The Algebra of Fuzzy Truth Values - Semantic Scholar

The Algebra of Fuzzy Truth Values Carol L. Walker and Elbert A. Walker New Mexico State University Department of Mathematical Sciences Las Cruces, NM 88003, USA

Abstract The purpose of this paper is to give a straightforward mathematical treatment of algebras of fuzzy truth values for type-2 fuzzy sets. Key words: Type-2 fuzzy sets, convolutions, algebras, Kleene algebras, De Morgan algebras, normal functions, convex functions, type-2 t-norms.

Introduction Type-2 fuzzy sets— that is, fuzzy sets with fuzzy sets as truth values— seem destined to play an increasingly important role in applications. They were introduced by Zadeh [24], extending the notion of ordinary fuzzy sets. Mendel’s book [15] has a section (1.6) on literature on type-2 fuzzy sets, and a rather extensive bibliography. Also, in [2], [11], [12], [16], [17], [18], and [21] are discussions of both theoretical and practical aspects of type-2 fuzzy sets. Especially in [16], there is an attempt to make fuzzy sets of type-2 easy to understand and use. Our emphasis is on the theoretical side. In the papers and books we have consulted on the subject, we …nd generally that the notation and the mathematical treatment of the theory are unnecessarily complicated. So we endeavor here to give a straightforward treatment of the mathematical basics of type-2 fuzzy sets that uses standard mathematical notation. We also present the material in a setting that includes both the …nite and continuous cases. In Section 1, we discuss two operations with functions— pointwise operations and convolutions. Both of these operations are intrinsic to the de…nition of operations on type-2 fuzzy sets and on their algebra of fuzzy truth values. In Section 2, we introduce the notion of type-2 fuzzy sets and establish some notation. We de…ne the basic operations and order relations and derive their basic properties. In Section 3, we discuss some of the subalgebras of the algebra Preprint submitted to Elsevier Science

5 December 2003

of fuzzy truth values. There is a multitude of interesting such subalgebras. We establish some criteria for these subalgebras to be lattices, distributive lattices, Kleene algebras, De Morgan algebras, and so forth. We describe the relationship between interval-valued type-2 fuzzy sets and classical intervalvalued fuzzy sets, which have been used interchangeably in applications. In Section 4, convolution with respect to _ and ordinary product is discussed. In Section 5, we brie‡y discuss the notion of triangular norms for type-2 fuzzy sets. Finally, in Section 6, we make some concluding remarks and raise some open questions.

1

Pointwise Operations and Convolutions

First, consider the following situation. Let T be a set with a binary operation on it, and let S just be a set. The set M ap(S; T ) of all functions from S into T automatically inherits a binary operation, which we also denote by , given by the equation (f g)(s) = f (s) g(s) See, for example, Equations (2). This mapping f be given pointwise.

g from S into T is said to Q

The set M ap(S; T ) may also be viewed as the Cartesian product T S = s2S T of copies of T , one copy for each element of S. The operation then becomes Q coordinatewise: for two elements fts gs2S and fus gs2S in s2S T , fts gs2S fus gs2S = fts us gs2S . These are just two ways of viewing the same thing. We’ll stick to the pointwise view. There is an important observation to be made here. The set T may have many operations on it, binary operations, unary operations, and nullary operations, for example. The operations may satisfy various equations, some of the binary operations may be commutative, or associative, some operations may distribute over others, and so on. But since the corresponding operations on M ap(S; T ) are pointwise, these equations will also hold in M ap(S; T ). We illustrate all this now with a pertinent speci…c example. A fuzzy subset A of a set S is a mapping A : S ! [0; 1]. The set S has no operations on it. So operations on the set M ap(S; [0; 1]) of all fuzzy subsets of S come from operations on [0; 1]. Common operations on [0; 1] of interest in fuzzy theory are ^, _, and 0 given by x ^ y = minfx; yg x _ y = maxfx; yg x0 = 1 x

2

(1)

The constants 0 and 1 are generally considered as part of the algebraic structure, technically being nullary operations. The operations ^ and _ are binary operations, and 0 is a unary operation. So the algebra basic to fuzzy set theory is ([0; 1]; _; ^;0 ; 0; 1). Of course, there are operations on [0; 1] other than these that are of interest in fuzzy matters, for example the t-norms and t-conorms that are discussed in Section 5. The corresponding operations on the set M ap(S; [0; 1]) of all fuzzy subsets of S are given pointwise by the formulas (A ^ B)(s) = A(s) ^ B(s) (A _ B)(s) = A(s) _ B(s) A0 (s) = (A(s))0

(2)

and the two nullary operations are given by 1(s) = 1 and 0(s) = 0 for all s 2 S: We use the same symbols, namely ^, _, 0 , 0, 1 for the pointwise operations on the elements of M ap(S; [0; 1]) as for the operations on [0; 1]. This is a standard practice, to which we adhere. The following equations hold in the algebra I = ([0; 1]; _; ^;0 ; 0; 1). (1) (2) (3) (4) (5) (6) (7) (8) (9)

x ^ x = x; x _ x = x (idempotent) x ^ y = y ^ x; x _ y = y _ x (commutative) x ^ (y ^ z) = (x ^ y) ^ z; x _ (y _ z) = (x _ y) _ z (associative) x ^ (x _ y) = x; x _ (x ^ y) = x (absorption laws) x^(y _ z) = (x ^ y)_(x ^ z); x_(y ^ z) = (x _ y)^(x _ z) (distributive) 0 _ x = x; 1 ^ x = x (identities) x00 = x (involution) (x ^ y)0 = x0 _ y 0 ; (x _ y)0 = x0 ^ y 0 (De Morgan’s laws) x ^ x0 y _ y 0 (Kleene inequality)

This is summarized by saying that I is a bounded distributive lattice with an involution 0 that satis…es De Morgan’s laws and the Kleene inequality. That is, I is a Kleene algebra. Thus (M ap(S; [0; 1]); _; ^;0 ; 0; 1) is also a Kleene algebra. The point is this. If the operations on an algebra T satisfy some law, for example an associative law, then the algebra M ap(S; T ) furnished with the corresponding pointwise operations will satisfy that law also. The converse also holds: if an equation holds in M ap(S; T ), then it holds in T . So to determine what equations hold in M ap(S; T ), it su¢ ces to determine those that hold in T . In particular, the equations that hold in the algebra (M ap(S; [0; 1]); _; ^;0 ; 0; 1) of fuzzy subsets of a set S are exactly those that hold in the algebra I. For example, x ^ y = y ^ x for all x; y 2 I so we know that A ^ B = B ^ A for all fuzzy sets A; B 2 M ap(S; [0; 1]). There is also an order relation

on [0; 1], and its relation to _ and ^ is that 3

x y if and only if x ^ y = x, or equivalently if and only if x _ y = y. This is a lattice order, that is, a partial order in which any two elements have a least upper bound and a greatest lower bound. This gives a lattice order relation on M ap(S; [0; 1]) pointwise by A B if and only if A(x) B(x), for all x, and this is equivalent to A ^ B = A, and to A _ B = B. The situation for type-2 fuzzy sets is the same except that fuzzy subsets of type-2 are mappings into a more complicated object than [0; 1], namely into the fuzzy subsets M ap(J; [0; 1]) of a lattice J. The operations on the type2 fuzzy subsets M ap(S; M ap(J; [0; 1])) of a set S will come pointwise from operations on M ap(J; [0; 1]). This paper is about operations on this latter set. This is where the di¢ culty of type-2 fuzzy sets lies. Operations are put on M ap(J; [0; 1]) using operations on both the domain J and the range [0; 1] of a mapping in M ap(J; [0; 1]). The equations that hold for the operations we put on M ap(J; [0; 1]) will automatically hold for the set of type-2 fuzzy subsets of a set S. In other words, the equations that hold for type-2 fuzzy sets are exactly those that hold for M ap(J; [0; 1]). The lattices that we consider are bounded and linearly ordered and come equipped with an involution. Bounded means that they have a smallest element and a largest element. We will denote these by 0 and 1, respectively. The involution is a one-to-one order reversing mapping 0 with the property that x00 = x. The operations are denoted the same as the operations on [0; 1], so J = (J; ^; _;0 ; 0; 1). In practice, J is usually the interval [0; 1] itself or a …nite subset of [0; 1] that includes the endpoints. Sometimes J is taken to be the extended reals or some bounded subset of the reals that satis…es appropriate conditions. A standard method used to construct operations on M ap(J; [0; 1]) is that of convolution. This concept is so central in what follows that we give it special attention. It is a mathematical concept that is used in many contexts, a standard way to combine functions in appropriate situations. Here is a basic situation. Let U and V be sets, and suppose that U has a binary operation on it, V has a binary operation N on it, and V has another appropriate operation H with which one may de…ne a binary operation on the set M ap(U; V ) by the formula (f g) (u) = x H (f (x) N g (y)) (3) y=u Here, H is applied to the set of all f (x) N g (y) for which x y = u. Thus H is an operation on subsets of V . You will see many concrete examples of such convolutions in the ensuing pages. This method of de…ning an operation on M ap(U; V ) from operations on both U and V is called convolution. It can be applied to n-ary operations on U , in particular to unary operations and to nullary operations on U . Sometimes, the same symbol is used for the operation created on M ap(U; V ) as that for 4

the operation on U that is used to create it— that is, the symbol might be used for that operation instead of the new symbol . But we will generally use a di¤erent symbol for the convoluted operation. The notion of convolution of functions is ubiquitous in mathematics. Perhaps the most common examples are polynomial multiplication, where a polynomial is viewed as a function from the nonnegative integers to the reals with …nite support and the kth coe¢ cient of the product is given by X

(p q) (k) =

p (i) q (j)

i+j=k

and convolution integrals, such as (f

g) (t) =

Z

1

f (x) g (t

x) dx

1

The object of interest to us is the set M ap(J; [0; 1]), the set of truth values for type-2 fuzzy sets. In the context of type-2 fuzzy sets, convolution is sometimes referred to as “Zadeh’s extension principle.”The two operations on the range [0; 1]— that is, on the set corresponding to V in the discussion above— are _ and ^ (except in Section 4 where we will use _ and product on [0; 1]). Since _ can be applied to any subset of [0; 1], being the least upper bound of that set, there is no problem of the convolution formula making sense for any operation on J. We can even apply _ to the empty set, the least upper bound of the empty subset of [0; 1] being 0. We will put operations on M ap(J; [0; 1]) that are of interest in type-2 fuzzy set theory, and develop some of their algebraic properties. Many of our results are known, but our treatment seems simpler and less computational than those heretofore. It follows a systematic pattern, putting this topic in the framework of algebras and their subalgebras. We have presented many of these results in abbreviated form in [22] and in [23].

2

Type-2 Fuzzy Sets

From now on, we use the letter I to denote either the unit interval [0; 1] or its associated algebra ([0; 1] ; _; ^;0 ; 0; 1), and J will denote either a bounded linearly ordered set with an involution or the associated algebra (J; _; ^;0 ; 0; 1). Both I and J are Kleene algebras— bounded distributive lattices satisfying the De Morgan laws and the inequality x ^ x0 y _ y 0 . De…nition 1 Let S be a set. A type-2 fuzzy subset of S is a mapping A : S ! M ap(J; I) 5

So for a set S, the set of all type-2 fuzzy subsets of S is M ap(S; M ap(J; I)). the elements of M ap(J; I) are ordinary fuzzy subsets of J: In the context of type-2 fuzzy sets, they are called fuzzy truth values, or membership grades of type2 sets. Operations on this set of type-2 fuzzy sets will come pointwise from operations on M ap(J; I). For any operation on I, we can put the corresponding pointwise operation on M ap(J; I), but the domain J also has operations on it, so we may use convolution to construct operations on M ap(J; I). As pointed out above, the operation _ on I, which takes the maximum of two elements of I, actually can be applied to any subset of I, taking the supremum, or least upper bound of that set. Similarly for the operation ^. Let f and g be in M ap(J; I). The elements f t g and f u g of M ap(J; I) are de…ned by the following equations. They are convolutions of the operations _ and ^ on the domain J with respect to the operations _ and ^ on I. W

(f t g) (x) = y_z=x (f (y) ^ g(z)) W (f u g) (x) = y^z=x (f (y) ^ g(z))

(4)

We will denote the convolution of the unary operation 0 on J of elements of M ap(J; I) by . The formula for it is f (x) =

W

y 0 =x f (y)

= f (x0 )

(5)

For f 2 M ap(J; I), f 0 denotes the function given by f 0 (x) = (f (x))0 : Denote by bold 1 the element of M ap(J; I) de…ned by 8 > < 0 if x 6= 1

(6)

1(x) = >

: 1 if x = 1

Denote by bold 0 the map de…ned by

0(x) =

8 > < 1 if x = 0

(7)

> : 0 if x 6= 0

These elements 0 and 1 of M ap(J; I) can be considered nullary operations, and can be obtained by convolution of the nullary operations 1 and 0 on J, but we skip the details of that explanation. Note the following: f

=f

f0 = f

f 00 = f

0

0 =1 1 =0

6

(8)

2.1 The Algebra (M ap(J; I); t; u; ; 0; 1) At this point, we have the algebra M = (M ap(J; I); t; u; ; 0; 1) with the operations t, u, , 0, and 1 obtained by convolution using the operations _; ^;0 ; 0; 1 on J, and _ and ^ on I. Whatever equations the algebra M satis…es, such as commutative laws or associative laws, will be automatically satis…ed by the set of all fuzzy type-2 subsets of S with the corresponding pointwise operations. Our main objective is to study the algebra M and some of its subalgebras. This is the basic algebra for type-2 fuzzy set theory. (In Section 4, we look at an algebra obtained using the operations _ and multiplication on I.) Although we are interested in the algebra M, the set M ap(J; I) also has the pointwise operations _, ^, 0 , 0, 1 on it coming from operations on the range I, and M ap(J; I) is a De Morgan algebra under these operations. In particular, it is a lattice with order given by f g if f = f ^ g, or equivalently, if g = f _ g. We are at liberty to use these operations in deriving properties of the algebra M. Pointwise operations are simpler than convolutions, so if a convolution can be expressed in terms of pointwise operations, it probably should be done. We will use the order relation on M extensively. One of our objectives is to express the operations t and u in terms of the simpler pointwise ones. We make heavy use of two auxiliary unary operations on M ap(J; I) which enable us to express the operations t and u in terms of pointwise ones. These unary operations also appear in [6], [22], and [23]. There are two bene…ts. First, it makes some computations much easier by replacing computations with t and u by computations with pointwise operations _ and ^ on M ap (J; I), so with computations in a lattice. Second, it provides some insight into the e¤ect of convolutions. For example, f t g is a function from J to I. What does its graph look like? Giving f t g in terms of pointwise operations on known functions readily provides such a graph. Now we de…ne these auxiliary unary operations and give some of their elementary properties and relations with other operations. Throughout, we carefully distinguish between a function f and its image f (x) at a point x. De…nition 2 For f 2 M, let f L and f R be the elements of M de…ned by f L (x) = _y

x f (y)

and f R (x) = _y

x f (y)

(9)

The function f L is monotone increasing and f R is monotone decreasing, as illustrated by the following …gures in which J = I. 7

fL

f

fR

The following proposition establishes relationships among the auxiliary pointwise operations _, ^, L, and R that we use for M ap (J; I) and the operation of M. Proposition 3 The following hold for all f; g 2 M. (1) f (2) (3) (4) (5) (6) (7) (8)

f L; f L

f R. R

f L = f L; f R = f R. (f L )R = (f R )L . (f )L = f R ; (f )R = f L . (f ^ g) = f ^ g ; (f _ g) = f _ g . (f _ g)L = f L _ g L ; (f _ g)R = f R _ g R . (f ^ g)L f L ^ g L ; (f ^ g)R f R ^ g R . f g implies f L g L and f R g R .

The proofs are immediate. Note that in item 3, (f L )R , which we will write as f LR or as f RL , is the constant function which takes the value _x2J f (x) everywhere. With respect to the point-wise operations _ and ^, M ap(J; I) is a lattice, and these operations on M ap(J; I) are easy to compute with compared to the convolution operations t and u de…ned in Equation (4). The following theorem expresses each of the operations t and u directly in terms of pointwise operations in two alternate forms. Similar pointwise expressions also appear in [1], [3], [4], and [6]. Theorem 4 The following hold for all f; g 2 M. f t g = f ^ g L _ f L ^ g = (f _ g) ^ f L ^ g L

(10)

f u g = f ^ g R _ f R ^ g = (f _ g) ^ f R ^ g R

(11)

8

Proof. Let f; g 2 M. (f t g) (x) =

W

=(

y_z=x

W

(f (y) ^ g(z))

x_z=x

(f (x) ^ g(z))) _ W

= (f (x) ^ ( = f (x) ^

x_z=x g(z)))

W

z x g(z)

_

_

W

y_x=x

W

W

(f (y) ^ g(x))

y_x=x f (y)

y x f (y)

= f (x) ^ g L (x) _ f L (x) ^ g(x) =

f ^ g L (x) _

f L ^ g (x) =

^ (g(x))

^ (g(x)) f ^ gL _ f L ^ g

(x)

So we have f t g = f ^ g L _ f L ^ g . Now f ^ gL _ f L ^ g = =

f ^ gL _ f L ^

f ^ gL _ g

f _ f L ^ gL _ f L

= f L ^ gL _ f L

^ (f _ g) ^ g L _ g

^ (f _ g) ^ g L

= f L ^ (f _ g) ^ g L = f L ^ g L ^ (f _ g) In a totally analogous manner, we get the formulas stated for f u g. Here are some elementary consequences. Corollary 5 The following hold for f; g 2 M . (1) (2) (3) (4) (5) (6) (7) (8) (9)

f t f = f; f u f = f f t g = g t f; f u g = g u f 1 u f = f; 0 t f = f f t 1 = f L and f u 1 = f R (1 is the constant function 1.) f u 0 = f t 0 = 0 (0 is the constant function 0.) (f t g) = f u g ; (f u g) = f t g f L t gL = f L ^ gL = f L t g = f t gL f R u gR = f R ^ gR = f R u g = f u gR If f g, then f t g = f ^ g L , and f u g = f ^ g R .

These are easy corollaries of the previous theorem. To prove item 7, for example, f L t g L = f L ^ g LL _ f LL ^ g L = f L ^ g L f t g L = f ^ g LL _ f L ^ g L = f L ^ g L

One should notice that the pointwise formula for the binary operation u follows from that of t and item 6, which is, in turn, an immediate consequence of the 9

de…nition of

and the formula for t, as follows:

f u g = (f u g)

= (f t g ) = (f _ g ) ^ f

= (f _ g) ^ f

L

^g

L

L

^g

L

= (f _ g) ^ f R ^ g R

= (f _ g) ^ f R ^ g R = f ^ g R _ f R ^ g

Proposition 6 The following hold for f; g 2 M. (f t g)L = f L t g L

(f u g)R = f R u g R

(f t g)R = f R t g R

(12)

(f u g)L = f L u g L

Proof. By Corollary 5, f L t g L = f L ^ g L , so for the …rst, we only need to show that (f t g)L = f L ^ g L . Now, L

(f t g) (x) = =

W

(f t g) (y) =

y x

W

u_v x

W

W

(f (u) ^ g (v))

u_v=y

y x

W

(f (u) ^ g (v)) =

!

u x

!

f (u) ^

W

v x

!

g (v) = f L (x) ^ g L (x)

so (f t g)L = f L ^ g L . The second follows from the …rst and duality, and Corollary 5, as follows: (f u g)R = (f u g) L

= f

^g

R L

= (f t g )

R

= (f t g )L

= f R ^ gR

= f R ^ gR

= f R ^ gR

For the third, R

(f t g) (x) = =

W

y x

W

(f t g) (y) =

u_v x

=

W

y x

(f (u) ^ g (v)) W

u x; v2[0;1]

W

u_v=y

!

!

(f (u) ^ g (v))

(f (u) ^ g (v)) _

W

v x; u2[0;1]

!

(f (u) ^ g (v))

= f R (x) ^ g RL (x) _ f RL (x) ^ g R (x) Thus (f t g)R = f R ^ g RL _ f RL ^ g R , and this is the formula for f R t g R . The last equality follows from duality. This completes the proof. We will use extensively the formulas in (12). Corollary 7 The associative laws hold for t and u. That is, f t (g t h) = (f t g) t h and f u (g u h) = (f u g) u h 10

(13)

Proof. Let f; g; h 2 M. Then (f t g) t h = =

h

h

h

(f _ g) ^ f L ^ g L

(f _ g) ^ f L ^ g L

= (f _ g _ h) ^

i

h

_ h ^ (f t g)L ^ hL i

_h ^

f L ^ gL _ h

h

i

i

f L ^ g L ^ hL

i

^ f L ^ g L ^ hL

= (f _ g _ h) ^ f L ^ g L ^ hL = f t (g t h)

(f u g) u h = ((f u g) u h) = f u (g u h)

= ((f t g ) t h ) = (f t (g t h ))

Thus the associative laws hold for both t and u. In the course of the proof, we had the identity (f t g) t h = (f _ g _ h) ^ f L ^ g L ^ hL

(14)

It will be of interest later to notice the following consequences. Corollary 8 The following hold for f1 ; f2 ; : : : ; fn 2 M . f1 t f2 t

t fn = (f1 _ f2 _

f1 u f2 u

u fn = (f1 _ f2 _

_ fn ) ^ (f1 )L ^ (f2 )L ^

_ fn ) ^ (f1 )R ^ (f2 )R ^

^ (fn )L

(15)

^ (fn )R

Proposition 9 The operations t and u distribute over _. That is, f t (g _ h) = (f t g) _ (f t h) f u (g _ h) = (f u g) _ (f u h)

(16)

Proof. Let f; g; h 2 M. f t (g _ h) = f ^ (g _ h)L _ f L ^ (g _ h) = f ^ g L _ hL

_

fL ^ g _ fL ^ h

= f ^ g L _ f ^ hL _ f L ^ g _ f L ^ h

(f t g) _ (f t h) = f ^ g L _ f L ^ g _ f ^ hL _ f L ^ h Thus t distributes over _. Now u distributes over _ since f u (g _ h) = [f u (g _ h)] = [f t (g _ h )] = [(f t g ) _ (f t h )] = [(f u g) _ (f u h)] Thus both operations distribute over the maximum.

Various distributive laws do not hold in M ap(J; I): t and u do not distribute over ^; _ distributes over neither t nor u, and similarly, ^ distributes over 11

neither t nor u; and t and u do not distribute over each other. There are easy examples to show this. But we do have the following. Lemma 10 For all f , g, h 2 M, the following hold. h

i

h

f t (g u h) = [f ^ g L ^ hRL ] _ f ^ g RL ^ hL _ f L ^ g ^ hR h

i

h

i

_ f L ^ gR ^ h h

i

h

i

i

h

(f t g) u (f t h) = f ^ g L ^ hRL _ f ^ g RL ^ hL _ f L ^ g ^ hR h

_ f L ^ g R ^ h _ f L ^ f R ^ g ^ hRL h

i

_ f R ^ f L ^ g RL ^ h h

i

h

i

h

i

h

i

i

i

f u (g t h) = f ^ g R ^ hRL _ f ^ g RL ^ hR _ f R ^ g ^ hL h

i

h

i

_ f R ^ gL ^ h h

i

h

(f u g) t (f u h) = f ^ g R ^ hRL _ f ^ g RL ^ hR _ f R ^ g ^ hL h

_ f R ^ g L ^ h _ f R ^ f L ^ g ^ hRL h

i

_ f L ^ f R ^ g RL ^ h

i

i

Proof. Let f , g, h 2 M. h

i

h

i

f t (g u h) = f ^ (g u h)L _ f L ^ (g u h) h

= f ^ g L u hL h

i

h

i

_ f L ^ (g u h)

i

g L ^ hRL _ g RL ^ hL

= f^

h

_ fL^

g ^ hR _ g R ^ h

= f ^ g L ^ hRL _ f ^ g RL ^ hL _ f L ^ g ^ hR _ f L ^ g R ^ h h

i

h

i

(f t g) u (f t h) = (f t g) ^ (f t h)R _ (f t g)R ^ (f t h) = = _ = _ _ _

nh

nh

f ^ gL _ f L ^ g

f ^ gL _ f L ^ g

i

i

^ f R t hR ^

h

f R ^ g RL _ f RL ^ g R

h

f L ^ g ^ f R ^ hRL

h

f RL ^ g R ^ f ^ hL

h

h

f ^ g L ^ f R ^ hRL

f R ^ g RL ^ f ^ hL

12

i

h

i

_

i

_

i

_

i

_

o

_

h

^

h

f ^ hL _ f L ^ h

i

h

f L ^ g ^ f RL ^ hR

i

h

f RL ^ g R ^ f L ^ h

i

h

i

f R t g R ^ (f t h)

f R ^ hRL _ f RL ^ hR

h

i

f ^ g L ^ f RL ^ hR

f R ^ g RL ^ f L ^ h

i

i

io

h

i

h

i

h

= f ^ g L ^ hRL _ f ^ g L ^ hR _ f L ^ f R ^ g ^ hRL h

i

h

i

h

i

_ f L ^ g ^ hR _ f ^ g RL ^ hL _ f R ^ f L ^ g RL ^ h h

i

h

i

_ f ^ g R ^ hL _ f L ^ g R ^ h h

i

h

i

h

= f ^ g L ^ hRL _ f ^ g RL ^ hL _ f L ^ g ^ hR i

h

h

i

i

h

i

i

_ f L ^ g R ^ h _ f L ^ f R ^ g ^ hRL _ f R ^ f L ^ g RL ^ h f u (g t h) = (f u (g t h)) =f f ^g

_ f

L

L

^h

^g ^h

= (f t (g u h )) RL

R

_ f ^g

_ f

L

RL

R

^g

L

^h

^h g

= f ^ g R ^ hRL _ f ^ g RL ^ hR _ f R ^ g ^ hL _ f R ^ g L ^ h

(f u g) t (f u h) = ((f u g) t (f u h)) h

=f f ^g h

_ f h

_ f h

L

L

L

^h

^g ^h ^f

R

R

RL

i

i

h

_ f ^g

h

_ f

^g ^h i

= ((f t g ) u (f t h )) L

RL

h

i

^g

h

R

_ f

RL

^h

^h

R

i

^f i

L

h

L

i

^g

RL

= f ^ g R ^ hRL _ f ^ g RL ^ hR _ f R ^ g ^ hL h

i

h

i

h

i

^h g i

i

_ f R ^ g L ^ h _ f R ^ f L ^ g ^ hRL _ f L ^ f R ^ g RL ^ h This establishes the pointwise formulas. Thus we have Theorem 11 The following hold for f; g; h 2 M. f t (g u h) f u (g t h)

(f t g) u (f t h) (f u g) t (f u h)

(17)

The absorption laws state that f u (f t g) = f f t (f u g) = f

(18)

They do not hold in general, as we will see. However, strangely enough, the following equality holds [17]. Thus if one absorption law holds, then so does the other. 13

Proposition 12 For f; g 2 M, f t (f u g) = f u (f t g) = f ^ g LR _ f R ^ f L ^ g

(19)

Proof. We use Lemma 10. f t (f u g) = f ^ f L ^ g RL _ f ^ f RL ^ g L _ f L ^ f ^ g R _ f L ^ f R ^ g = f ^ g RL _ f L ^ f R ^ g

f u (f t g) = f ^ f R ^ g RL _ f ^ f RL ^ g R _ f R ^ f ^ g L _ f R ^ f L ^ g = f ^ g LR _ f R ^ f L ^ g

It follows that f t (f u g) = f u (f t g) for all f; g 2 M. It is easy to see that the absorption laws do not always hold. For example, taking f to have all its values larger than the sup of the values of g, we get f u (f t g) = f ^ g LR _ f R ^ f L ^ g = g LR

(20)

which is constant with value the sup of the values of g, and has no value in common with f . The function f constant with value 1, and g constant with value anything less than 1, will work for this counterexample. Before proceeding further, we want to contrast the mathematical notation we are using with other notation fairly common in the literature. For a type2 fuzzy subset A of a set X, we use standard mathematical notation for a function A : X ! M ap (J; I)

and if B is another type-2 fuzzy subset of X, the union of A and B is (A t B) (x) (u) =

_

v_w=u

(A (x) (v) ^ B (x) (w))

The other notation in the literature expresses a type-2 fuzzy subset of X as A~ =

Z

x2X

Z

u2Jx

A

(x; u) = (x; u)

RR

“where denotes union over all admissible x and u,”and Jx = fu 2 J : A(x)(u) 6= 0g. And if B is a second type-2 fuzzy subset of X, they write A[B ,

Z

x2X

14

A[B

(x) =x

where, after replacing A (x) by fx and B (x) by gx , A[B

(x) =

Z

u2Jxu

Z

w2Jxw

(fx (u) ^ gx (w))/ (u _ w) =

A

(x) t

B

(x)

This introduction of multiple layers of nonstandard notation seems unnecessary.

2.2 Two Order Relations

In a lattice with operations _ and ^, a partial order is given by a b if a ^ b = a, or equivalently if a _ b = b. Think, for example, of the unit interval with its usual order. This gives a lattice order— that is, a partial order in which any two elements have a least upper bound and a greatest lower bound. The algebra M = (M ap(J; I); t; u; ; 0; 1) is not a lattice under the operations t and u because the absorption laws given in Equation (18) fail. For example, according to Equation (20), for f constant with value 1, and g constant with value less than 1, f u (f t g) = g 6= f . However, the operations t and u each have the requisite properties to de…ne a partial order. De…nition 13 f v g if f u g = f ; f

g if f t g = g.

Proposition 14 The relations v and transitive, thus are partial orders.

are re‡exive, antisymmetric, and

This is immediate from f uf = f = f tf , and the commutative and associative laws for u and t. These two partial orders are not the same, and neither implies the other. For example, f v 1, but it is not true that f 1. Proposition 15 The following hold for f; g 2 M. (1) Under the partial order v, any two elements f and g have a greatest lower bound. That greatest lower bound is f u g. (2) Under the partial order , any two elements f and g have a least upper bound. That least upper bound is f t g. Proof. For two elements f and g, note that f u g v f and f u g v g since f u g u f = f u g, and similarly f u g v g. Therefore f u g is a lower bound of both f and g. Suppose that h v f and h v g. Then h = h u f = h u g, and so h u (f u g) = h u g = h. Thus h v f u g. Therefore f u g is the greatest lower bound of f and g. Item 2 follows similarly.

15

Proposition 16 The pointwise criteria for v and (1) f v g if and only if f R ^ g (2) f g if and only if f ^ g L

f g

are these:

gR. f L.

Proof. If f v g then f u g = f = (f _ g) ^ f R ^ g R , whence f f = f ^ g R _ f R ^ g , whence f R ^ g

g R . Also

f . Conversely, if f R ^ g

f g R , then f u g = f ^ g R _ f R ^ g = f _ f R ^ g = f so f v g. Item 2 follows similarly. Note: These partial orders could be de…ned on the maps using the pointwise criteria without even considering convolutions. Then the proof that v is a partial order is the following. Since f R ^ f f f R , the order is re‡exive. R R R Suppose that f ^ g f g and g ^ f g f R . Then g = g ^ f R f , so the order is antisymmetric. Suppose that f R ^ g f g R and g R ^ h g hR . Then f ^ hR = f ^ g R ^ hR = f ^ g R = f since g R hR . Now R R R R R R R R R f f ^g =f ^g^h f ^ g ^ h ^ h = f ^ g ^ h = f ^ h. Thus the order is transitive. The following proposition is obtained easily from the pointwise criteria for these partial orders. (See (6) and (7) for de…nition of 1 and 0.) Proposition 17 The following hold for f; g 2 M. (1) (2) (3) (4) (5) (6)

3

f v 1 and 0 f . f v g if and only if g f . If f and g are monotone decreasing, then f v g if and only if f If f is monotone decreasing, then f v g if and only if f g R . If f and g are monotone increasing, then f g if and only if g If g is monotone increasing, then f g if and only if g f L .

g. f.

Subalgebras of Type-2 Fuzzy Sets

In modeling vague concepts in the fuzzy set spirit, one must choose an algebra of values. For example, in the classical case that algebra is ([0; 1]; _; ^;0 ; 0; 1), and a fuzzy subset of a set S is a mapping into this algebra. Interval-valued fuzzy sets are mappings into the algebra ([0; 1][2] ; _; ^;0 ; 0; 1), where [0; 1][2] = f(a; b) : a; b 2 [0; 1]; a (a; b) _ (c; d) = (a _ c; b _ d) (a; b) ^ (c; d) = (a ^ c; b ^ d) (a; b)0 = (b0 ; a0 ) 0 = (0; 0) and 1 = (1; 1) 16

bg

(21)

Here, each pair (a; b) corresponds to a closed subinterval of [0; 1]. This is a De Morgan algebra. Its fundamental mathematical properties may be found in [7]. Type-2 fuzzy sets are mappings into the algebra M. This latter algebra is the one for which we have been deriving properties. We will see in Theorems 21 and 28 that in the case J = [0; 1], M contains as subalgebras isomorphic copies of the algebras ([0; 1]; _; ^;0 ; 0; 1) and ([0; 1][2] ; _; ^;0 ; 0; 1), or more generally, for any J, M contains as subalgebras isomorphic copies of the algebras (J; _; ^;0 ; 0; 1) and (J [2] ; _; ^;0 ; 0; 1). This fully legitimizes the claim that type2 fuzzy sets are generalizations of type-1 and of interval-valued fuzzy sets. But M contains many other subalgebras of interest. This section examines several of these subalgebras. De…nition 18 A subalgebra of an algebra is a subset of the algebra that is closed under the operations of the original algebra. Any subalgebra A of M will give rise to a subalgebra of type-2 fuzzy sets, namely those maps in M ap(S; M ap(J; I)) whose images are in A. Computations in M are quite complicated, and the fact that distributive and absorption laws do not hold in M compounds the computational di¢ culties. Thus most, perhaps all, applications of type-2 fuzzy sets are restricted to subalgebras A of M that enjoy nicer algebraic properties than M itself. In Mendel’s book [15], the emphasis is on interval type-2 fuzzy sets, and he even states “Today, there does not seem to be a rational reason for not choosing type-2 interval sets...” When more general type-2 fuzzy sets are used to model linguistic uncertainties, it seems likely that the functions in M that will be used for this modeling will be patterned after type-1 membership functions, most of which are convex in the sense that we de…ne in Section 3.4 and normal in the sense that we de…ne in Section 3.3. Both classical and interval-valued fuzzy sets, when viewed as subalgebras of M, fall into this category. The triangular, trapezoidal, Gaussian, and sigmoidal membership functions are all convex and usually taken to be normal as well. For all of these reasons, it is important to know the algebraic properties of subalgebras of M. A subalgebra of M is a lattice if and only if the absorption laws in Equation (18) hold. It is well known in lattice theory that the two partial orders de…ned by join and meet operations coincide if and only if the absorption laws hold. Proposition 19 A subalgebra A of M satis…es the absorption laws if and only if v = . Proof. Let f; g 2 A and assume that the absorption laws f = f u (g t f ) and g = g t (g u f ) hold for f; g 2 A. If f v g, this means f u g = f , so that g = g t (g u f ) = g t f , and f g. On the other hand, if f g, this means f t g = g, so that f = f u (g t f ) = f u g, whence f v g. Now assume that the 17

two partial orders coincide. Then f t g = (f t f ) t g = f t (f t g) implying that f f t g, and thus that f v f t g. But this means that f = f u (f t g). By a similar proof, f = f t (f u g). 3.1 The Lattice J

Type-1 fuzzy sets take values in the unit interval [0; 1]. To realize type-1 fuzzy sets as special type-2 fuzzy sets, we realize the algebra (I; _; ^;0 ; 0; 1) as a subalgebra of (M ap(I; I); t; u; ; 0; 1). We can do this more generally, embedding (J; _; ^;0 ; 0; 1) as a subalgebra of M = (M ap(J; I); t; u; ; 0; 1). De…nition 20 For each a 2 J, its characteristic function is the function a : J ! I that takes a to 1 and the other elements of J to 0. These characteristic functions of points in J are clearly in one-to-one correspondence with J, but much more is true. Theorem 21 The mapping a 7! a is an isomorphism from (J; _; ^;0 ; 0; 1) into the subalgebra of M consisting of its characteristic functions of points. We will denote this subalgebra of M by J (or by I in the special case that J = I). It should be noted that a _ b is not the characteristic function of a _ b. In fact, if a < b, 8 > < 1 if x = a or x = b

(a _ b) (x) = >

: 0 otherwise

while the characteristic function of a _ b = b takes the value 1 only at b. It is a routine exercise to show the following. Proposition 22 For a; b 2 J, (1) The characteristic function of a _ b is a t b. (2) The characteristic function of a ^ b is a u b. (3) The characteristic function of a0 is a . Of course, if one wants to use type-1 fuzzy sets, they would simply be taken to be maps into J or I, and not as maps into this subalgebra of M. There would be no need to cloud the issue with type-2 sets. The subalgebra J of characteristic functions of points in J is a Kleene algebra since it is isomorphic to (J; _; ^;0 ; 0; 1). One might ask if J is contained in any larger Kleene subalgebra of M. We answer that question later in this paper. For now, we consider the question of whether or not J is contained in a larger chain inside M. 18

Lemma 23 Let A be a linearly ordered sublattice of M containing J. Let f 2 A and a 2 J. If a v f , then f (x) = 0 for 0 x < a and f R (a) = 1. If f a, then f (x) = 0 for a < x 1 and f L (a) = 1. Proof. Suppose that a v f . Then by Proposition 16, aR ^ f aR (x) =

8 > < 1 if 0

x

> : 0 if a < x

a

f R . Now

a 1

is the characteristic function of the interval [0; a]. Thus for 0 x < a, aR (x) ^ f (x) = f (x) a (x) = 0. And a (a) = 1 f R (a). On the other hand, if L L f a, we have a ^ f a f . This says that for a < x 1, aL (x) ^ f (x) = f (x) a (x) = 0. And a (a) = 1 f L (a). Lemma 24 Let A be a linearly ordered sublattice of M containing J. Suppose f 2 A satis…es f (a) 6= 0 and f (b) 6= 0 for some a < b and suppose further that there is no element of J strictly between a and b. Then a v f v b and the support of f is exactly fa; bg. Proof. Because A is a lattice, by Proposition 19, the two partial orders coincide. If b v f , then, by Lemma 23, f (a) = 0; and if f v a, then by Lemma 23, f (b) = 0. Thus, since A is linearly ordered, we must have a v f v b. Now by Lemma 23, a v f implies that f (x) = 0 for all x < a and f v b implies that f (x) = 0 for all x > b. Call a; b 2 J an adjacent pair if a < b and there is no element of J strictly between a and b. For an adjacent pair a; b 2 J and 2 (0; 1) I, de…ne functions

gab (x) =

8 > > > 1 > >
> > > > : 0 8 > < 1

fab (x) = >

: 0

if x = a hab (x) =

if x = b otherwise

if x = b or x = a

8 > > > 1 > >
> > > > : 0

otherwise

otherwise

Lemma 25 If a < b is an adjacent pair in J, and


x, and f 6= x, then x v f v y and the support of f is exactly fx; yg. Similarly if x has an adjacent point y < x. Again, because 0 v f , we have 0 (0) = 1 f R (0), so either f (x) = 1 or f (y) = 1. Thus f 2 fgab g [ fhab g [ ffab g. It follows that A J. The following corollary is immediate because I contains no adjacent pairs. In fact, this corollary applies to any J having no adjacent pairs. Corollary 27 The only linearly ordered subalgebra of M containing I is I itself.

3.2 Intervals

The usual interpretation of interval type-2 fuzzy sets corresponds to the subalgebra J[2] of M consisting of those functions in M ap(J; I) that take all elements of a closed interval of J to 1 and its complement to 0. Such an interval can be 20

identi…ed by its two endpoints. In this context, the elements of J are simply closed intervals for which the two endpoints coincide. In particular, 1 and 0 are in this subalgebra. The empty set is not an interval. This subalgebra is of particular interest in applications of type-2 fuzzy sets. It seems that most applications of type-2 fuzzy sets are restricted to these interval type-2 ones. In the discussion of type-2 fuzzy sets in [15], the emphasis is almost entirely on interval type-2 fuzzy sets. The elements of the subalgebra J[2] will be called simply intervals. In the previous section, we de…ned the characteristic function a of an element a 2 J. A pair (a; b) 2 J [2] is identi…ed with the function aL ^ bR 2 J[2] . Note the following. aL ^ bL = (a t b)L and aR _ bR = (a t b)R (22) Now for intervals f = aL ^ bR and g = cL ^ dR , we have f t g = aL ^ bR t cL ^ dR = = =

h

h

aL ^ bR ^ cL ^ dR i

h

L

_

aL ^ bR

aL ^ bR ^ cL _ aL ^ cL ^ dR i

aL ^ cL ^ bR _

h

aL ^ cL ^ dR

i

L

^ c L ^ dR

i

= aL ^ cL ^ bR _ dR = aL t cL ^ bR _ dR

= (a t c)L ^ (b t d)R

This says that the t-union of the intervals [a; b] and [c; d] is the interval [a _ c; b _ d]. That is, we can compute t coordinatewise. The formula for f u g can be gotten similarly, and is (a u c)L ^ (b u d)R , corresponding to the interval [a ^ c; b ^ d]. It can also be obtained from the formula for f t g using duality. This is summed up by the following theorem. Theorem 28 The mapping (a; b) 7! aL ^ bR is an isomorphism from the the algebra (J [2] ; _; ^;0 ; 0; 1) into the subalgebra J[2] of M consisting of closed intervals. The upshot of all this is that if interval type-2 sets are used, one may as well just use interval-valued type-1 fuzzy sets, that is, maps of a set into J [2] , the intervals of J, and combining as indicated in Equation (21). The theory of these has been worked out in detail in [7] for J = I. This subalgebra of fuzzy type-2 intervals is a De Morgan algebra under t; u; ; 0; 1. Note also that J[2] is a subalgebra of 2J , the subalgebra of all subsets of [0; 1], which is discussed in Section 3.7. There is another obvious interpretation of the phrase “interval type-2 fuzzy 21

sets,” namely the characteristic functions of all intervals, including open, closed, and half-open/half-closed intervals. This distinction vanishes when working with M ap (J; I) for …nite chains J, but could be of interest in other cases.

3.3 Normal Functions

The usual de…nition of normality of an element f of M ap(J; I), is that f (x) = 1 for some x. We use a slightly weaker de…nition that coincides with the usual de…nition in the …nite case. De…nition 29 An element f of M ap(J; I) is normal if the least upper bound of f is 1. There are convenient ways to express this condition in terms of our operations on this algebra. The proof of the following proposition is immediate from de…nitions. Proposition 30 The following four conditions are equivalent: (1) (2) (3) (4)

f is normal. f RL = 1. f L (1) = 1. f R (0) = 1.

Proposition 31 The set N of all normal functions is a subalgebra of M. Proof. Here is the veri…cation that it is closed under t. (f t g)L (1) = f L t g L (1) = f L ^ g L (1)

= f L (1) ^ g L (1) =1 The other veri…cations are just as easy. The subalgebra N contains the non-empty subsets of J as characteristic functions, and it contains the subalgebra J[2] of intervals. Note also that the Kleene algebra J in Theorem 26 consists of normal functions. Normal functions will play an important role later in this paper. 22

3.4 Convex Functions

The subalgebra C of convex functions is a particularly interesting one. It contains the subalgebra J[2] of intervals, which in turn contains the subalgebra J corresponding to the lattice J. It also contains the Kleene algebra J of Theorem 26. De…nition 32 An element f of M ap(J; I) is convex if whenever x then f (y) f (x) ^ f (z).

y

z,

There is a convenient way to express this condition in terms of our operations on this algebra. Proposition 33 An element f of M ap(J; I) is convex if and only if f = f L ^ f R. Proof. Suppose that f = f L ^ f R and x and f R (y) f R (z), we get

y

z. Then since f L (y)

f L (x)

f (y) ^ f (x) ^ f (z) = f L ^ f R (y) ^ f L ^ f R (x) ^ f L ^ f R (z)

= f L (y) ^ f R (y) ^ f L (x) ^ f R (x) ^ f L (z) ^ f R (z) = f L (x) ^ f R (x) ^ f L (z) ^ f R (z) = f (x) ^ f (z)

Thus f (y) f (x) ^ f (z). Suppose that f is convex. Then f (y) f (x) ^ f (z) for all x y and for all z y. Thus f (y) f R (y) ^ f L (y) = f R ^ f L (y), so f f R ^ f L . But it is always true that f f R ^ f L . Thus f = f R ^ f L . How do convex functions come about? Here is a simple description of convex functions in terms of monotone functions. Proposition 34 A function is convex if and only if it is the minimum of a monotone increasing function and a monotone decreasing one. Proof. We have seen that a convex function f is the minimum of an increasing function and a decreasing one, namely f = f L ^ f R : Let f be an increasing function and g a decreasing one, and suppose that x y z. Then ((f ^ g) (y)) ^ ((f ^ g) (x)) ^ ((f ^ g) (z)) = f (y) ^ g(y) ^ f (x) ^ g(x) ^ f (z) ^ g(z) = f (x) ^ g(x) ^ f (z) ^ g(z) so that (f ^ g) (y)

((f ^ g) (x)) ^ (f (z) ^ g(z)). Hence f ^ g is convex. 23

Proposition 35 The set C of convex functions is a subalgebra of M. Proof. It is clear that 0 and 1 are convex. Suppose that f is convex. Then f = f L ^ f R ; and f = f L ^ f R = f L ^ f R = f L ^ f R : Hence f is convex. Now suppose that f and g are convex. Then f t g = f L ^ g _ f ^ gL

= f L ^ gL ^ gR _ f L ^ f R ^ gL = f L ^ gL ^ f R _ gR

Thus by Proposition 34, f t g is convex. Now f u g is convex since (f u g) = (f t g ) is convex. Theorem 36 The distributive laws f t (g u h) = (f t g) u (f t h) f u (g t h) = (f u g) t (f u h)

(23)

hold for all g; h 2 M if and only if f is convex. Proof. From Theorem 10, we have f t (g u h) = f ^ g L ^ hRL _ f ^ g RL ^ hL _ f L ^ g ^ hR _ f L ^ g R ^ h

(f t g) u (f t h) = f ^ g L ^ hRL _ f ^ g RL ^ hL _ f L ^ g ^ hR _ f L ^ g R ^ h _ f L ^ f R ^ g ^ hRL _ f R ^ f L ^ g RL ^ h

If f is convex, then f L ^ f R = f , and the last two terms in the expression for (f t g) u (f t h) are smaller than the …rst two terms for that expression. Thus f t (g u h) = (f t g) u (f t h). The other distributive law follows similarly. Now suppose f t (g u h) = (f t g) u (f t h) holds for all g and h. Then f ^ g L ^ hRL _ f ^ g RL ^ hL _ f L ^ g ^ hR _ f L ^ g R ^ h

= f ^ g L ^ hRL _ f ^ g RL ^ hL _ f L ^ g ^ hR _ f L ^ g R ^ h

_ f L ^ f R ^ g ^ hRL _ f R ^ f L ^ g RL ^ h

Letting h be the function that is 1 everywhere, and g = 0, we get f ^ 0L _ f ^ 0RL _ f L ^ 0 _ f L ^ 0R

= f ^ 0L _ f ^ 0RL _ f L ^ 0 _ f L ^ 0R

_ f L ^ f R ^ 0 _ f R ^ f L ^ 0RL and so

f = f _ fR ^ fL = fR ^ fL 24

Similarly, if f u (g t h) = (f u g) t (f u h) holds for all g and h, then f is convex. Using f u g R = f R u g from Corollary 5, note that f u g R t hR = f u (g t h)R = f R u (g t h)

= f R u g t f R u h = f u g R t f u hR

and similarly f t g L u hL = f t (g u h)L = f L t (g u h)

= f L t g u f L t h = f t g L u f t hL

Thus the distributive law f u (g t h) = (f u g) t (f u h) holds for any f if both g and h are monotone decreasing, and f t (g u h) = (f t g) u (f t h) holds for any f if both g and h are monotone increasing.

3.5 Convex Normal Functions

The set L of functions that are both convex and normal is clearly a subalgebra of M, being the intersection of the subalgebra C of convex functions and the subalgebra N of normal functions. The absorption laws hold by the following Proposition. Proposition 37 If f is convex and g is normal, then f t (f u g) = f u (f t g) = f

(24)

Proof. We showed in Proposition 12 that f t (f u g) = f u (f t g) = f ^ g LR _ f R ^ f L ^ g Since f is convex and g is normal, we have f ^ g LR _ f R ^ f L ^ g = f _ (f ^ g) = f Thus the absorption law holds for convex normal functions. The distributive laws hold by Theorem 36. From all the other properties of M, the subalgebra L of convex normal functions is a bounded distributive lattice, and is an involution that satis…es De Morgan’s laws with respect to t and u: Thus we have 25

Theorem 38 The subalgebra L of M consisting of all the convex normal functions is a De Morgan algebra. Both De Morgan algebras and Kleene algebras are, among other things, bounded, distributive lattices. Theorem 39 If A is a subalgebra of M that is a lattice with respect to t and u and that contains J, then the functions in A are normal and convex, that is, A L. Thus the subalgebra L of all convex normal functions is a maximal lattice in M. Proof. Let f 2 A. Since A is a lattice, the absorption laws hold, so in particular, a u (a t f ) = a and f u (f t a) = f for all a 2 J. By Proposition 12, a = a u (a t f ) = a ^ f LR _ aL ^ aR ^ f so that a (a) = 1 = f LR ^ f (a) implies that f (a) = f RL = 1, so f is normal. Again by Proposition 12, f = f u (f t a) = f ^ aLR _ f L ^ f R ^ a which implies that f (a) = f (a) _ f L (a) ^ f R (a) = f L (a) ^ f R (a). Since this holds for all a 2 J, we have that f = f L ^ f R , that is, f is convex. A subalgebra of M is a Kleene algebra if it is a bounded, distributive lattice with an involution, satisfying the De Morgan laws and the inequality f u f v g t g for all f; g in the subalgebra. For every x 2 J, either x x0 or x x0 because J is linearly ordered. De…ne J1 = fx 2 J : x x0 g and J2 = fx 2 J : x x0 g. For example, I1 = [0; 1=2] and I2 = [1=2; 1]. It is easy to see that if x 2 J1 and y 2 J2 , then x y. If J contains adjacent elements c < c0 — meaning there are no elements strictly between c and c0 , then let J1+ = J1 [ fc0 g and J2+ = J2 [ fcg. If there are no such elements, J1+ = J1 and J2+ = J2 . For f 2 M the support of f is the set Supp (f ) = fx 2 J : f (x) > 0g. Call f an upper function if Supp (f ) J2+ , and a lower function if Supp (f ) J1+ . Note that J1 [ J2 = J and there are three possibilities for J1+ \ J2+ . Either J1+ \ J2+ = ?, J1+ \ J2+ = fcg where c = c0 , or J1+ \ J2+ = fc; c0 g where c < c0 . Theorem 40 The set K of all of the convex normal upper and lower functions is a Kleene subalgebra of M. 26

Proof. By Propositions 37 and 19, the absorption laws hold in K and the two partial orders v and coincide. Moreover, by Theorem 36, the distributive laws hold in K. The unary operations 0 and 1 are convex normal lower and upper functions, respectively. Because K L, we know that the properties of convex and normal are preserved under all of the operations of M. For any f 2 K, it is easy to see that one of f and f is upper and the other is lower. Thus it remains only to show that for f; g 2 K, each of f t g and f u g is either upper or lower and to show the Kleene inequalities are satis…ed in K. Since (f t g) (x) (f _ g) (x) and (f u g) (x) (f _ g) (x) then if f; g are both upper, both f t g and f u g have support contained in J2+ , and if f; g are both lower, both f t g and f u g have support contained in J1+ . If f is a lower function and g is an upper function, then (f t g) (x) g L (x) = 0 for x 2 = J2+ since (f t g) (x) = (f (x) _ g (x)) ^ f L (x) ^ g L (x), so Supp (f t g) J2+ . Thus f t g is an upper function. By a dual argument, f u g is a lower function. Thus K is closed under the operations t and u. It remains to show the Kleene inequalities hold. One way to express the Kleene inequality (f u f ) v (g t g ) is by the equation (f u f ) t (g t g ) = (g t g ) (25) and this is what we will prove. Since one of f and f is a lower function and the other is an upper function, we know that h = f u f is a lower function and k = g t g is an upper function. Also, by Theorem 4, (h t k) (x) = (h (x) _ k (x)) ^ hL (x) ^ k L (x) If x 2 = J1+ , then h (x) = 0 and hL (x) = 1, so that (h t k) (x) = k (x) ^ k L (x) = k (x) And if x 2 = J2+ , then k L (x) = 0, so (h t k) (x)

k L (x) = 0 = k (x)

Thus, if x 2 = J1+ \ J2+ then (h t k) (x) = k (x) and Equation (25) holds. If + + J1 \ J2 = ? we are …nished. To prove the Kleene equation (25) for the case J1+ \ J2+ 6= ?, we need only check Equation (25) for x = c and x = c0 . We …rst show that if J1+ \ J2+ 6= ?, and f and g are in K, then

By Theorem 4,

(f u f ) (c)

(f u f ) (c0 )

(26)

(g t g ) (c)

(g t g ) (c0 )

(27)

(f u f ) (c) = (f (c) _ f (c0 )) ^ f R (c) ^ f L (c0 ) (f u f ) (c0 ) = (f (c0 ) _ f (c)) ^ f R (c0 ) ^ f L (c) 27

If f is a lower function, because J1+ contains the support of f , we have f (c0 ) = f R (c0 ) f R (c) and f L (c) f L (c0 ) so that (f u f ) (c) (f u f ) (c0 ). If f is an upper function, then f is a lower function and we get the same result. Thus (26) holds for any f 2 K. And, by a dual argument, (27) holds for any g 2 K. If J1+ \J2+ = fcg with c = c0 , the inequalities (26) and (27) are trivially true. Now h (c) h (c0 ) implies that hL (c) = hL (c0 ) = 1. Also, k L (c) = k (c) so that (h t k) (c) = (h (c) _ k (c)) ^ hL (c) ^ k L (c) = (h (c) _ k (c)) ^ k (c) = k (c) and k (c)

k (c0 ) implies k L (c0 ) = k (c0 ), so that

(h t k) (c0 ) = (h (c0 ) _ k (c0 )) ^ hL (c0 ) ^ k L (c0 ) = (h (c0 ) _ k (c0 )) ^ k (c0 ) = k (c0 ) Thus h (x) t k (x) = k (x) for every x 2 J. It follows that f u f v g t g for all f; g 2 K, and K is a Kleene algebra. Theorem 41 If A is any Kleene algebra containing J, then A is a maximal Kleene algebra in M.

K. Thus K

Proof. Suppose A is a Kleene algebra containing J. Since A is, in particular, a lattice containing J, Theorem 39 says that the elements of A are normal and convex. We only need to show that elements of A are either upper or lower functions. By Proposition 19, the partial orders v and are the same. For simplicity, we will use the symbol v for both. Let f 2 A and a 2 J1 . Then a 2 A and a t a = a . By the Kleene inequality, f uf vata =a

and

a=aua vf tf

By Proposition 22, a is the characteristic function of a0 , so a (a0 ) = 1 and a (x) = 0 for x 6= a0 . By Proposition 16, f u f v a is equivalent to (f u f ) ^ a

L

a

(f u f )L

Now by (11), f u f = f ^ f R _ f R ^ f . Since a L (x) = aR (x0 ) = 1 for x0 a, or equivalently for x a0 , and a (x) = 0 for x 6= a0 , the condition L (f u f ) ^ a a implies that f (x) ^ f

R

(x) = 0 = f R (x) ^ f (x)

for all x > a0

(28)

Again by Proposition 16, a v f t f is equivalent to aR ^ (f t f )

a

(f t f )R

Now by (10), f t f = f ^ f L _ f L ^ f . Since aR (x) = 1 for x a (x) = 0 for x 6= a, the condition above implies that f (x) ^ f

L

(x) = 0 = f L (x) ^ f (x) 28

for all x < a

a, and (29)

Note that (28) and (29) hold for all a 2 J1 . Suppose there are elements u 2 J1 and v 2 J2 with f (u) 6= 0 and f (v) 6= 0. Since 0 is order reversing, v 2 J2 implies that v 0 2 J1 . Suppose there exists a 2 J1 with v 0 < a. Then by (28), v > a0 implies f (v) ^ f R (v) = 0, so we have f R (v) = f L (v 0 ) = 0, which implies that f (x) = 0 for all x v 0 . This forces u > v 0 . Suppose there exists b 2 J1 with u < b. Then by (29), f (u) ^ f L (u) = 0, so we have f L (u) = f R (u0 ) = 0, which implies that f (x) = 0 for all x u0 . This forces v < u0 , which is incompatible with u > v 0 . Thus if u is not the largest element of J1 and v is not the smallest element of J2 , then either f (u) = 0 or f (v) = 0. This says that Supp (f ) J1+ or Supp (f ) J2+ , so f is either an upper or lower function, as claimed. In [12], one of the main results is Theorem 1, page 330. We note that this theorem in our context is immediate from our Corollary 8, using normality and convexity. Also the hypotheses of their Corollary 1 implies immediately the pointwise criteria for f1 v f2 v v fn and for f1 f2 fn . We note that the hypothesis of normality is not needed for their Corollary 1. A more general version of their Corollary 1 is the following which requires no special hypothesis. For the proof, just observe that if f v g, then by de…nition f = f u g, and if f g, then by de…nition f t g = g. Proposition 42 Let functions f1 ; f2 ; : : : ; fn 2 M . If f1 v f2 v n n u fi = f1 . If f1 f2 fn , then t fi = fn .

i=1

v fn , then

i=1

3.6 Endmaximal Functions

In [21], Nieminen considered the notions of endmaximal and of b-maximal functions and showed the collection of such to have some nice algebraic properties. We present here those results, using mainly our pointwise formulas for t and u, avoiding tedious computations. We also provide a counter example to one of the claims in [21]. De…nition 43 An element f of M ap(J; I) is endmaximal if f L = f R . An immediate consequence is that f L = f R = f RL . The de…nition just says that the function assumes its maximum at its endpoints, in our case at 0 and at 1. First, notice that 0 and 1 are not endmaximal, so the set of endmaximal functions cannot be a subalgebra of M. But it is a subalgebra of (M ap(J; I); t; u; ). Proposition 44 The set E of endmaximal functions is closed under t, u, and . 29

Proof. Since f L = f R and f R = f L , it follows that the set of endmaximal functions is closed under . Suppose that f and g are endmaximal. Then (f t g)L = f L t g L = f RL t g RL = f R t g R = (f t g)R

(f u g)R = f R u g R = f RL u g RL = f L u g L = (f u g)L

Thus the set of endmaximal functions is closed under both t and u. Nieminen [21] states that the normal endmaximal functions form a distributive subalgebra of (M ap(J; I); t; u). But it seems that normality is not needed. The following proposition follows immediately from Lemma 10 and the fact that f L = f R = f LR . Proposition 45 The endmaximal functions form a distributive subalgebra of the algebra (M ap(J; I); t; u; ). Nieminen also shows that the endmaximal functions satisfy a modular law. His proof is highly computational. We use our Lemma 10. Proposition 46 The endmaximal functions satisfy (1) f u (g t (f u h)) = (f u g) t (f u h) (2) f t (g u (f t h)) = (f t g) u (f t h) Proof. For item 1, we calculate both sides using Lemma 10. f u (g t (f u h)) = f ^ g R ^ (f t h)RL _ f ^ g RL ^ (f t h)R _ f R ^ g ^ (f t h)L _ f R ^ g L ^ (f t h) = f ^ g RL ^ f RL ^ hRL

_ f RL ^ g ^ f RL ^ hRL _ f RL ^ g RL ^

_ f ^ g RL ^ f RL ^ hRL

f ^ hRL _ f RL ^ h

= f ^ g RL ^ hRL _ f ^ g RL ^ hRL _ f RL ^ g ^ hRL

_

h

f RL ^ g RL ^ f ^ hRL

i

_

h

f RL ^ g RL ^ f RL ^ h

= f ^ g RL ^ hRL _ f ^ g RL ^ hRL _ f RL ^ g ^ hRL h

i

h

i

_ f ^ g RL ^ hRL _ f RL ^ g RL ^ h

= f ^ g RL ^ hRL _ f RL ^ g ^ hRL _ f RL ^ g RL ^ h 30

i

and (f u g) t (f u h) = f ^ g R ^ hRL _ f ^ g RL ^ hR _ f R ^ g ^ hL _ f R ^ g L ^ h

= f ^ g RL ^ hRL _ f ^ g RL ^ hRL

_ f RL ^ g ^ hRL _ f RL ^ g RL ^ h

= f ^ g RL ^ hRL _ f RL ^ g ^ hRL _ f RL ^ g RL ^ h

Item 2 follows similarly, or by using duality. De…nition 47 [21] A function f : I ! I is left-maximal (b-maximal) if f L = f LR . Theorem 48 If g and h are left-maximal, then so are g t h and g u h. Proof. If g and h are left-maximal, (g t h)L = g L t hL = g LR t hLR = g L t hL

R

= g L u hL

= (g t h)L R

R

= (g u h)L

= (g t h)LR and (g u h)L = g L u hL = g LR u hLR

R

= (g u h)LR .

One distributive law holds for left-maximal functions. Theorem 49 If g and h are left-maximal, then f u(g t h) = (f u g)t(f u h). Proof. By Lemma 10, f u(g t h) = f ^ g RL ^ hR _ f ^ g R ^ hRL _ f R ^ g ^ hL _ f R ^ g L ^ h and (f u g) t (f u h) = f ^ g RL ^ hR _ f ^ g R ^ hRL _ f R ^ g ^ hL _ f R ^ g L ^ h _ f R ^ g ^ hRL _ f R ^ g RL ^ h

But since g RL = g L and hRL = hL , these two expressions are equal. In [21], Nieminen claims in Theorem 4 that the left-maximal fuzzy grades constitute a distributive t-semilattice, meaning that if f; g; h are left-maximal, and h f t g, then there exists f1 f and g1 g such that h = f1 t g1 . This theorem is false, as shown by the following counterexample. Take J = I and de…ne f (x) =

8 > < 1 if x = 0 > : 3=4 if x 6= 0

g (x) =

31

8 > < 1=2 if x = 0 > : 0 if x 6= 0

h (x) = 1

Then f L = hL = h = 1 and g L = 1=2. Also f t g = f ^ g L _ f L ^ g = 1=2 = g L

(f t g) t h = g L t h = g L ^ hL _ g L ^ h

= gL ^ 1 _ gL ^ 1 = gL = f t g

so we have h f t g. The proof of Theorem 4 in [21] concludes at one point that h f _ g. If this were the case, the theorem would follow. However, in this example, this inequality fails. Now suppose that we have f1 f and L L g1 g. This means that g1 t g = (g1 _ g) ^ g1 ^ g = g so that (g1 (0) _ g (0)) ^ g1L (0) ^ g L (0) = (g1 (0) _ 1=2) ^ g1L (0) ^ 1=2 = g (0) = 1=2 yielding g1L (0) x > 0 we have

1=2, which implies that g1L (x)

1=2 for any x. Also, for

(g1 (x) _ g (x)) ^ g1L (x) ^ g L (x) = (g1 (x) _ 0) ^ g1L (x) ^ 1=2 = g1 (x) ^ 1=2 = g (x) = 0 which implies that g1 (x) = 0 = g (x) for x > 0. We also have f1 t f = (f1 _ f ) ^ f1L ^ f L = f so that (f1 (0) _ f (0)) ^ f1L (0) ^ f L (0) = (f1 (0) _ 1) ^ f1L (0) ^ 1 = f (0) = 1 yielding f1L (0) = 1, which implies that f1L (x) = 1 for any x. Also, for x > 0 we have (f1 (x) _ f (x)) ^ f1L (x) ^ f L (x) = (f1 (x) _ 3=4) ^ 1 ^ 1 = f1 (x) _ 3=4 = f (x) = 3=4 which implies that f1 (x)

3=4 = f (x) for all x > 0. But now for x > 0,

(f1 t g1 ) (x) = (f1 (x) _ g1 (x)) ^ f1L (x) ^ g1L (x) f1 (x) _ g1 (x) f (x) _ g (x) = 3=4 < 1 = h (x) In other words, f1 t g1 = h is not possible. 3.7 Subsets of the Unit Interval Consider the set 2J = M ap(J; f0; 1g) M ap(J; I). It is very easy to see that 2J is a subalgebra of M. For example, if f and g map into f0; 1g, then f t g = 32

f ^ g L _ f L ^ g clearly does also. These elements can be identi…ed with the subsets of J via the correspondence f 7! f 1 (1). If such an identi…cation is made, the following are clear. f _g =f [g

and f ^ g = f \ g

f L = fx : x

some element of f g

f R = fx : x

some element of f g

Making this identi…cation, the formulas for t and u become f t g = (f [ g) \ f L \ g L = f \ g L [ f L \ g

f u g = (f [ g) \ f R \ g R = f \ g R [ f R \ g Also, 1 corresponds to the set f1g, 0 to the set f0g, and the complement of f to the map f 0 . This is not the same as f . We have not investigated the subalgebra (M ap(J; f0; 1g); t; u; ; 0; 1). Its elements are the subsets of J; but the operations t and u are not union and intersection. It satis…es all the equations that M satis…es, being a subalgebra of it, but maybe some others as well. This subalgebra is actually closed under several other operations. For example, f L and f R are in this subalgebra whenever f is. The constant function with value 1, and the constant function with value 0, are in the subalgebra. It should also be noted that this subalgebra is closed under the pointwise operations of _ and ^. 3.8 Functions With Finite Support Functions with …nite support are those for which Supp (f ) = fx 2 J : f (x) > 0g is …nite. It is easy to see that the set F of all functions in M with …nite support is a subalgebra of M. For example, if f and g have …nite support, then Supp (f t g) and Supp (f u g) are both …nite, because Supp (f t g) = Supp Supp (f u g) = Supp

f ^ gL _ f L ^ g

f ^ gR _ f R ^ g

Supp (f ) [ Supp (g) Supp (f ) [ Supp (g)

Also, Supp (f ) = fx0 : x 2 Supp (f )g is …nite. Both 0 and 1 have …nite support. The set F is also closed under the operations ^ and _. Note that although f L and f R are not in F, the functions f L ^ g and f R ^ g have …nite support for g 2 F, so are in F. The set of …nite subsets of J yields a subalgebra of F. Of course, if J is 33

…nite, F = M. If J = I, F does not contain any convex functions except singletons, and the subalgebra of M consisting of all convex normal functions in F coincides with I. We have not investigated this subalgebra of M beyond these observations.

4

Convolutions Using Product

In this section, we consider convolution of functions in M ap(J; I), where instead of using _ and ^ on I, we use _ and ordinary product. As before, we only assume that J is a bounded chain with an involution. Here are the basic operations we consider. De…nition 50 For f; g 2 M ap (J; I), d and e are de…ned by the convolutions W

(f d g) (x) = y_z=x (f (y)g(z)) W (f e g) (x) = y^z=x (f (y)g(z))

(30)

The other operations on M ap(J; I) that we consider are the same as before, namely the unary operation and the two constants 0 and 1: Our concern here is with the algebra P = (M ap(J; I); d; e; ; 0; 1): And principal tools are the auxiliary unary operations L and R. Theorem 51 The following hold for all f; g 2 M ap (J; I) f d g = f Lg _ f gL f e g = f Rg _ f gR

(31)

The proofs are straightforward. For functions f and g in M ap(J; I), f g of course means the function given pointwise by the operation of multiplication on [0; 1], that is, (f g)(x) = f (x)g(x). Some algebraic properties of P are developed in [17] and in [21] for J …nite. Remark 52 The algebra P is constructed by convolution operations on J using I with the operations _ and ordinary multiplication . Since the algebra (I; _; ) is isomorphic to the algebra (I; _; 4), where 4 is any strict t-norm, the results we obtain for P will be the same as if we were using an arbitrary strict t-norm 4 instead of . We have not looked at convolutions with respect to join and t-norms that are not strict. Theorem 53 The following hold for f; g 2 M ap (J; I). (1) d and e are commutative and associative. 34

(2) 0 d f = f ; 1 e f = f . (3) (f d g) = f e g ; (f e g) = f d g . Proof. The commutative laws are obvious from the pointwise formulas for d and e. The associative laws follow almost immediately from the de…nition of the operations. (f d (g d h)) (x) = = Similarly

W

y_z=x

f (y) (g d h) (z) = W

W

y_z=x u_v=z

W

y_z=x

f (y)g(u)h(v) =

((f d g) d h) (x) =

W

y_u_v=z

f (y)

W

u_v=z

W

y_u_v=z

g(u)h(v)

f (y)g(u)h(v)

f (y)g(u)h(v)

The associative law for e follows from that of d and item 3 of this Theorem. Item 2 is easy from (31). It should be noted however, that 0 e f 6= 0 and 1 d f 6= 1. We prove the …rst half of item 3. (f d g) (x) = f L g _ f g L

(x) = f L g _ f g L (x0 )

= f L (x0 ) g (x0 ) _ f (x0 ) g L (x0 ) = f L (x) g (x) _ f (x) g L (x)

=f

R

(x) g (x) _ f (x) g

R

(x) = f

= (f e g ) (x)

R

g _f g

R

(x)

The proof of the second half of item 3 is similar. The associative law for d is the equation f d (g d h) = (f d g) d h

(32)

and from (31), we get f L g L h _ f L ghL _ f g L h

L

_ f ghL

L

= f L ghL _ f g L hL _ f L g

L

h _ f gL

Taking f = 1, the function that is 1 everywhere, we get the equation g L h _ ghL _ g L h

L

_ ghL

L

= ghL _ g L hL _ g L h _ g L h

from which follows gLh

L

_ ghL

g L h _ ghL

L

= g L hL

L

= g L hL

(g d h)L = g L hL

But also

g L d hL = g LL hL _ g L hLL = g L hL _ g L hL = g L hL 35

L

h

From this and duality, we have the equations (g d h)L = g L d hL = g L hL

(33)

(g e h) = g e h = g h R

R

R

R R

The operations d and e are not idempotent. The idempotent law holds only in the special cases described in the following proposition. Proposition 54 The following hold in P. (1) f d f = f if and only if f (0) = 1; equivalently if and only if f L = 1. (2) f e f = f if and only if f (1) = 1; equivalently if and only if f R = 1. Proof. Let f 2 M ap (J; I). Then f d f = f Lf _ f f L = f Lf = f if and only if f L (x) = 1 for all x 2 J, which is equivalent to f (0) = 1. Item 2 follows similarly. Note that in the terminology of Nieminen [21], both d and e are idempotent for f if and only if f is endmaximal and normal. But the two operations d and e coincide for normal endmaximal functions, in fact, f L g _f g L = f R g _f g R = f _ g. Proposition 55 The normal functions form a subalgebra of P. Proof. We already know that 0 and 1 are normal, and that if f is normal, then so is f . So we only need to show that if f and g are normal, then so are f d g and f e g. A function f is normal if and only if f L (1) = 1. Now if f and g are normal, then by (33), (f d g)L (1) = g L hL (1) = g L (1)hL (1) = 1 It follows that f d g is normal, and dually that f e g is normal since f e g = (f e g)

= (f d g )

is normal. The following proposition was left as an unsolved problem in [17], and was settled in the a¢ rmative by [21] for the case that J was …nite. We make no such restriction on J here. Proposition 56 The convex functions form a subalgebra of P. 36

Proof. Again, we need only to show that f d g and f e g are convex whenever f and g are convex. For a function f to be convex means that f = f L ^ f R . We have f d g = f Lg _ f gL = f L gL ^ gR _ f L ^ f R gL = f LgL ^ f LgR _ f LgL ^ f RgL

= f LgL ^ f LgR _ f RgL Similarly

f e g = f RgR ^ f LgR _ f RgL The following two lemmas …nish the proof. Lemma 57 If f is monotone and g is convex, then f ^ g is convex. Proof. Suppose that f is monotone increasing. Then f = f L and g = g L ^ g R . We have f ^ g = f L ^ g L ^ g R = f L ^ g L ^ g R , which is convex by Proposition 34. Similarly if f is monotone decreasing. Lemma 58 For any functions f and g, f L g R _ f R g L is convex. n

o

Proof. Let S = x : f L (x) f R (x) and T = fx : g L (x) g R (x)g. Because both S and T are intervals of the chain J containing 0, we may suppose, without lost of generality, that S T . If x 2 S, we see that f L (x) =

W

W

f (y)

y x

so that f RL =

f (z) = f R (x)

z x

W

f (w)

f R (x)

w2J

and we must have f R (x) = f RL for x 2 S. Similarly, for x 2 T , g R (x) = g RL ; for x 2 J S , f L (x) = f RL ; and for x 2 J T , g L (x) = g RL : We have respectively 8 > > > f L g RL _ f RL g L (z) if z 2 S > >
f RL g RL

if z 2 T

> > > > : f RL g R _ f R g RL (z) if z 2 J

S T

This function is monotone increasing for z 2 S, on T S it is constant and greater or equal to the …rst part, and when z 2 J T , it is monotone decreasing and less or equal to the second part. Thus f L g R _ f R g L is convex. Note that the function f L g R _ f R g L has the general shape in the diagram below. It need not, of course, be continuous, but it is increasing on the left, 37

‡at on the top, and decreasing on the right.

f LgR _ f RgL Since the intersection of subalgebras is again a subalgebra, Propositions 55 and 56 yield the following. Theorem 59 The set of normal convex functions forms a subalgebra of P. We described several subalgebras of M in the early part of this paper for which the functions took values in f0; 1g, such as the algebra J of characteristic functions of points of J which is isomorphic to J; J[2] the closed-interval type-2 fuzzy sets, which is isomorphic to the algebra of interval-valued fuzzy sets; and 2J = M ap (J; f0; 1g) which is isomorphic to an algebra of subsets of J (not the usual algebra of subsets). We note that these same algebras are subalgebras of P due to the observation that (M ap (J; f0; 1g) ; t; u; ; 0; 1) = (M ap(J; f0; 1g); d; e; ; 0; 1) This follows immediately from the fact that minimum and product coincide on the set f0; 1g. 5

T-norms for Type-2 Fuzzy Sets

In this section, we restrict ourselves to the case J = I; and look at convolutions with respect to _ and ^ of a general class of binary operations on I, namely, the class of t-norms, and their duals, t-conorms. De…nition 60 A binary operation 4 : I (1) 4 is commutative and associative. (2) x 4 (y _ z) = (x 4 y) _ (x 4 z). (3) x 4 1 = x. 38

I ! I is a t-norm if

A binary operation 5 : I

I ! I is a t-conorm if

(4) 5 is commutative and associative. (5) x 5 (y _ z) = (x 5 y) _ (x 5 z). (6) x 5 1 = 1. A t-norm 4 and t-conorm 5 are dual with respect to 0 if (7) (x 4 y)0 = (x0 5 y 0 ). Notice that item (2) implies that a t-norm is increasing in each variable, and that x 4 (y ^ z) = (x 4 y) ^ (x 4 z) Similar properties hold for t-conorms. There is an extensive theory of these operations, and they play a signi…cant role in fuzzy theory and practice. See, for example [13] and [20]. In this section, we will examine the convolution of these operations, which probably should be called type-2 t-norms, and type-2 t-conorms. We will take convolutions with respect to _ and ^, as in the following proposition. Note that the idempotent t-norm “^” has already been dealt with, resulting in the operation u. The other two basic t-norms, product xy and Lukasiewicz (x + y 1) _ 0, will give new binary operations on M. Proposition 61 Let 4 be a t-norm, and let N be its convolution (f N g) (x) =

W

y4z=x

(f (y) ^ g(z))

Let 5 be the t-conorm dual to 4 with respect to the negation 0 , and let H be its convolution W (f H g) (x) = (f (y) ^ g(z)) y5z=x

Then for f; g; h 2 M ap (I; I) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

N is commutative and associative. f N 1 = f. f N 1 = f R: f N (g _ h) = (f N g) _ (f N h). If g h, then (f N g) (f N h). (f N g) = f H g . H is commutative and associative. f H 0 = f. f H (g _ h) = (f H g) _ (f H h). If g h, then (f H g) (f H h). 39

The proofs are routine. From now on, we assume that t-norms and t-conorms are continuous. An equivalent condition is that if y 4 z x, then x = y1 4 z1 ; with y1 y and z1 z; and similarly for t-conorms. Proposition 62 The following hold for f; g 2 M ap (I; I). (1) (2) (3) (4)

(f N g)R = f R N g R (f N g)L = f L N g L (f H g)R = f R H g R (f H g)L = f L H g L

Proof. To prove item 1, consider (f N g)R (x) = =

W

W

w xy4z=w

W

y4z x

f R N g R (x) =

(f (y) ^ g(z))

W

f R (y) ^ g R (z)

y4z=x

=

W

W

u y

y4z=x

If y 4 z

x, then x = y1 4 z1 ; with y1 f (y) ^ g(z)

(f (y) ^ g(z))

W

u y1

!

W

f (u) ^

f (u) ^

W

g(v)

v z

y and z1 !

!!

z: So !

g(v)

v z1

Thus (f N g)R f R N g R . If y 4 z = x and u y and v z, then u 4 v x and f (u) ^ g(v) (f N g)R (x). Thus (f N g)R = f R N g R . Item 3 is proved similarly. To prove item 2, (f N g)L = (f N g)L

= (f H g )R = f

= f L H gL

= f L N gL

R

Hg

R

= f L N gL

and item 4 is gotten similarly. The distributive laws in the following theorem are analogous to the laws f 4 (g ^ h) = (f 4 g) ^ (f 4 h) f 5 (g ^ h) = (f 5 g) ^ (f 5 h)

f 4 (g _ h) = (f 4 g) _ (f 4 h) f 5 (g _ h) = (f 5 g) _ (f 5 h)

for t-norms and t-conorms. 40

Theorem 63 The distributive laws f N (g u h) = (f N g) u (f N h) f H (g u h) = (f H g) u (f H h)

f N (g t h) = (f N g) t (f N h) f H (g t h) = (f H g) t (f H h)

hold for all g; h 2 M ap (J; I) if and only if f is convex. Proof. We prove the second identity holds if and only if f is convex. The rest follows immediately from this and the four identities (f t g) = f u g (f N g) = f H g

(f u g) = f t g (f H g) = f N g

It is easy to see that the following hold (I): (f N (g t h)) (x) =

_

y4(u_v)=x

(II): ((f N g) t (f N h)) (x) =

(f (y) ^ g (u) ^ h (v)) _

(p4q)_(s4t)=x

(f (p) ^ g (q) ^ f (s) ^ h (t))

and clearly (I) (II). Assume that f is convex, and let (p 4 q) _ (s 4 t) = x. To show that (I) (II), we want y such that y 4 (q _ t) = (y 4 q) _ (y 4 t) = x and f (y) ^ g (q) ^ h (t) f (p) ^ g (q) ^ f (s) ^ h (t) If p 4 q = s 4 t = x, let y = p ^ s. Then (y 4 q) _ (y 4 t) = x and either f (y) = f (p) or f (y) = f (s). In either case f (y) ^ g (q) ^ h (t)

f (p) ^ g (q) ^ f (s) ^ h (t)

Otherwise, we may as well assume that p 4 q < x and s 4 t = x. If s 4 q x, then (s 4 q) _ (s 4 t) = s 4 t = x and, taking y = s, we have the same inequality as above. On the other hand, if s 4 q > x, then s 4 q > s 4 t implies q > t. Thus we have p4q <x<s4q so there is a y with p < y < s and y 4 q = x. Then t < q implies that y 4 t y 4 q = x so that (y 4 q) _ (y 4 t) = (y 4 q) = x Now since f is convex, f (y)

f (p) ^ f (s), so that

f (y) ^ g (q) ^ h (t)

f (p) ^ g (q) ^ f (s) ^ h (t) 41

as desired. It follows that (I) (II), and hence (I) = (II) when f is convex. Suppose f N (g t h) = (f t g) N (f t h) holds for all g and h. Then, in particular, f N (1 t 1) = (f N 1) t (f N 1) The left side is

f N (1 t 1) = f N 1 = f

and the right side is

(f N 1) t (f N 1) = f t f R = f _ f R ^ f L ^ f RL = fR ^ fL

Thus f is convex. On the unit interval, t-norms are increasing in each variable. For convex functions, type-2 t-norms behave in a similar way with respect to the orders v and . Corollary 64 If f is convex and g v h, then f Ng v f Nh If f is convex and g

and

f Hg v f Hh

and

f Hg

h, then

f Ng

f Nh

f Hh

Proof. f N g = f N (g u h) = (f N g) u (f N h); whence f N g v f N h. The other parts follow similarly.

5.1

T-Norms on the Subalgebra [0; 1]

As we have seen, a copy of the algebra I = ([0; 1]; _; ^;0 ; 0; 1) is contained in the algebra M = (M ap(I; I); t; u; ; 0; 1), namely the characteristic functions a for a 2 [0; 1]. The formula (a N b) (x) =

W

y4z=x

a(y) ^ b(z)

says that a N b is the characteristic function of a4b, as it should be. It is clear that the t-norm N acts on the subalgebra of characteristic functions in exactly the same was as the t-norm 4 acts on the algebra [0; 1]: More precisely, Theorem 65 The mapping a ! a is an isomorphism from the algebra (I; 4) = ([0; 1]; _; ^; 4;0 ; 0; 1) into the algebra (M; N) = (M ap(I; I); t; u; N; ; 0; 1). 42

5.2 T-Norms on the Subalgebra of Intervals Consider the subalgebra of intervals, which are represented by functions of the form aL ^ bR with a b: From the formula aL ^ bR N cL ^ dR (x) =

W

y4z=x

aL ^ bR (y) ^ cL ^ dR (z)

we see that aL ^ bR N cL ^ dR (x) = 1 only for those x = y 4 z such that y 2 [a; b] and z 2 [c; d]: Thus the smallest x can be is a 4 c and the largest is b 4 d, and for any value in between, aL ^ bR N cL ^ dR (x) = 1: Recalling that the characteristic function of a 4 c is a N c, this means that aL ^ bR N cL ^ dR = (a N c)L ^ (b N d)R , which in turn is aL N cL ^

bR N dR : So t-norms on this subalgebra are calculated coordinatewise on the endpoints of the intervals. In [7], t-norms were de…ned on the set [0; 1][2] , and the requirements resulted in exactly that t-norms were calculated coordinatewise on the endpoints of the intervals. The fact that convolutions of t-norms result in coordinate-wise calculations is interesting indeed. These considerations apply to any t-norm, in particular to multiplication. Theorem 66 The mapping (a; b) ! aL ^ bR is an isomorphism from the algebra ([0; 1][2] ; _; ^; 4;0 ; 0; 1) into the algebra (M ap(J; I); t; u; N; ; 0; 1). 6

Notes and Questions

We hope that our treatment is mathematically palatable and sparks some interest in this topic. In a subsequent paper, we study the automorphisms of M, especially in the case J = I. There are many questions one can ask about the algebra M. Others besides those in the fuzzy community have studied algebras similar to M. In particular, the algebra in Section 3.7 has been studied, but we have no good references. We close with some speci…c questions and problems. (1) Generalize to M ap(V; W ) with V and W appropriate lattices. (2) f ! f R ^ f L is a natural way to convexify a function. What properties does this convexi…cation have? (3) For what binary operations on M is it true that (f g)R = f R g R ? Same question for L. (4) Does the algebra M have a subalgebra that is a non-distributive lattice? (5) Other subalgebras of M to consider: 43

(a) The subalgebra of concave functions. One can probably just dualize the results for convex functions. (b) In the case J = I, the subalgebra of piece-wise linear functions. (6) When is t increasing in each variable with respect to v; When is u increasing in each variable with respect to ? (7) Type-2 t-norms introduced in Section 5 merit further study.

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