Extended lattice-ordered structures for L-fuzzy sets and L-fuzzy numbers

Fuzzy Sets and Systems 54 (1993) 81-90 North-Holland

81

Extended lattice-ordered structures for L-fuzzy sets and L-fuzzy numbers Yuang-Cheh Hsueh Department of Computer and Information Science, National Chiao Tung University, Hsinchu, 300 Taiwan, ROC Received March 1992 Revised June 1992

Abstract: An L-fuzzy set A is a mapping of a set X into another set L. The set L is called the true set of A and X is called the universe of A. It has been discussed that a better structure for the truth set L is a complete lattice-ordered monoid. Then operations on L can be directly extended to operate on L x and a complete lattice-ordered monoid can be obtained. In this paper, we consider operations on L x which are extended from operations on X by the extension principle. We will obtain a similar complete lattice-ordered monoid constituted by extended operations. Moreover, a notion of L-fuzzy numbers is proposed. Then, extended operations on L-fuzzy numbers are discussed and a distributively lattice-ordered structure is developed for L-fuzzy numbers. Keywords: L-fuzzy set; L-fuzzy number; generalized extension principle; extended operation; extended structure; clc-monoid, dlc-monoid, tc-group.

1. Introduction

The notion of fuzzy sets is first introduced by Zadeh [31] in 1965 and is generalized to the notion of L-fuzzy sets by Goguen [11] in 1967. Let X and L be two nonempty sets. An L-fuzzy set A is a mapping of X into L. The set X is called the universe of A. Let L x denote the set of all L-fuzzy sets with universe X. In order to obtain operations on L x, it has been discussed in detail [12, 13] that a better structure for L is a complete lattice-ordered monoid. Definition 1.1 (Birkhoff [2], p. 327). A structure (L, v, A, *, T, ±, t) is called a complete latticeordered monoid or cl-monoid if (L, v, A, T, ±) is a complete lattice, where q- is the greatest element and ± is the least element, (L, *, t) is a monoid, where t is the identity, and the following infinite distributive laws are satisfied: (ID1) a v Aa~s bo = A v i s (a v b~), (ID2) a A Vows bo = Vows (a A bo), (ID3) a * Vows bo = Vows (a * bo) and ( V o , s bo) • a = Vows (bo * a), for arbitrary subset S of L. Since most monoids arising in applications are commutative, in this paper, we will confine ourselves to complete lattice-ordered commutative monoids or clc-monoids. In a clc-monoid (L, v, A, *, T, ±, t), it should be noted that a * ± = ± for all a e L. The element defined by Vc*b<-aC is called [2] the residual of a by b and is denoted by a : b. The pair ( *, :) forms a Galois connection or an adjunction [4] and possesses many useful algebraic properties. These algebraic properties have been used extensively in mathematical morphology [14, 24, 29]. They have also been used in solving fuzzy equations [5, 25]. A binary operation o on [0, 1] is called [17, 26, 27] a t-norm if it is isotone, commutative, associative Correspondence to: Dr. Yuang-Cheh Hsueh, Department of Computer and Information Science, National Chiao Tung University, Hsinchu, 300 Taiwan, ROC. 0165-0114/93/$06.00 ~) 1993----Elsevier Science Publishers B.V. All rights reserved

82

Yuang-Cheh Hsueh

/ E x t e n d e d lattice-ordered structures

and satisfies 0 o 0 = 0 and a o l = a for all a e [0, 1]. Note that if ([0, 1], v , A, * ,1, 0, 1) is a clc-monoid then * is a t-norm. Observation 1.2. I f (L, v , A , T, ±) is a completely distributive lattice [2], i.e., a complete lattice satisfying the infinite distributive laws (ID1) and (ID2), the the structure (L, v, A, A, T, Z, T) is a clc-monoid.

This observation is particularly important in the work of Hsueh [15] where he extends mathematical morphology on binary images to/-images. It is also the key for generalizing the extension principle and defining L-fuzzy numbers in this paper. 2. E x t e n d e d operations and structures for L-fuzzy sets

All operations in a clc-monoid (L, v , A , *, T , ±, t) can be directly extended from L to L x pointwise. Then, it is easy to see that the structure (L x, v , A , *, T , _1_, t) is also a clc-monoid. The residual A : B of A by B is given by (A : B)(x) = A(x) : B(x). On the other hand, extension of operations on X to operations on L x is usually done by the principle of extension [30, 32]. Let (L, v , A, *, T, _L, t) be a clc-monoid. Given any n-ary operation ~ on X, the principle of extension says that ~ can be extended to an operation A I~ on L x defined by:

A 1~ (Ao, A1 . . . . , An-1)(Y) =

V

y = ea(xo,Xl ..... x . _ l )

(A0(xo)

^ AI(Xl) A''"

^

An_l(Xn_l) ).

However, since (L, v , A, A, T, _L, 3-) is a clc-monoid whenever (L, v , ^ , *, q-, ±, t) is, if we consider the operation A appearing in the above formula as the operation in the monoid (L, A, q-), then in a more general sense, any n-ary operation q~ on X can be extended to an operation * I~ on L x defined by: • ],(A0, A1, • • • , A . - 1 ) ( y ) =

V

y=q,(Xo,Xl..... x,,-t)

(Ao(xo) *AI(Xl) * ' ' "

*A.-I(x.-1)).

Example 2.1. Consider the clc-monoid ([-o% oo], v , ^, +, 0% - ~ , 0) and the set V which is the underlying set of a vector space. Then for any two elements A, B in [-0% o0]v we have

(A + I+ B ) ( y ) =

V

A(xo) + B(Xl)

y =X0+Xl

for all y e V, where x0 + xl is the vector sum of Xo and Xl. The operation + l+ is conventionally written as ~ in mathematical morphology [7, 28] and is referred to as the Minkowski addition. Consider the commutative monoid ([0, 1], ^ , 1) and the clc-monoid ([0, 1], v, A, A, 1,0, 1). Mizumoto and Tanaka [18] show that the extended operation A 1,, on [0, 1] 1°'1] is idempotent, commutative, and associative. If we consider the commutative monoid ([0, 1], -, 1), they also show that the extended operation A 1. is commutative and associative, but not idempotent. In general, we will show that, for arbitrary commutative monoid (X, o, r/) and clc-monoid (L, v, A, *, T, .1_, t), the structure (L x, v, ^, .1 o, 3-, ±, tin ) is a clc-monoid. Such a structure with operations obtained from the generalized principle of extension will be called an extended lattice-ordered structure for L-fuzzy sets. Let (X, o, r/) be a commutative monoid and (L, v, ^, *, T, ±, t) be a clc-monoid. It is easy to see that the extended operation *lo is associative and commutative. Moreover, by defining tin as tin(x)=

t i f x = r/, _L otherwise,

we can easily show that t In is an identity in L x. We summarize the above results as a lemma.

Yuang-Cheh Hsueh ! Extended lattice-ordered structures

83

Lemma 2.2. If (X, o, rl) is a commutative monoid and (L, v , A, *, -I-, 1, ~) is a clc-monoid, then (L x, *[o, tin ) is a commutative monoid. The following two theorems are essentially due to Hsueh [15]. We omit their proofs. Theorem 2.3. I f (X, o, 77) is a commutative monoid and (L, v, A, *, T, _L, t) is a clc-monoid, then the structure (L x, v, A, *lo, -I-, ±, tl~ ) is a clc-monoid. Theorem 2.4. In the clc-monoid (L x, v, A, *[o, T, _L, el,), the residual A :lo B is given by

(A:loB)(x)= A A(y):B(z). y ~X°Z

Corollary 2.5. ff (X, o,-1, ~) is an Abelian group, then the residual A:]oB in the clc-monoid (L x, v, A, * [o, T, l , tin )/s given by

(A:IoB)(x)= A A(y):B(z). X ~yoz

-

3. Extended structures for L-fuzzy numbers Consider [0, 1]-fuzzy sets with the universe R, the set of all real numbers. A [0, 1]-fuzzy set A is called normal if there is a x e R such that A ( x ) = 1; A is called convex if for all x, y, z e R with x<-y <~z, A ( x ) A A ( z ) < ~ A ( y ) . Dubois and Prade [4] call a convex and normal [0, 1]-fuzzy set A a fuzzy number if, in addition, A is piecewise continuous. Goetschel and Voxman [9, 10] call a convex and normal [0, 1J-fuzzy set a fuzzy number if, in addition, A is upper semi-continuous or, equivalently, each cross section of A is a closed interval. The latter property is also required in Kaufmann and Gupta's definition of fuzzy numbers [16], while the upper semi-continuity assumption is also used in Dubois and Prade's latter definition of fuzzy numbers [6]. In this paper, any convex and normal [0, 1]-fuzzy set will be called a f u z z y number. Then, based on Observation 1.2, we will generalize the notion of fuzzy numbers to that of L-fuzzy numbers. Before doing so, we will make the following observations. First, observe that (R, v, A) is a chain (a totally ordered set), but is not complete. Moreover, the structures (~, +, - , 0) and (R\{0},., -1, 1) are Abelian groups. Next, let At([10, 1]~) denote the set of all fuzzy numbers. Under the assumption of upper semi-continuity, it is shown [10] that the structure (At[(0, 1]R, Air, A I^ ) is a distributive lattice. However, consider the following example. Example 3.1. In [0, 1], let a * b = a + b - ab, the probabilistic sum of a and b. It is known that the structure ([0, 1], v, A, *, 1, 0, 0) is a clc-monoid. Define a fuzzy number A by A(x)=

i

x -x

-l
Then

(A*Iv A ) ( - ½ )

= _½y~,,yA(x) , A ( y ) = ½• ½ = 3 C A ( - ½ )

and

(A*IvA)(1)= ~:xvy V A(x),A(y)

= 1 , ½ = 14=A(½).

84

Yuang-Cheh Hsueh / Extended lattice-orderedstructures

It should be clear that, in a clc-monoid (L, v , A, *, T, 3-, t), if a * a : k a or -l-*aCa, then the operation *Iv is not idempotent on L R. A clc-monoid (L, v , ^ , *, -I-, ±, t) is called integral if t = T. It should be pointed out that if ([0, 1], v, ^ , *, 1, 0, 1) is an integral clc-monoid with * being idempotent, then a * b = a ^ b for all a, b in [0, 1]. See Bellman and Giertz [1] for details. In a word, we have to assume that the clc-monoid (L, v, ^ , *, T, 3-, t) is integral and the operation * is idempotent in order that *Iv and *1^ will possibly become lattice operations on L-fuzzy numbers. The first observation motivates us to consider the following general replacement for the universe R.

Definition 3.2. A structure (M, v, A, o, r/) will be called a distributively lattice-ordered commutative monoid or dlc-monoid if (M, v, ^ ) is a lattice, (M, o, 7/) is a commutative monoid, and the following distributive laws are satisfied: (D1) a v ( b A c ) = ( a v b ) A ( a v c ) ; (D2) a A ( b v c ) = ( a A b ) v ( a A c ) ; (D3) ao(b v c) = (aob) v (aoc), for all a, b, c in M. Definition 3.3. If (G, o, -1, 7) is an Abelian group, then the structure (G, v, ^, o, -~, r/) will be called a dlc-group if it is a dlc-monoid. Definition 3.4. If (M, v, A) is a chain, then the dlc-monoid (M, v, ^, o, r/) will be called a totally ordered commutative monoid or tc-monoid. Similarly the dlc-group (G, v, ^, o, -1, r/) will be called a tc-group, if (G, v, A) is a chain.

Example 3.5. The structure (R, v , ^ , +, - , 0) is a tc-group while the structure (R\{0}, v , ^ , -, -1, 1) is not, since the distributive law (D3) is violated by • on R\{0}. However, if we let E+ = {x • E Ix > 0}, the the structure (R ÷, v , ^ , . , -1, 1) is a tc-group. Now, we are in the position to introduce the notion of L-fuzzy numbers. Definition 3.6. Given a tc-group (G, v, A, o, -1, r/) and an integral clc-monoid (L, v, A, *, T, 3-, T) such that * is idempotent, An L-fuzzy set A in L c will be called normal if A ( x ) = T for some x e G; A will be called convex if for all x, y, z e G with x <<-y ~ z, A ( x ) * A ( z ) <- A ( y ) . Definition 3.7. Let (G, v, A, o,-1, ?7) be a tc-group and (L, v, ^ , *, T, 3-,-1-) be an integral clc-monoid such that * is idempotent. Then a convex and normal L-fuzzy set will be called an L-fuzzy number.

Remark. Other generalizations of fuzzy numbers are also considered by Rodabaugh [19-23] and Drossos and Markakis [3]. Their approaches are quite different from the one given above. Lemma 3.8. Let ( G, v , A) be a chain and ( L, v , A, *, T, 3_, T) be an integral clc-monoid such that * is idempotent. Then the extended operations *l v and *1 ^ on L c preserve normality and convexity. Proof. Let A and B be two L-fuzzy sets in L ~. If A and B are normal, then there are elements a and b in G such that A ( a ) = T and B ( b ) = T. Let c = a v b. Then (A*IvB)(c)=

V c=xvy

A ( x ) * B ( Y ) > ~ A ( a ) * B ( b ) = T.

Yuang-ChehHsueh / Extendedlattice-orderedstructures Thus,

A,Iv B is normal.

85

Next, if A and B are convex, then for any x<-y<~zinG,

(A * Iv B)(x)* (A,Iv B)(z)= (x=VvvA(u)* B(v))* (zVv A(s) * B(t)) =(V

\v~x t~z

A ( x ) * B ( v ) * A ( z ) * B ( t ) ) v (~<~xZ(u)* B(x)* A(z)* B(t))

v ( V a(x)* B(V)* A(s)* B(z)) v(~<~xA(U)*B(x)*A(s)*B(z) ). \13~X S<<.z

s~z

Consider the first term V . . . . , , z A ( x ) * B(v)*A(z)*B(t). Since A and B are convex, A(x)*A(z) A(y). Moreover, if y < t, then since v ~ y <-t, B(v) * B(t) <-B(y). If t ~
= V A(s),B(t) = ( A , Iv B)(y). y=svt

Thus, A*I v B is also convex. This shows that the operation *l v preserves normality and convexity. Dually, we also have that the operation *l,, preserves normality and convexity. [] Let J~(L G) denote the set of all L-fuzzy numbers with the universe G, where (G, v, ^, o, -1, ~?) is a tc-group and (L, v, A, *, T, _1_,T) is an integral clc-monoid such that * is idempotent. Then, Lemma 3.8 says that *Iv and *1^ are operations on Jf(LG). To show they are lattice operations, we need to show the idempotent, commutative, associative, and absorption laws do hold for them.

Theorem 3.9. Let ( G, v, A) be a chain and (L, v, A, *, T, _L, T) be an integral clc-monoid such that * is idempotent. Then the structure (~¢'(LG), *l v, *l ^) is a lattice.

Proof. See the Appendix. Theorem 3.10. In the lattice (Jf(LG), *]v, *l ^), the following distributive laws hold:

A,Iv(B,I^ c)=(A ,IvB),[^(A,IvC) and A*[^(B*I^ C)=(A*I^B),Iv(A*[^ C) That is, the lattice (~f(LG), *Iv, ,]^) a distributive lattice. Proof. See the Appendix. Example 3.11. Since (~, v, A) is a chain and ([0, 1], v, ^ , A, 1, 0) is a clc-monoid, we have that the structure (~'([0, 1]R), ^Iv, A I^) is a distributive lattice.

Now, consider the extended operation .1o on L c. Since (G, o, -1, T/) is a group, we note that

(A*IoB)(x)= V A(u)*B(u-l°x) • u~G

Theorem 3.12. Let (G, v, ^, o,-1, T1) be a tc-group and (L, v, ^, *, T, 3_, T) be an integral

clc-monoid such that * is idempotent. Then the extended operation *1o on L c preserves normality and convexity. Proof. See the Appendix.

Yuang-Cheh Hsueh / Extendedlattice-orderedstructures

86

By Lemma 2.1, we have that (L a, * Io, t In) is a commutative monoid. Since t In is normal and convex, the structure (W(LC), *lo, tin) is also a commutative monoid.

Theorem 3.13. Let (G, v, h, o,-~, r/) be a tc-group and (L, v, h, *, T, ±,-F) be an integral clc-monoid such that * is idempotent. Then the structure (W(LC), *Iv, *l^, *1o, tin) is a dlc-monoid. Proof. See the Appendix. Example 3.14. Since (E, v, h, +, --, 0) and (R+, v, h , . , --1, 1) are tc-groups, and ([0, 1], v, h h, 1,0,1) is a clc-monoid, we have that the two structures (W([0,1]R), hi,, , hl^,hl÷,ll0) and (W([0, 1JR÷), hi,,, hi^, hi', 111) are dlc-monoids. Remark. In the dlc-monoid (W(LG), *Iv, *l^, *lo, tl,), define A*I<
Then the

C

(C*IoB)*I,,A

if it exists. Similar to the works of Dubois and Prade [5] and Sanchez [25], the notion of residuals can be used to find the maximal solution of the L-fuzzy equation A * IoB = C if there is one.

Acknowledgement The author wishes to thank the referees for their valuable comments and directions to literature of which he was unaware.

Appendix Proof of Theorem 3.9. That the commutative and associative laws hold follow directly from Lemma 2.1. It remains for us to show that the idempotent and absorption laws hold. For all A in W(LC), since

A(x)=A(x)*A(x)<'A(x) * V A(Y)<~A(x) * T =A(x), y<~x

we have

(A*lvA,(x , =x=yvzA(y,* A ( z , = (A(x)*y~<~xA ( y , ) v (A(X,*z~<~x A(z)) = A ( x ) * V A(y) = A ( x ) *A(x) = A(x). y<-~-x

Thus, *l,, is idempotent. Similarly, we have that *l^ is idempotent. Next, for any A and B in W(La),

((a*lvB)*l^m)(x)= V x=yvz

= V

V A(u),B(v)*A(z) y=u^o

A(u),B(v),A(z)v

x=uvly x<_z

= V

A(x).B(v).A(z)v

ll~.x~, z

v x<_.u V A(u).A(x)v = A(x),

V

A(u),B(v),A(x)

x<-uvl)

V

A(u).n(x).A(z)

u<~.x<<.z

V

u.::x<_~

A(u),B(v).A(x)

Yuang-ChehHsueh / Extendedlattice-orderedstructures

87

since V , ~ , A(u)* A(x) = A(x) and all other terms are less than or equal to A(x), for all x in G. Thus, ( a , I v B)*[^ a = a . The proof for the equality (A,l^ B)*[v A = A is similar. Thus, we conclude that the structure (W(LC), *Iv, *[^) is a lattice. []

.N'(LG), V A(y)*B(s)*C(t)

Proof of Theorem 3.10. For any A, B, and C in

(A*lv(B*l^ c))(x)= V x=yvz

Z=SAt

= V A ( x ) * B ( s l v V A(xl,B(s),C(t) S <~X

t<-x
v V A(y),B(x),C(t)v y~x
V A(y),B(s),C(x) y~x~s

=A(x)*(VxB(S)Vt~x
((z*lvn)*l^(A*lvC))(x)= V x~y

V A(u),B(s),A(v),C(t)

^Z y=UVS Z=UVt

= V A(u)*B(s)*A(v)*C(t)v V A(u)*B(s)*A(v)*C(t) X~Uvs x~U vl

x=UVt x<-U VS

= V A(u)*B(x)*A(v)v V A(u)*B(x)*A(v)*C(t) V~X~t

v V A(x),B(s),A(v)v

V A(x),B(s),A(o),C(t)

v V A(u)*A(v)*C(x)v V A(u)*B(s)*A(v)*C(x) O<_X~U

U<_X U~X<~S

v V A(ul*A(x)*C(t)v V A(ul*B(s)*A(x)*C(t) t<~x<~u

t~x u<_X<_S

for all x in G. Observe that

V A(u)*B(x)*A(v)~A(x)*B(x)<'A(x) * V B(s) u<_X<.U

S<_X

and

V A(x)*B(s)*A(v)=A(x)* V B(s). S<-X<-U

S~X

Thus

V A(u)*B(x)*A(v)v V A(x)*B(s)*A(v)=A(x)*V B(s). Similarly,

V A(u)*A(v)*C(x)v V A(u) *A(x) * C(t) = A(x) * V c(t). t~X

Yuang-ChehHsueh / Extendedlattice-orderedstructures

88 Moreover,

V Z(u)*B(x)*A(v)*C(t)=B(x), u<<-x o~x<<_t

V Z(u)*C(t), u<
V A(x)*B(s)*A(v)*C(t)=A(x)* $<<-x ~x~t

V B(s)*C(t)<-A(x)*V B(s), $~x~l

V A(u)*B(s)*A(v)*C(x)=C(x), t3~X

s<<.X

V A(u)*B(s), U<
V A(u)*B(s)*A(x)*C(t)=A(x)* t<<-x u ~ x <<_$

V B(s)*C(t)<-A(x)*VC(t). t<~x<<$

t<< x

These equalities and inequalities imply

((A*I~B)*I^(A*[v C))(x)=A(x)* V B(s) v B(x)* V A(u)*C(t) s<~x

u<~x<<_t

vA(x)* V C(t)vC(x)* V A(u)*B(s) t~x

u<<.x<<_$

=A(x)* V B(s) v B(x)* V A ( u ) * C ( t ) v A ( x ) , s
u<<.x<<_t

V B(s)*C(t) t<~x<<_s

v A(x)* V B(s) * C(t) v C(x)* V A(u) * B(s) t~x $ <~-x

u<<.x~ $

V A(u)*C(t)vA(x)*

=A(x)*VB(s)vB(x), = ( A * [ v (B*[ ^

V

B(s)*C(t)vC(x),

V A(u)*B(s)

C))(x).

Thus, A , v(B* ^ C) = (A*lv B)*I^(A*I~ C). In a s i m i l a r w a y , w e can p r o v e t h e o t h e r e q u a l i t y A * ^ ( B * ^ C ) = ( A * ^ B ) * 1,, ( A * I ^ C). T h e n , t h e t h e o r e m h a s b e e n p r o v e d . [] Proof of Theorem 3.12. F o r a n y n o r m a l L - f u z z y sets A, B in L c, if A(a) = -l- a n d B(b) = T , let

c =aob.

Then

(A *[o B)(c)

>!A(a) * B(b) = T.

T h u s , t h e o p e r a t i o n *[o p r e s e r v e s n o r m a l i t y . N e x t , s u p p o s e A a n d B a r e c o n v e x L - f u z z y sets. F o r a n y x, y, z in G with x ~< y ~< z. T h e n ,

(m *[on)(x)*(A*lon)(z)= V a(u)*B(u-l°x) * V A(v)*B(v-l°z) ueG

v~G

= ~/ A ( u ) * B ( u - ~ o x ) * A ( v ) * B ( v - l o z ) . u, IIEG

C a s e 1:

u-~ox<~v-loz. Ifu-~ox<~u-loy<_v-~oz,

then

V A(u)*B(u-l°x)*A(v)*B(v-l°z) <- V A(u)*B(u-I°Y)=(A u, veG

If

u-lox <~v-~oy <-v-~oz,

*[°B)(y).

ueG

then

k/ A(u) * B(u-1 ox) * A(v) * B(v -1 oz) <- k/ A(o) * B(o-1 oy) = (A * IoB)(y). U,1)EG

13EG

O t h e r w i s e , i.e. v - 1 o y ~< u - ~o x ~< v - ~o z ~< u - 1 o y, let s in G b e such t h a t u - 1 o x ~< s - 1 o y <~ v - ~o z. T h e n , since v -1 oy < s -~ oy ~< u -~ oy, w e h a v e u ~< s ~< v. T h u s ,

V A(u) • B(u -1 ox) • A(v) • B(v-lo z) <~ V A(s) • B(s -1 oy) = (A*Io B)(y). u,o~G

seG

Yuang-Cheh Hsueh / Extended lattice-ordered structures

89

Case 2: v - ~ o z <~u-~ox. Similar to case 1, we have

V

A(u)*B(u-'°x)*A(v)*B(v-l°z)

= (A*[oB(Y).

U,~EG

C o m b i n i n g all cases, we have (A,l° B ) ( x ) * (A*[o B ) ( z ) <~(A *lo B ) ( y ) for all x, y, z with x ~ < y ~ < z . Thus, the convexity is p r e s e r v e d by the o p e r a t i o n * Io. []

Proof of Theorem 3.13. It remains for us to s h o w the validity o f the following distributive law: A*

Io(B*Iv c)

= (A • IoB) *l ~ (A *lo C).

F o r each x in G, we have ( ( A , l o B ) , l ~ (A,[o C))(x) : x:Yvz ( y ~

=

A ( u ) , B ( v ) ) , ( z ~ o t A ( s ) , C(t))

V A ( u ) • B ( v ) , A ( s ) • C(t) x =(uoo)v (sot) V A ( s ) • B ( v ) * C(t) x =(soy)v (s.t)

V

A(s)*B(v)*C(t)

x = s o ( v v t)

=

(A*lo(a*lv C))(x).

O n the o t h e r hand,

V

((A * IoB) *l ,, ( a *lo C))(x) =

x=uvv

V A(s-I°u)*B(s)*A(t-I°o)*C(t) s,I~G

= V V A(s-l°x)*B(s)*A(t-l°v)*C(t) v V V A(s-l°u)*B(s)*A(t-l°x)*C(t) V<-X s , t ~ G

u<-x S , l e G

= V A ( s -1 ox) • B(s) , A ( t -lo v) • C(t) v V A ( s-1 ox) • B(s) , A ( t -lo v) • C(t) s<<-t

t~s

v V A ( s - l ° u ) * B(s) , A ( t -1 ox) • C(t) v V A ( s-1 ou) • B(s) , A ( t -1 ox) • C(t). u~x S~t

u~x t<-~S

If v ~
A ( t -1 o v) * A ( s -1 ox) <~A ( t -1 o x ). Similarly, if u ~< x and t ~< s, t h e n A ( s - 1 o u) * A (t - 1 o x) ~
= V

INS

V A(u-lox)*B(s)*C(t)

uffG U=$Vt

-- ( A , l o ( n , l v T h e p r o o f of the t h e o r e m is n o w c o m p l e t e d .

c))(x). []

References [1] [2] [3] [4] [5]

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90

Yuang-Cheh Hsueh / Extended lattice-ordered structures

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