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A remark on constructing t-norms
Fuzzy Sets and Systems 41 (1991) 195-199 North-Holland
195
A remark on constructing t-norms* Janos C. Fodor Computer Center, E6tvOs Lordnd University, P.O. Box 157, H-1502 Budapest 112, Hungary Received July 1988 Revised June 1989
Abstract: In this paper we give a method to construct t-norms from t-norms. Our procedure is motivated by the Appendix of [4].
Keywords: Fuzzy connectives; t-norm; t-conorm
1. Introduction It is well known from the literature that t-norms and t-conorms are used very often in fuzzy set theory (see e.g. [5] for references). Applications to practical problems require the use of, in a sense, the 'most appropriate' t-norm or t-conorm. For this reason, the construction of new t-norms seems to be an important tool not only for the theory but also for the applications. The Appendix of [4] Schweizer and Sklar presented a method to construct t-norms from t-norms. In the present paper we generalize their procedure with the aid of some results of [5] and [2].
2. Background Let I = [0, 1] and Io = (0, 1). In this paper we will use the terminology of [5], i.e: (a) A function T : I x l - - - > I is said to be a t-norm iff T is commutative, associative, non-decreasing and T(a, 1 ) = a
Va~l.
(1)
A t-norm T is Archimedean iff T is continuous and T(a, a) < a
Va ~ Io.
(2)
An Archimedean t-norm T is strict iff T is strictly increasing in Io x Io. (b) A function S : I x I---> I is said to be a t-conorm or Archimedean or strict iff S has the same properties as T in part (a) with the following modifications: S(O, a) = a
Va ~ I,
(1')
S(a, a) > a
Va ~ Io.
(2')
* This work has been partially supported by OTKA-27-5-606. 0165-0114/91/$03.50 © 1991--Elsevier Science Publishers B.V. (North-Holland)
196
J. C. Fodor
Assume now that T is a continuous t-norm and S is a continuous t-conorm. Then we can define the following operations in I (see [2, 3, 5]): a err b = sup{x; T(a, x) <~b} and a COsb = inf{x; b <- S(a, x)}. We list some properties of mr and COs which will be used in the following part of the paper (more details can be found in [2] and [3]).
Lemma 1. (a) I f bl <- b2 then a trr bl <~a trr b2 and a COsbl <~a COsb2. (b) S ( a, a COsb) >- b; equality holds iff a <~b. (c) T(a, a olr b) <<-b; equality holds iff a >i b. (d) If T is strict then a o:r T(a, b) = b. (e) I f S is strict then a COsS(a, b) = b. (f) If S is strict then a COs 1 = 1. Proof. See [2, 3]. We will also use the following result of Climescu [1].
Theorem 1 [1]. Let (A, F) and (B, G) be semigroups. If the sets A and B are disjoint and if H is the mapping defined on (A t_JB) x (A U B) by (x,y)
H(x, y) =
Ii
LG(x,y)
ifx~A,y•A, ifx•A,y•B, if x • B, y c A , i/xeB,
yeB,
then (A t.J B, H ) is a semigroup.
3. The construction Assume that T~, T2 are any t-norms, T is a strict t-norm and S is a strict t-conorm. Moreover, let ~, • Io. We can define two functions in the following way:
Ul(a, b) = T[;t, Tl(;t acra, ;t olrb)]
for a, b • [0, A)
U2(a, b) = S[~,, T2(X ms a, ~. COsb)]
for a, b • [~,, 1].
and One can easily see that
0 <~ Ul(a, b) < ~, for a, b e [0, ~,)
(3)
Z <~U2(a, b) ~< 1 for a, b • [~., 1].
(4)
and
Constructing t-norms
197
Finally, let
f Ul(a, b) min(a, b) / min(a, b) l
l:(a, b)
1
U2(a, b)
if a • [0, ).), b • [0, ~.), if a e [0, 30, b e [)., 1],
if a ~ [3., 1], b ~ [0, ~,),
(5)
if a ~ [)., 1], b e [~,, 1].
Theorem 2. (a)
r is a t-norm. (b) If T1 and T2 are Archimedean then • is Archimedean. (c) If T1 and T2 are strict then • is strict.
Proof. (a) Commutativity and monotonicity follow from (3)-(5). z(a, 1 ) = a is obvious when a ~ [0, ~,). Assume now that a ~ [~,, 1]. Then r(a, 1) = U2(a, 1) = S[)., T2(~. tOs a, 3. to s 1)]
= S[A, T2(), tOs a, 1)] = S(~., ; ~ t O s a ) = a , from L e m m a l(b), (f). For associativity, it is sufficient to show that U1 and U2 are associative (by T h e o r e m 1 and (3), (4)). Using L e m m a l(d), we get ~. OCTT[)., TI(), OCTa, 3. OCTb)] = TI(~. Oct a, ~. Oct b). So (by associativity of T0 we have
UI(UI(a, b), c) = T[X, TI(~. OCTT(X, TI(X Oct a, 2. Oct b)), X Oct c)] = T[~., TI(TI(~. 0
= Ul(a, Ul(b, c)). It follows from L e m m a l(e) that ~. ~ s S[)., T2(). O)s a, ;t O)s b)] = T2(~. ms a, ~. O)s b). Hence, one can see the associativity property for/52 as for/51. Thus we have that is a t-norm. (b) The reader can readily verify that $ is continuous when T1 and T2 are continuous. Moreover, if a e [0, ,~) then
r(a, a) = Ul(a, a) = T[)., TI(~. OCTa, ,~, OCTa)] < T(3., Z OCTa ) ----a because T1 is A r c h i m e d e a n and T is strict. If a e [),, 1] then
g(a, a) = U2(a , a) = S[3., T2(~. ms a, ,~ ms a)] < S(),, ). ms a) = a by L e m m a l(b). Hence we get part (b). (c) This part of the assertion follows from simple calculations. Thus our theorem is proved.
J.C. Fodor
198
Example. Let To(a, b) = ab, So(a, b) = a + b - ab. T h e n
atrr°b=
1
if a<~b,
b/a
ifa>b
and
at°s°b=
0
ifa>~b
(b-a)/(1-a)
ifa
It is obvious that To and So are strict. O n e can conclude that T h e o r e m 8 of [4] is a particular case of our T h e o r e m 2 with T = To and S = So. In the next t h e o r e m we present a o:~ b. Theorem 3. If T1 and T2 are continuous t-norms then a o~ b = t i
[~" (~" a~r a) trrl (~, ter b)]
[ S [ ~ , (~, tOs a) a~r2 (). tOs b)]
if if if if
aa<~b' > b and a ~ [0, ~), a e [~., 1], b e [0, ~.),
a > b and a, b e [~,, 1].
Proof. T h e case a ~< b is obvious. Assume now that a > b. (i) Let a • [0, ~.). In this case there exists no x • [)., 1] such that b I> r(a, x) = a. Thus if r(a, x) ~< b then x • [0, ).). H e n c e we get that r(a, x) = Ul(a, x), i.e., a a~ b = s u p ( x ; T[~., TI(A trra, A trrx)] <- b}. This is equivalent to ~, act b = TI[~, trr a, Z oct ( a t r r b)], because T and T~ are continuous t-norms. (Z 0or a) ar~ (). a~r b) = ). trr (a o~r b), i.e., iff
(6) (6)
holds
if and
only
if
a • b = T[A, (7~ acra) trr~ (Z arb)]. (ii) Assume that a • [~,, 1] and b • [0, ~,). If b < x ~ Z then z(a, x) = x, i.e., it is impossible that r(a, x) ~< b. Let now x • [,1,, 1]. T h e n b I> z(a, x) = UE(a, x) >- Z is a contradiction. Thus a a~ b -- b in this case. (iii) Let a, b • [~,, 1]. If x • [0, ~,) then z(a, x) = x < b. Thus a a~ b = sup{x; S[~., T2(A to s a, ~. tOs x)] ~< b} = inf{x; S[~,, T2(~, tOs a, ~, tOs x)]/> b}. This is true if and only if T2[~, ~os a, ~. tos (a a~ b)] = ). tOs b, because ~, ~< b. T h e last equality is equivalent to (Z tOs a) ocr~(;~ tOs b) = 2~tOs (a tr, b ), i.e., a a~, b = S[A, (~. tos a) crr~ (~, tOs b)]. Thus our t h e o r e m is proved.
Constructing t-norms
199
References [1] A.C. Climescu, Sur l'6quation fonctionelle de l'associativit6, Bull. Inst. Politehn. lasi 1 (1946) 1-16. [2] J.C. Fodor, On inclusion and equality of fuzzy sets, in: A. Iv~inyi, Ed., Proc. of the 4th Conf. of Program Designers (ELTE, Budapest, 1988) 249-254. [3] M. Miyakoshi and M. Shimbo, Solutions of composite fuzzy relational equations with triangular norms, Fuzzy Sets and Systems 16 (1985) 53-63. [4] B. Schweizer and A. Sklar, Associative functions and abstract semigroups, Publ. Math. Debrecen 10 (1963) 69-81. [5] S. Weber, A general concept of fuzzy connectives, negations and implications based on t-norms and t-conorms, Fuzzy Sets and Systems 11 (1983) 115-134.
195
A remark on constructing t-norms* Janos C. Fodor Computer Center, E6tvOs Lordnd University, P.O. Box 157, H-1502 Budapest 112, Hungary Received July 1988 Revised June 1989
Abstract: In this paper we give a method to construct t-norms from t-norms. Our procedure is motivated by the Appendix of [4].
Keywords: Fuzzy connectives; t-norm; t-conorm
1. Introduction It is well known from the literature that t-norms and t-conorms are used very often in fuzzy set theory (see e.g. [5] for references). Applications to practical problems require the use of, in a sense, the 'most appropriate' t-norm or t-conorm. For this reason, the construction of new t-norms seems to be an important tool not only for the theory but also for the applications. The Appendix of [4] Schweizer and Sklar presented a method to construct t-norms from t-norms. In the present paper we generalize their procedure with the aid of some results of [5] and [2].
2. Background Let I = [0, 1] and Io = (0, 1). In this paper we will use the terminology of [5], i.e: (a) A function T : I x l - - - > I is said to be a t-norm iff T is commutative, associative, non-decreasing and T(a, 1 ) = a
Va~l.
(1)
A t-norm T is Archimedean iff T is continuous and T(a, a) < a
Va ~ Io.
(2)
An Archimedean t-norm T is strict iff T is strictly increasing in Io x Io. (b) A function S : I x I---> I is said to be a t-conorm or Archimedean or strict iff S has the same properties as T in part (a) with the following modifications: S(O, a) = a
Va ~ I,
(1')
S(a, a) > a
Va ~ Io.
(2')
* This work has been partially supported by OTKA-27-5-606. 0165-0114/91/$03.50 © 1991--Elsevier Science Publishers B.V. (North-Holland)
196
J. C. Fodor
Assume now that T is a continuous t-norm and S is a continuous t-conorm. Then we can define the following operations in I (see [2, 3, 5]): a err b = sup{x; T(a, x) <~b} and a COsb = inf{x; b <- S(a, x)}. We list some properties of mr and COs which will be used in the following part of the paper (more details can be found in [2] and [3]).
Lemma 1. (a) I f bl <- b2 then a trr bl <~a trr b2 and a COsbl <~a COsb2. (b) S ( a, a COsb) >- b; equality holds iff a <~b. (c) T(a, a olr b) <<-b; equality holds iff a >i b. (d) If T is strict then a o:r T(a, b) = b. (e) I f S is strict then a COsS(a, b) = b. (f) If S is strict then a COs 1 = 1. Proof. See [2, 3]. We will also use the following result of Climescu [1].
Theorem 1 [1]. Let (A, F) and (B, G) be semigroups. If the sets A and B are disjoint and if H is the mapping defined on (A t_JB) x (A U B) by (x,y)
H(x, y) =
Ii
LG(x,y)
ifx~A,y•A, ifx•A,y•B, if x • B, y c A , i/xeB,
yeB,
then (A t.J B, H ) is a semigroup.
3. The construction Assume that T~, T2 are any t-norms, T is a strict t-norm and S is a strict t-conorm. Moreover, let ~, • Io. We can define two functions in the following way:
Ul(a, b) = T[;t, Tl(;t acra, ;t olrb)]
for a, b • [0, A)
U2(a, b) = S[~,, T2(X ms a, ~. COsb)]
for a, b • [~,, 1].
and One can easily see that
0 <~ Ul(a, b) < ~, for a, b e [0, ~,)
(3)
Z <~U2(a, b) ~< 1 for a, b • [~., 1].
(4)
and
Constructing t-norms
197
Finally, let
f Ul(a, b) min(a, b) / min(a, b) l
l:(a, b)
1
U2(a, b)
if a • [0, ).), b • [0, ~.), if a e [0, 30, b e [)., 1],
if a ~ [3., 1], b ~ [0, ~,),
(5)
if a ~ [)., 1], b e [~,, 1].
Theorem 2. (a)
r is a t-norm. (b) If T1 and T2 are Archimedean then • is Archimedean. (c) If T1 and T2 are strict then • is strict.
Proof. (a) Commutativity and monotonicity follow from (3)-(5). z(a, 1 ) = a is obvious when a ~ [0, ~,). Assume now that a ~ [~,, 1]. Then r(a, 1) = U2(a, 1) = S[)., T2(~. tOs a, 3. to s 1)]
= S[A, T2(), tOs a, 1)] = S(~., ; ~ t O s a ) = a , from L e m m a l(b), (f). For associativity, it is sufficient to show that U1 and U2 are associative (by T h e o r e m 1 and (3), (4)). Using L e m m a l(d), we get ~. OCTT[)., TI(), OCTa, 3. OCTb)] = TI(~. Oct a, ~. Oct b). So (by associativity of T0 we have
UI(UI(a, b), c) = T[X, TI(~. OCTT(X, TI(X Oct a, 2. Oct b)), X Oct c)] = T[~., TI(TI(~. 0
= Ul(a, Ul(b, c)). It follows from L e m m a l(e) that ~. ~ s S[)., T2(). O)s a, ;t O)s b)] = T2(~. ms a, ~. O)s b). Hence, one can see the associativity property for/52 as for/51. Thus we have that is a t-norm. (b) The reader can readily verify that $ is continuous when T1 and T2 are continuous. Moreover, if a e [0, ,~) then
r(a, a) = Ul(a, a) = T[)., TI(~. OCTa, ,~, OCTa)] < T(3., Z OCTa ) ----a because T1 is A r c h i m e d e a n and T is strict. If a e [),, 1] then
g(a, a) = U2(a , a) = S[3., T2(~. ms a, ,~ ms a)] < S(),, ). ms a) = a by L e m m a l(b). Hence we get part (b). (c) This part of the assertion follows from simple calculations. Thus our theorem is proved.
J.C. Fodor
198
Example. Let To(a, b) = ab, So(a, b) = a + b - ab. T h e n
atrr°b=
1
if a<~b,
b/a
ifa>b
and
at°s°b=
0
ifa>~b
(b-a)/(1-a)
ifa
It is obvious that To and So are strict. O n e can conclude that T h e o r e m 8 of [4] is a particular case of our T h e o r e m 2 with T = To and S = So. In the next t h e o r e m we present a o:~ b. Theorem 3. If T1 and T2 are continuous t-norms then a o~ b = t i
[~" (~" a~r a) trrl (~, ter b)]
[ S [ ~ , (~, tOs a) a~r2 (). tOs b)]
if if if if
aa<~b' > b and a ~ [0, ~), a e [~., 1], b e [0, ~.),
a > b and a, b e [~,, 1].
Proof. T h e case a ~< b is obvious. Assume now that a > b. (i) Let a • [0, ~.). In this case there exists no x • [)., 1] such that b I> r(a, x) = a. Thus if r(a, x) ~< b then x • [0, ).). H e n c e we get that r(a, x) = Ul(a, x), i.e., a a~ b = s u p ( x ; T[~., TI(A trra, A trrx)] <- b}. This is equivalent to ~, act b = TI[~, trr a, Z oct ( a t r r b)], because T and T~ are continuous t-norms. (Z 0or a) ar~ (). a~r b) = ). trr (a o~r b), i.e., iff
(6) (6)
holds
if and
only
if
a • b = T[A, (7~ acra) trr~ (Z arb)]. (ii) Assume that a • [~,, 1] and b • [0, ~,). If b < x ~ Z then z(a, x) = x, i.e., it is impossible that r(a, x) ~< b. Let now x • [,1,, 1]. T h e n b I> z(a, x) = UE(a, x) >- Z is a contradiction. Thus a a~ b -- b in this case. (iii) Let a, b • [~,, 1]. If x • [0, ~,) then z(a, x) = x < b. Thus a a~ b = sup{x; S[~., T2(A to s a, ~. tOs x)] ~< b} = inf{x; S[~,, T2(~, tOs a, ~, tOs x)]/> b}. This is true if and only if T2[~, ~os a, ~. tos (a a~ b)] = ). tOs b, because ~, ~< b. T h e last equality is equivalent to (Z tOs a) ocr~(;~ tOs b) = 2~tOs (a tr, b ), i.e., a a~, b = S[A, (~. tos a) crr~ (~, tOs b)]. Thus our t h e o r e m is proved.
Constructing t-norms
199
References [1] A.C. Climescu, Sur l'6quation fonctionelle de l'associativit6, Bull. Inst. Politehn. lasi 1 (1946) 1-16. [2] J.C. Fodor, On inclusion and equality of fuzzy sets, in: A. Iv~inyi, Ed., Proc. of the 4th Conf. of Program Designers (ELTE, Budapest, 1988) 249-254. [3] M. Miyakoshi and M. Shimbo, Solutions of composite fuzzy relational equations with triangular norms, Fuzzy Sets and Systems 16 (1985) 53-63. [4] B. Schweizer and A. Sklar, Associative functions and abstract semigroups, Publ. Math. Debrecen 10 (1963) 69-81. [5] S. Weber, A general concept of fuzzy connectives, negations and implications based on t-norms and t-conorms, Fuzzy Sets and Systems 11 (1983) 115-134.