four properties of a t-norm, the only property which we do not want to keep fully is the rst one, that f& (1; x) = x for all x, because as a particular case of this property, for x = 1, we get f& (1; 0) = 0, and we want f&(1; 0) > 0. Not all particular cases of this property need to be dismissed: e.g., when x = 1, i.e., when both conditions are absolutely true, we still want to be able to conclude that the conjunction A&B is absolutely true, i.e., that f& (1; 1) = 1. Similarly, if neither of the conditions A and B is satis ed at all, we expect the conjunction A&B to be absolutely false, so we must have f&(0; 0) = 0. Thus, instead of the rst condition from the de nition of a t-norm, we should only require that f&(0; 0) = 0 and f& (1; 1) = 1. It turns out that for continuous operations, the two conditions f& (0; 0) = 0 and f&(1; 1) = 1 are equivalent to the three conditions from the de nition of a nullnorm: Theorem8. A continuous commutative associative monotonic function f : [0; 1]  [0; 1] ! [0; 1] is a nullnorm if and only if f (0; 0) = 0 and f (1; 1) = 1.

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3. Non-Commutative Nullnorms

Similarly to t-norms and t-conorms, nullnorms can be used to combine degrees of certainty. For this intended application, commutativity is not always a reasonable requirement. Indeed, the combination f (x; y) may re ect, e.g., the upgrade in our degree of certainty x when we get a new piece of evidence whose degree of certainty is y. In this case, there is an asymmetry between the old and the new knowledge:  we may put more emphasis on the old knowledge, or  we may put more emphasis on the new knowledge. In both cases, commutativity is not desired. The following result describes all possible non-commutative nullnorms. De nition 5. A continuous associative monotonic function f : [0; 1]  [0; 1] ! [0; 1] is called a generalized nullnorm if f (0; 0) = 0 and f (1; 1) = 1. Theorem 1. For every generalized nullnorm f (x; y), there exist two numbers g  G for which f (x; y) has one of the following two forms: 1:  for x; y  G, f (x; y) is equal to a t-norm;  for x  G and y  G, we have f (x; y) = G;  for g  x  G, we have f (x; y) = x;  for x  g and y  g, we have f (x; y) = g;  for x; y  g, f (x; y) is equal to a t-conorm. 2:  for x; y  G, f (x; y) is equal to a t-norm;  for y  G and x  G, we have f (x; y) = G;  for g  y  G, we have f (x; y) = y;  for y  g and x  g, we have f (x; y) = g;  for x; y  g, f (x; y) is equal to a t-conorm.

Nullnorms of the rst type can be visualized by the following table: x 2 [0; g] x 2 [g; G] x 2 [G; 1] y 2 [G; 1] g x t-norm y 2 [g; G] g x G y 2 [0; g] t-conorm x G Nullnorms of the second type can be visualized by the following table: x 2 [0; g] x 2 [g; G] x 2 [G; 1] y 2 [G; 1] G G t-norm y 2 [g; G] y y y y 2 [0; g] t-conorm g g

6 Non-Commutative Nullnorms

Before we prove the theorem, let us make some comments. Comment 1. Non-commutativity means that one of the values is given a priority:

 In the rst case, priority is given to the old value x. For example, if the new information is not de nite (i.e., if y is in the middle zone [g; G]), then f (x; y) = x, which means that we keep the old value x.

 In the second case, priority is given to the new value y. For example, if originally, we did not have a strong opinion about a statement (i.e., if x was is the middle zone [g; G]), then we f (x; y) = y, which means that we take the new degree y as the result of our belief update. Comment 2. In general, g  G. As we can see from Theorem 1, when g = G, we get a commutative operation, i.e., a regular nullnorm. Non-commutative case corresponds to g < G. The very term \nullnorm" comes from the fact that in algebra, an element g for which f (g; x) = x for every x is called a null element. In a commutative nullnorm, there is only one null element { g. In a non-commutative nullnorm, when g < G, we thus have a whole interval [g; G] of null elements. The existence of the whole interval of null elements means that instead of a single neutral (\don't know") value, we have the whole interval of dierent values describing, so to say, dierent \shades" of this neutral (\don't know") attitude. If old and new knowledge are inconsistent, e.g., if x < g and y > G or if x > G and y < g, then the resulting degree of certainty is \we don't know". The exact \shade" of this \do not know" depends on whether we give priority to the old or to the new knowledge:

 In the rst case, we trust the old knowledge more than the new one. In this

case, it is natural to have more belief in a statement when the old knowledge supports it (and the new knowledge refutes it) than in the opposite situation when the new knowledge supports the statement while the old knowledge refutes it. This idea is consistent with our formulas:

 when the old knowledge supports a statement (x > G) and the new knowledge supports it (y < g), the resulting value is f (x; y) = G;  when the old knowledge refutes a statement (x < g) and the new knowledge supports it (y > G), the resulting value is f (x; y) = g < G.  In the second case, we trust the new knowledge more than the old one (e.g.,

because the old knowledge may be outdated). In this case, it is natural to have more belief in a statement when the new knowledge supports (and the

International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 7

old knowledge refutes it) than in the opposite situation when the old knowledge supports the statement while the new knowledge refutes it. This idea is consistent with our formulas:

 when the old knowledge refuted a statement (x < g) and the new knowledge supports it (y > G), the resulting value is f (x; y) = G;  when the old knowledge supported a statement (x > G) and the new knowledge refutes it (y < g), the resulting value is f (x; y) = g < G. Comment 3. Nullnorms are generalizations of t-norms and t-conorms which have null elements. Both t-norms and t-conorms have not only null elements, they also have unit elements, i.e., elements e for which f (e; x) = x for all x: for a t-norm, 1 is the unit element, while for a t-conorm, 0 is the unit element. It is worth mentioning that there is another generalization of t-norms and t-conorms in which an arbitrary number g 2 [0; 1] can serve as a unit element; this generalization is called a uninorm (see, e.g.,2 3 5 12 13 ). In view of our result about non-commutative nullnorms, it would be interesting to investigate non-commutative uninorms and see if they happen to have the entire interval of unit elements. Comment 4. In the idempotent case, all possible non-commutative operations were described in4 ; ; ;

4. Proof of Theorem 1

;

1: In the description of a nullnorm, the number g is equal to f (0; 1). Since a generalized nullnorm is not necessarily commutative, let us consider two numbers (which may be dierent): f (0; 1) and f (1; 0). Let us denote the smallest of these numbers by g, and the largest of these two numbers by G. Let us show that when the number f (0; 1) is the smallest, i.e., when f (0; 1) = g  G = f (1; 0), we get a generalized nullnorm of the rst type. (Similarly, one can prove that when the number f (1; 0) is the smallest, we get a generalized nullnorm of the second type.) 2: Let us rst consider the values for f (x; y) for x; y  G. 2:1: Let us rst show that f (G; 0) = G. Indeed, by the de nition of G, f (G; 0) = f (f (1; 0); 0). By associativity, f (G; 0) = f (1; f (0; 0)). By the de nition of a generalized nullnorm, f (0; 0) = 0, hence f (G; 0) = f (1; 0) = G. 2:2. By monotonicity, we can now conclude that if x; y  G, then f (x; y)  f (G; 0) = G. Thus, for values x; y 2 [G; 1], we have f (x; y) 2 [G; 1]. Hence, the restriction f of a function f to the interval [G; 1] is an operation f : [G; 1]  [G; 1] ! [G; 1]. Since the original function f is continuous, associative, and monotonic, its restriction f is also continuous, associative, and monotonic. G

G

8 Non-Commutative Nullnorms

2:3. Let us show that the element 1 is a unit element for the operation f , i.e., that f (1; x) = f (x; 1) = x for all x  G. 2:3:1. Let us rst show that f (1; x) = x for every x 2 [G; 1]. Indeed, let x 2 [G; 1]. By the de nition of a generalized nullnorm, we have f (1; 1) = 1. By de nition of G, we have f (1; 0) = G. Thus, from G  x  1, we conclude that f (1; 0)  x  f (1; 1). Since f (1; t) is a continuous function, we can apply the Intermediate Value Theorem and conclude that there exists an element t 2 [0; 1] for which f (1; t) = x. Thus, f (1; x) = f (1; f (1; t)). By associativity, we get f (1; x) = f (f (1; 1); t). By the de nition of a generalized nullnorm, f (1; 1) = 1, so f (1; x) = f (1; t). By our choice of t, this means that f (1; x) = x. 2:3:2. Let us now show that f (x; 1) = x for every x 2 [G; 1]. Indeed, let x 2 [G; 1]. By the de nition of a generalized nullnorm, we have f (1; 1) = 1. By de nition of g, we have f (0; 1) = g. Thus, from G  x  1, and from g  G, we conclude that g  x  1, i.e., f (0; 1)  x  f (1; 1). Since f (t; 1) is a continuous function, we can apply the Intermediate Value Theorem and conclude that there exists an element t 2 [0; 1] for which f (t; 1) = x. Thus, f (x; 1) = f (f (t; 1); 1). By associativity, we get f (x; 1) = f (t; f (1; 1)). By the de nition of a generalized nullnorm, f (1; 1) = 1, so f (x; 1) = f (t; 1). By our choice of t, this means that f (x; 1) = x. 2:4. Let us show that the element G is a null element for the operation f , i.e., that f (G; x) = f (x; G) = G for all x  G. 2:4:1. Let us rst show that f (G; x) = G for all x 2 [G; 1]. We will actually prove a stronger statement: that f (G; x) = G for all x 2 [0; 1]. Due to monotonicity of the operation f , it is sucient to prove this for two cases: for x = 1 and for x = 0; then, the desired equality for an arbitrary x 2 [0; 1] will follow from the fact that G = f (G; 0)  f (G; x)  f (G; 1) = G. For x = 1, the desired equality f (G; 1) = G follows from the already proven fact that 1 is a unit element for the interval [G; 1], and G belongs to this interval. Let us show that f (G; 0) = G. Indeed, by the de nition of G, we have G = f (1; 0), hence f (G; 0) = f (f (1; 0); 0). By associativity, we get f (G; 0) = f (1; f (0; 0)). By the de nition of a generalized nullnorm, f (0; 0) = 0, so f (G; 0) = f (1; 0) = G. 2:4:2. Let us now show that f (x; G) = G for all x 2 [G; 1]. Indeed, we have already shown that f (G; x) = x for all x 2 [G; 1], hence f (G; G) = G. We also know that 1 is a unit element on [G; 1], hence f (1; G) = G. So, from monotonicity, we can conclude that f (x; G) = G for all x between G and 1. 2:5. We have shown that the operation f is a continuous associative operation on the interval [G; 1]. We have also shown that the lower endpoint G of this interval is a null element, and the upper endpoint 1 of this interval is a unit element. Theorem G

G

G

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B from9 enables us then to conclude that f is an ordinal sum of Archimedean operations of this same type. Hence, f is also commutative11, and f is a t-norm. G

G

G

3: Let us now consider the values for f (x; y) for x; y  g. 3:1: Let us rst show that f (g; 1) = g. Indeed, by the de nition of g, f (g; 1) = f (f (0; 1); 1). By associativity, f (g; 1) = f (0; f (1; 1)). By the de nition of a generalized nullnorm, f (1; 1) = 1, hence f (g; 1) = f (0; 1) = g. 3:2. By monotonicity, we can now conclude that if x; y  g, then f (x; y)  f (g; 1) = g. Thus, for values x; y 2 [0; g], we have f (x; y) 2 [0; g]. Hence, the restriction f of the function f to the interval [0; g] is an operation f : [0; g]  [0; g] ! [0; g]. Since the original f is continuous, associative, and monotonic, its restriction f is also continuous, associative, and monotonic. 3:3. Let us show that the element 0 is a unit element for the operation f , i.e., that f (0; x) = f (x; 0) = x for all x  g. 3:3:1. Let us rst show that f (0; x) = x for every x 2 [0; g]. Indeed, let x 2 [0; g]. By the de nition of a generalized nullnorm, we have f (0; 0) = 0. By de nition of g, we have f (0; 1) = g. Thus, from 0  x  g, we conclude that f (0; 0)  x  f (0; 1). Since f (0; t) is a continuous function, we can apply the Intermediate Value Theorem and conclude that there exists an element t 2 [0; 1] for which f (0; t) = x. Thus, f (0; x) = f (0; f (0; t)). By associativity, we get f (0; x) = f (f (0; 0); t). By the de nition of a generalized nullnorm, f (0; 0) = 0, so f (0; x) = f (0; t). By our choice of t, this means that f (0; x) = x. 3:3:2. Let us now show that f (x; 0) = x for every x 2 [0; g]. Indeed, let x 2 [0; g]. By the de nition of a generalized nullnorm, we have f (0; 0) = 0. By de nition of G, we have f (1; 0) = G. Thus, from 0  x  g, and from g  G, we conclude that 0  x  G, i.e., f (0; 0)  x  f (1; 0). Since f (t; 0) is a continuous function, we can apply the Intermediate Value Theorem and conclude that there exists an element t 2 [0; 1] for which f (t; 0) = x. Thus, f (x; 0) = f (f (t; 0); 0). By associativity, we get f (x; 0) = f (t; f (0; 0)). By the de nition of a generalized nullnorm, f (0; 0) = 0, so f (x; 0) = f (t; 0). By our choice of t, this means that f (x; 0) = x. 3:4. Let us show that the element g is a null element for the operation f , i.e., that f (g; x) = f (x; g) = g for all x  g. 3:4:1. Let us rst show that f (g; x) = g for all x 2 [0; g]. We will actually prove a stronger statement: that f (g; x) = g for all x 2 [0; 1]. Due to monotonicity of the operation f , it is sucient to prove this for two cases: for x = 1 and for x = 0; then, the desired equality for an arbitrary x 2 [0; 1] will follow from the fact that g = f (g; 0)  f (g; x)  f (g; 1) = g. g

g

g

g

10 Non-Commutative Nullnorms

For x = 0, the desired equality f (g; 0) = g follows from the already proven fact that 0 is a unit element for the interval [0; g], and g belongs to this interval. Let us show that f (g; 1) = g. Indeed, by the de nition of g, we have G = g(0; 1), hence f (g; 1) = f (f (0; 1); 1). By associativity, we get f (g; 1) = f (0; f (1; 1)). By the de nition of a generalized nullnorm, f (1; 1) = 1, so f (g; 1) = f (0; 1) = g. 3:4:2. Let us now show that f (x; g) = g for all x 2 [0; g]. Indeed, we have already shown that f (g; x) = g for all x 2 [0; g], hence f (g; g) = g. We know that 0 is a unit element on [0; g], hence f (0; g) = g. So, from monotonicity, we can conclude that f (x; g) = g for all x between 0 and g. 3:5. We have shown that the operation f is a continuous associative operation on the interval [0; g]. We have also shown that the lower endpoint 0 of this interval is a unit element, and the upper endpoint g of this interval is a null element. ' Theorem B from9 enables us then to conclude that f is an ordinal sum of Archimedean operations of this same type. Hence, f is also commutative11, and f is a t-conorm. g

G

g

g

4: Let us now show that if x  G and y  G, then f (x; y) = G. Let us rst x x  G. We want to prove the desired equality for all y 2 [0; G]. Due to monotonicity, it is sucient to prove it for y = 0 and for y = G. In other words, to prove the desired equality for all x and y, it is sucient to prove that for every x 2 [G; 1], we have f (x; 0) = G and f (x; G) = G. Due to the same monotonicity, instead of proving these two equalities for all x 2 [G; 1], it is sucient to prove them for x = G and x = 1. Thus, to prove the desired equality for all x and y, it is sucient to prove that it is sucient to prove the following four equalities: f (G; 0) = G, f (1; 0) = G, f (G; G) = G, and f (1; G) = G. These four equalities follow from what we have already proven. Indeed:

 In Part 2:4:1, we have already shown that f (G; x) = G for all x 2 [0; 1], so f (G; 0) = f (G; G) = G.  By the de nition of G, we have f (1; 0) = G.  Finally, in Part 2:3:1, we have shown that f (1; x) = x for every x 2 [G; 1], hence f (1; G) = G. The equalities are proven. 5: Let us now show that if x  g and y  g, then f (x; y) = g. Let us rst x x  g. We want to prove the desired equality for all y 2 [g; 1]. Due to monotonicity, it is sucient to prove it for y = g and for y = 1. In other words, to prove the desired equality for all x and y, it is sucient to prove that for every x 2 [0; g], we have f (x; g) = g and f (x; 1) = g. Due to the same monotonicity, instead of proving these two equalities for all x 2 [0; g], it is sucient to prove them for x = 0 and x = g. Thus, to prove the

International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 11

desired equality for all x and y, it is sucient to prove that it is sucient to prove the following four equalities: f (0; g) = g, f (g; g) = g, f (0; 1) = g, and f (g; 1) = g. These four equalities follow from what we have already proven. Indeed:

 In Part 3:3:1, we have shown that f (0; x) = x for every x 2 [0; g], hence f (0; g) = g.  In Part 3:4:1, we have already shown that f (g; x) = g for all x 2 [0; 1], so f (g; g) = f (g; 1) = g.  By the de nition of g, we have f (0; 1) = g. The equalities are proven. 6: To complete the proof, we must show that if g  x  G, then f (x; y) = x. 6:1: Due to monotonicity, to prove this equality for arbitrary y 2 [0; 1], it is sucient to prove it for y = 0 and y = 1. In other words, it is sucient to prove that for every x 2 [g; G], we have f (x; 0) = x and f (x; 1) = x. 6:2: Let us rst show that f (x; 0) = x for all x 2 [g; G]. We will actually prove a stronger statement: that f (x; 0) = x for all x  G. Indeed, let 0  x  G. By the de nition of a generalized nullnorm, f (0; 0) = 0. In Part 2:1, we have shown that f (G; 0) = G. Thus, 0 = f (0; 0)  x  f (G; 0) = G. Since f (t; 0) is a continuous function, we can apply the Intermediate Value Theorem and conclude that there exists an element t 2 [0; G] for which f (t; 0) = x. Thus, f (x; 0) = f (f (t; 0); 0). By associativity, we get f (x; 0) = f (t; f (0; 0)). By the de nition of a generalized nullnorm, f (0; 0) = 0, so f (x; 0) = f (t; 0). By our choice of t, this means that f (x; 0) = x. 6:3: Let us rst show that f (x; 1) = x for all x 2 [g; G]. We will actually prove a stronger statement: that f (x; 1) = x for all x  g. Indeed, let g  x  1. By the de nition of a generalized nullnorm, f (1; 1) = 1. In Part 3:1 , we have shown that f (g; 1) = g. Thus, g = f (g; 1)  x  f (1; 1) = 1. Since f (t; 1) is a continuous function, we can apply the Intermediate Value Theorem and conclude that there exists an element t 2 [g; 1] for which f (t; 1) = x. Thus, f (x; 1) = f (f (t; 1); 1). By associativity, we get f (x; 1) = f (t; f (1; 1)). By the de nition of a generalized nullnorm, f (1; 1) = 0, so f (x; 1) = f (t; 1). By our choice of t, this means that f (x; 1) = x. The statement is proven, and so is the theorem.

Acknowledgments

This work was supported in part by NASA under cooperative agreement NCC5-209, by NSF grants No. DUE-9750858 and CDA-9522207, by the United Space Alliance, grant No. NAS 9-20000 (PWO C0C67713A6), by the Future Aerospace Science and Technology Program (FAST) Center for Structural Integrity of Aerospace Systems, eort sponsored by the Air Force Oce of Scienti c Research, Air Force Materiel

12 Non-Commutative Nullnorms

Command, USAF, under grant number F49620-95-1-0518, and by the National Security Agency under Grant No. MDA904-98-1-0561.

References 1. T. Calvo, B. de Baets, and J. Fodor, \The functional equations of Frank and Alsina for uninorms and nullnorms", Fuzzy Sets and Systems, 2000 (to appear). 2. B. De Baets, \Uninorms; the known classes", In: D. Ruan, H. A. Abderrahim, P. D'hondt, and E. E. Kerre (eds.), Fuzzy Logic and Intelligent Technologies for Nuclear Science and Industry (World Scienti c, Singapore, 1998) 21{28. 3. B. De Baets and J. Fodor, \On the structure of uninorms and their residual implicators", Proceedings of the 18th Linz Seminar on Fuzzy Set Theory (Linz, Austria, 1997) 81{87. 4. J. C. Fodor, \An extension of Fung-Fu's theorem", Int. J. of Uncertainty, Fuzziness, and Knowledge-Based Systems 4, No. 3 (1996) 235{243. 5. J. Fodor, R. R. Yager, and A. Rybalov, \Structure of uni-norms", Internat. J. Uncertainty, Fuzziness, and Knowledge-Based Systems 5 (1997) 411{427. 6. E. P. Klement, R. Mesiar, and R. Pap, Triangular Norm (Kluwer Academic Publ., Dordrecht, 2000). 7. G. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications (Prentice Hall, Upper Saddle River, NJ, 1995). 8. M. Mas, G. Mayor, and J. Torrens, \t-Operators", Int. J. of Uncertainty, Fuzziness, and Knowledge-Based Systems 7, No. 1 (1999) 31{50. 9. P. S. Mostert and A. L. Shields, \On the structure of semigroups on a compact manifold with boundary", Ann, of Math. 65 (1957) 117{143. 10. H. T. Nguyen and E. A. Walker, First Course in Fuzzy Logic (CRC Press, Boca Raton, FL, 1999). 11. B. Schweizer and A. Sklar, Probabilistic metric spaces (North Holland, New York, 1983). 12. R. R. Yager, \Uninorms in fuzzy systems modeling", Fuzzy Sets and Systems, 2000 (to appear). 13. R. R. Yager and A. Rybalov, \Uninorm aggregation operators", Fuzzy Sets and Systems 80 (1996) 111{120. 14. H. H. Zimmerman and P. Zysno, \Latent connectives in human decision making", Fuzzy Sets and Systems 4 (1980) 37{51.