Answering An Open Problem on $T$-Norms for Type-2 Fuzzy Sets
Xinxing Wu Guanrong Chen
math.GM


]Answering an Open Problem on T-Norms for Type-2 Fuzzy Sets


X. Wu]Xinxing Wu
[X. Wu]School of Sciences, Southwest Petroleum University, Chengdu, Sichuan 610500, China
wuxinxing5201314@163.com

G. Chen]Guanrong Chen
[G. Chen]Department of Electrical Engineering, City University of Hong Kong,
Hong Kong SAR, China
gchen@ee.cityu.edu.hk

This work was supported by the National Natural Science Foundation of China
(No. 11601449), the Science and Technology Innovation Team of Education Department of Sichuan for Dynamical Systems and its
Applications (No. 18TD0013), and the Youth Science and Technology Innovation Team of Southwest
Petroleum University for Nonlinear Systems (No. 2017CXTD02).











This paper proves that a binary operation ⋆ on [0, 1], ensuring that the binary operation
⋏ is a t-norm or ⋎ is a t-conorm, is a t-norm,
where ⋏ and ⋎ are special convolution operations defined by

    (f⋏ g)(x)=sup{f(y)⋆ g(z): y▵ z=x},


    (f⋎ g)(x)=sup{f(y)⋆ g(z): y ▿ z=x},

for any f, g∈ Map([0, 1], [0, 1]), where ▵ and ▿ are
a continuous t-norm and a continuous t-conorm on [0, 1], answering negatively an open
problem posed in <cit.>. Besides, some characteristics of t-norm and t-conorm
are obtained in terms of the binary operations ⋏ and ⋎.


[
    [
    December 30, 2023
=====================





§ INTRODUCTION


In 1975, Zadeh <cit.> introduced the notion of type-2 fuzzy sets (T2FSs) – that is, fuzzy set with fuzzy sets as truth values
(simply, “fuzzy-fuzzy sets”) – being an extension of type-1 fuzzy sets (FSs) and interval-valued fuzzy sets (IVFSs), which was also equivalently
expressed in different forms by Mendel et al. (<cit.>–<cit.>). Because the truth values of T2FSs are fuzzy, they are more adaptable to a
further study of uncertainty than FSs and have been applied in many studies (<cit.>–<cit.>).
Mendel <cit.> summarized some important advances for FSs and T2FSs from 2001 to 2007.
Hu and Kwong <cit.> discussed t-norm operations of T2FSs and obtained a few properties of type-2 fuzzy numbers.
For better understanding of T2FSs, Aisbett et al. <cit.> translates their constructs to the language of functions in spaces.
Chen and Wang <cit.> used T2FSs to give a new technique for fuzzy multiple attributes decision making. Sola et al. <cit.> provided
a more general perspective for interval T2FSs and showed that IVFSs can be viewed as a special case of interval T2FSs.
Ruiz et al. <cit.> obtained two results for join and meet operations for T2FSs with arbitrary secondary
memberships. Recently, Wang  <cit.> introduced the notion of conditional fuzzy sets to characterize T2FSs.
Then, Wu et al. <cit.> presented a Jaccard similarity measure for general T2FSs, as
an extension of the Jaccard similarity measure for FSs and IVFSs.



Being an extension of the logic connective conjunction and disjunctionin classical two-valued logic,
triangular norms (t-norms) with the neutral 1 and triangular conorms (t-conorms) with the neutral 0
on the unit interval I=[0, 1] were introduced by Menger <cit.> in 1942 and by Schweizer and Sklar <cit.>
in 1961, respectively. Because t-norms and t-conorms have a close connection with fuzzy set theory
and order related theories, they play an important role in many fields, such as fuzzy set theory <cit.>, fuzzy logic <cit.>,
fuzzy systems modeling <cit.>, and probabilistic metric spaces <cit.>. Walker and Walker <cit.>
extended t-norms and t-conorms to the algebra of truth values of T2FSs. Then, Hernándes et al. <cit.> introduced the
notions of t_r-norm and t_r-conorm by adding some “restrictive axioms” (see Definition <ref> below) with
systematic analysis. In particular, they <cit.> proved that the following binary operation ⋏
(resp., ⋎) on the set of all normal and convex functions constructed by convolution is a t_r-norm (resp., a t_r-conorm).
Recently, we proved <cit.> that the fuzzy metric M of every stationary fuzzy metric space (X, M, ⋆) is uniformly continuous.



Throughout this paper, let I=[0, 1], Map(X, Y) be the set of all mappings
from X to Y, and `≤' denote the usual order relation in the lattice of real numbers.
In particular, let 𝐌=Map(I, I) and 𝐋 be the set of all normal and convex functions in 𝐌.

<cit.>
A t-norm on I is a binary operation ⋆: I^2→ I satisfying

  
  (T1) (commutativity/symmetry) x⋆ y=y⋆ x for x, y∈ I;
  
  (T2) (associativity) (x ⋆ y) ⋆ z=x⋆ (y⋆ z) for x, y, z∈ I;
  
  (T3) (increasing) ⋆ is increasing in each argument;
  
  (T4) (neutral element) 1⋆ x=x⋆ 1=x for x∈ I.

A binary operation ⋆: I^2→ I is a
t-conorm on I if it satisfies axioms (T1), (T2), and (T3) above;
axiom (T4'): 0⋆ x=x⋆ 0=x for x∈ I.



For any subset B of X, a special fuzzy set χ_B, which is called the characteristic function
of B, is defined as

    χ_B(x)=
    1,     x∈ B,
    
    0,     x∈ X∖ B.


<cit.>
A binary operation T: 𝐋^2 →𝐋 is a t_r-norm
(t-norm according to the restrictive axioms), if

  
  * T is commutative, i.e., T(f, g)=T(g, f) for f, g∈𝐋;
  
  * T is associative, i.e., T(T(f, g), h)=T(f, T(g, h)) for f, g, h∈𝐋;
  
  * T(f, χ_{1})=f for f∈𝐋 (neutral element);
  
  * letting f, g, h∈𝐋 such that g⊑ h;
  then, T(f, g)⊑ T(f, h) (increasing in each argument);
  
  * T(χ_[0, 1], χ_[a, b])=χ_[0, b];
  
  * T is closed on 𝐉;
  
  * T is closed on 𝐊;

where 𝐉 is the set of all characteristic functions of the elements of I,
and 𝐊 is the set of all characteristic functions of the
closed subintervals of I, i.e., 𝐉={χ_{x}: x∈ I},
𝐊={χ_[a, b]: 0≤ a ≤ b≤ 1}.

A binary operation S: 𝐋^2→𝐋 is a
t_r-conorm if it satisfies axioms (O1), (O2), (O4), (O6), and (O7) above;
axiom (O3'): S(f, χ_{0})=f; and axiom (O5'):
S(χ_[0, 1], χ_[a, b])=χ_[a, 1]. Axioms (O1), (O2), (O3), (O3'),
and (O4) are called “basic axioms”, and an operation that complies with
these axioms will be referred to as t-norm or t-conorm, respectively.


Convolution as a standard way to combine functions was used to
construct operations on Map(J, [0, 1]). Let ∘ and ▴ be two binary operations
defined on X and Y, respectively, and ▾ be an appropriate operation on Y.
Define a binary operation ∙ on the set Map(X, Y) by

    (f∙ g)(x)=▾{f(y) ▴ g(z): y∘ z=x}.

This method of defining an operation on Map(X, Y) from operations on X and Y is called
convolution.

<cit.>
Let ⋆ be a binary operation on I, ▵ be a
t-norm on I, and ▿ be a t-conorm on I. Define the binary operations
⋏ and ⋎: 𝐌^2→𝐌 as follows: for f, g∈𝐌,

    (f⋏ g)(x)=sup{f(y)⋆ g(z): y▵ z =x},

and

    (f⋎ g)(x)=sup{f(y)⋆ g(z): y ▿ z =x}.



In 2015, Hernándes et al. <cit.> proposed
the following open problem on the binary operations ⋏ and ⋎.

<cit.>
Apart from the t-norms, does there exist other binary operation
`⋆' on I such that `⋏' and `⋎' are, respectively,
a t_r-norm and a t_r-conorm on 𝐋?


This paper first gives a negative answer to Question <ref>, proving that, if a binary operation ⋆
ensures that ⋏ is a t_r-norm on 𝐋 or ⋎ is a t_r-conorm on 𝐋,
then ⋆ is a t-norm, i.e., ⋆ satisfies axioms (T1)–(T4). Then, it is proved that the following
are equivalent:

  
  * ⋆ is a t-norm on I;
  
  * ⋏ is a t_r-norm on 𝐋;
  
  * ⋏ is a t-norm on 𝐋;
  
  * ⋎ is a t_r-conorm on 𝐋;
  
  * ⋎ is a t-conorm on 𝐋.

Finally, analogous results on ▵ are presented when
the binary operation ⋆ is restricted to be a continuous t-norm.




§ PRELIMINARIES

A type-1 fuzzy set A in space X is a mapping from X to I, i.e.,
A∈ Map(X, I), and A(x) is called the degree of membership of an element x∈ X to the set A.
The two sets Ø and X are special elements in Map(X, I), with Ø(x)≡ 0 and X(x)≡ 1, respectively.
A fuzzy set A∈ Map(X, I) is normal if sup{A(x): x∈ I}=1.

<cit.>
A function f∈𝐌 is convex if, for any x≤ y≤ z, it holds that
f(y)≥ f(x)∧ f(z).


<cit.>
A type-2 fuzzy set A in space X is a mapping

    A: X→𝐌,

i.e., A∈ Map(X, 𝐌). For any x∈ X, A(x) is also called the degree of membership
of an element x∈ X to the set A.


<cit.>
The operations of ⊔ (union), ⊓ (intersection),  (complementation) on 𝐌 are
defined as follows: for any f, g∈𝐌,

    (f⊔ g)(x)=sup{f(y)∧ g(z): y∨ z=x},


    (f⊓ g)(x)=sup{f(y)∧ g(z): y∧ z=x},

and

    ( f)(x)=sup{f(y): 1-y=x}=f(1-x).



From <cit.>, it follows that 𝕄=(𝐌, ⊔, ⊓, , χ_{0}, χ_{1})
does not have a lattice structure, although ⊔ and ⊓ satisfy the De Morgan's laws with respect
to the given operation .

Walker and Walker <cit.> introduced the following partial order on 𝐌.
<cit.>
f⊑ g if f⊓ g=f; f≼ g if f⊔ g=g.


It follows from <cit.> that both ⊑ and ≼ are partial
orders on 𝐌. In <cit.>, it was proved that the subalgebra
𝕃=(𝐋, ⊔, ⊓, , χ_{0}, χ_{1})
is a bounded complete lattice. In particular, χ_{0} and χ_{1}
are the minimum and maximum, respectively.


For f∈𝐌, define f^L and f^R in 𝐌 by

    f^L(x)=sup{f(y): y≤ x},

and

    f^R(x)=sup{f(y): y≥ x}.

Clearly, f^L and f^R are monotonically increasing and
decreasing, respectively. The following properties of f^L
and f^R are obtained by Walker et al. (<cit.>).
<cit.>
For f, g∈𝐌,

  
  * f⊑ g if and only if f^R∧ g ≤ f≤ g^R;
  
  * f≼ g if and only if f∧ g^L≤ g≤ f^L;
  
  * f≤ f^L, f≤ f^R;
  
  * (f^L)^L=f^L, (f^R)^R=f^R;
  
  * (f^L)^R=(f^R)^L=sup f;



(<cit.>)
Let f, g∈𝐋. Then, f⊑ g if and only if
g^L≤ f^L and f^R≤ g^R.



For f∈𝐌, f^L(0)=f(0) and f^R(1)=f(1).



From the definitions of f^L and f^R, this holds trivially.




§ ANSWER TO THE OPEN PROBLEM




 §.§ Commutativity and Associativity of ⋆



Let ⋆ be a t-norm on I. Then, x⋆ y=1 if and only if x=y=1.



Let ▵ be a continuous t-norm on I and ⋆ be a binary operation on I.
Then,

    (f⋏ g)(1)=f(1)⋆ g(1).




Since ▵ is a t-norm, from Lemma <ref>, it follows that

    (f⋏ g)(1)=sup{f(y)⋆ g(z): y▵ z=1}=f(1)⋆ g(1).




Let ▵ be a continuous t-norm on I and ⋆ be a binary operation on I.
Then, 
  
  * ⋏ and ⋎ are commutative on 𝐋 if and only if ⋆ is commutative;
  
  * If ⋏ and ⋎ are associative on 𝐋, then ⋆ is
associative.



(1) The sufficiency follows from the proof of <cit.>. It remains to prove the necessity.
Suppose on the contrary that ⋆ is not commutative. Then, there exist u, v∈ I such that u⋆ v≠ v⋆ u.
Choose two functions f, g∈𝐌 as

    f(x)=(u-1)x+1,

and

    g(x)=(v-1)x+1,
 for x∈ I. It can be verified that
f, g∈𝐋, as f and g are decreasing.
Since ⋏ is commutative, applying Lemma <ref> yields that

    u⋆ v=f(1) ⋆ g(1)=(f⋏ g)(1)
       =(g⋏ f)(1)=g(1) ⋆ f(1)=v⋆ u,

which is a contradiction. Therefore,
⋆ is commutative.



(2) Suppose on the contrary that ⋆ is not associative. Then, there exist
u, v, w∈ I such that u⋆ (v⋆ w)≠ (u⋆ v) ⋆ w. Choose three
functions f, g, h∈𝐌 as

    f(x)=(u-1)x+1,


    g(x)=(v-1)x+1,

and

    h(x)=(w-1)x+1,
 for x∈ I.
It can be verified that u, v, w∈𝐋, as f, g, and h are decreasing. Since
⋏ is associative, applying Lemma <ref> yields that

    u⋆ (v⋆ w)=f(1) ⋆ (g⋏ h)(1)=(f⋏ (g⋏ h))(1)
       =
    ((f⋏ g) ⋏ h)(1)=(f⋏ g)(1) ⋆ h(1)=(u⋆ v)⋆ w,

which is a contradiction. Therefore,
⋆ is associative.





Similar results to Proposition <ref> are obtained by Hernández et al.
<cit.> under the assumption that ⋏ and ⋎ are commutative or associative
on 𝐌, which is stronger than the condition in Proposition <ref>.





 §.§ Neutral Element 1 for ⋆



For any fixed x∈ I, define 𝒲_x: I→ I by

    𝒲_x(t)=
    0,     t∈ [0, x),
    
    t,     t∈ [x, 1],

for t∈ I. It can be verified that 𝒲_x∈𝐋, as 𝒲_x is increasing
for x∈ I.





Let ▵ be a continuous t-norm on I and ⋆ be a binary operation on I.
If ⋏ is a t-norm on 𝐋, then
0⋆ x=x⋆ 0=0 for all x∈ I.



(1)
As χ_{1} is a neural element, by Lemma <ref>, one has

    0   =χ_{0}(1)=(χ_{1}⋏χ_{0})(1)
       =χ_{1}(1) ⋆χ_{0}(1)=1⋆ 0.


(2) Fix any x∈ (0, 1). From
𝒲_x(t)=(𝒲_x⋏χ_{1})(t)=sup{𝒲_x(y) ⋆χ_{1}(z):
y ▵ z=t}, it follows that, for any t∈ (0, x),

    0=𝒲_x(t)=sup{𝒲_x(y) ⋆χ_{1}(z):
    y ▵ z=t}.

Since ▵(x, -) is continuous on [0, 1], and ▵(x, 0)=0, ▵(x, 1)=x,
it follows from the intermediate value theorem that there exists some
z_1∈ (0, 1) such that ▵(x, z_1)=x▵z_1=t. This, together with (<ref>), implies that

    0≥𝒲_x(x)⋆χ_{1}(z_1)=x⋆ 0,  i.e.,  x⋆ 0=0.


(3) Note that 0=χ_{0}(1/4)=(χ_{0}⋏χ_{1})(1/4)
=sup{χ_{0}(y)⋆χ_{1}(z): y▵ z=1/4}.
Similarly to the proof of (2), it follows that there exists
y∈ (0, 1) such that y▵1/2=1/4. This implies that

    0≥χ_{0}(y) ⋆χ_{1}(1/2)=0 ⋆ 0,  i.e.,  0=0⋆ 0.



Summing up (1)–(3) and the commutativity of ⋆ (Proposition <ref>),
it follows that, for any x∈ [0, 1],

    x⋆ 0=0⋆ x=0.




Let ▵ be a continuous t-norm on I and ⋆ be a binary operation on I.
If ⋏ is a t-norm on 𝐋, then
1⋆ x=x⋆ 1=x for all x∈ I.



(1) Since χ_{1} is a neural element, from Lemma <ref>, it follows that

    1   =χ_{1}(1)=(χ_{1}⋏χ_{1})(1)
       =χ_{1}(1)⋆χ_{1}(1) =1⋆ 1.


(2) For any fixed x∈ (0, 1), x=𝒲_x(x)=(𝒲_x⋏χ_{1})(x)
=sup{𝒲_x(y)⋆χ_{1}(z): y▵ z=x}. For y, z∈ I with y▵ z=x,
consider the following two cases:


Case 1. If z=1, then y=x. This implies that 𝒲_x(y)⋆χ_{1}(z)=x⋆ 1;

Case 2. If z<1, then χ_{1}(z)=0. Applying Lemma <ref> gives that

    𝒲_x(y)⋆χ_{1}(z)=0.

Thus,

    x=sup{𝒲_x(y)⋆χ_{1}(z): y▵ z=x}=x⋆ 1.


The proof is completed by applying 0⋆ 1=0 and the commutativity of ⋆.




 §.§ Increasing in Each Argument for ⋆

For any fixed x∈ I, define 𝒱_x: I→ I by

    𝒱_x(t)=(x-1)t+1,   ∀ t∈ I.

It can be verified that 𝒱_x∈𝐋, as 𝒱_x
is decreasing for x∈ I. Clearly, functions f, g, and h constructed in Proposition <ref>
satisfy that f=𝒱_u, g=𝒱_v, and h=𝒱_w.




Applying the decreasing property of 𝒱_x immediately yields the following result.


For any x∈ I, 𝒱^L_x≡ 1 and 𝒱^R_x=𝒱_x.




For any x_1, x_2∈ I with x_1≤ x_2,
𝒱_x_1⊑𝒱_x_2.



Clearly, 𝒱_x_1≤𝒱_x_2. Applying Lemma <ref> yields that

    𝒱^L_x_2≤𝒱^L_x_1,

and

    𝒱^R_x_1≤𝒱^R_x_2.

This, together with Theorem <ref>, implies that

    𝒱_x_1⊑𝒱_x_2.







Let ▵ be a continuous t-norm on I and ⋆ be a binary operation on I.
If ⋏ is a t-norm on 𝐋, then for any y∈ (0, 1)
the functions ⋆^r_y and ⋆^l_y are increasing, where
⋆^r_y(x)=x⋆ y and ⋆^l_y(x)=y⋆ x for x∈ I.



It follows from Proposition <ref> that ⋆^r_y=⋆^l_y. So,
it suffices to prove that ⋆^r_y is increasing.

For any  0≤ x_1≤ x_2≤ 1, since ⋏ is increasing in each argument,
from Lemma <ref>, it follows that

    𝒱_x_1⋏𝒱_y⊑𝒱_x_2⋏𝒱_y.

In particular, by Theorem <ref>,

    (𝒱_x_1⋏𝒱_y)^R≤ (𝒱_x_2⋏𝒱_y)^R.

This, together with Lemmas <ref> and <ref>, implies that

    x_1⋆ y    =𝒱_x_1(1)⋆𝒱_y(1)
    =(𝒱_x_1⋏𝒱_y)(1)
       =(𝒱_x_1⋏𝒱_y)^R(1)
    ≤ (𝒱_x_2⋏𝒱_y)^R(1)
       =(𝒱_x_2⋏𝒱_y)(1)=
    𝒱_x_2(1)⋆𝒱_y(1)
       =x_2⋆ y,

i.e.,

    ⋆^r_y(x_1)=x_1⋆ y≤ x_2⋆ y=⋆^r_y(x_2).

Therefore, ⋆^r_y is increasing.




 §.§ Answer to Question <ref>



Let ▵ be a continuous t-norm on I and ⋆ be a binary operation on I.
If ⋏ is a t-norm on 𝐋, then ⋆ is a t-norm.



It follows directly from Proposition <ref>, and Lemmas <ref> and <ref>.




Similarly, the following result can be verified.




Let ▿ be a continuous t-conorm on I and ⋆ be a binary operation on I.
If ⋎ is a t-conorm on 𝐋, then ⋆ is a t-norm.



Theorems <ref> and <ref> show that a binary operation ⋆ on I,
ensuring that ⋏ is a t-norm (thus a t_r-norm) on
𝐋 or ⋎ is a t-conorm (thus a t_r-conorm) on
𝐋, must be a t-norm. This give a negative answer to Question <ref>.




Combining together Theorems <ref>, <ref>, and <cit.>,
one obtains the following.




Let ▵ be a continuous t-norm, ▿ be a continuous t-conorm, and
⋆ be a continuous binary operation on I. Then, the following statements are equivalent:

  
  * ⋆ is a t-norm on I;
  
  * ⋏ is a t_r-norm on 𝐋;
  
  * ⋏ is a t-norm on 𝐋;
  
  * ⋎ is a t_r-conorm on 𝐋;
  
  * ⋎ is a t-conorm on 𝐋.





§ FURTHER STUDY ON THE BINARY OPERATION ⋏


Let ⋆ be a continuous t-norm on I and ▵ be a surjective
binary operation on I. Define the binary operation
⋏: 𝐌^2→𝐌 as follows:
for f, g∈𝐌,

    (f⋏ g)(x)=sup{f(y)⋆ g(z): y▵ z =x}.

Here, the surjection assumption on ▵ is necessary, because (f⋏ g)(x)
is not well defined for every point x in I∖▵(I^2), if ▵
is not surjective. Motivated by Question <ref>, a fundamental question is:
Apart from the t-norms, does there exist other binary operation `▵' on
I such that `⋏' is a t_r-norm on 𝐋?

This section will also give a negative answer to this question.


For x_1, x_2∈ I, χ_{x_1}⊑χ_{x_2} if and only if x_1≤ x_2.



Firstly, it can be verified that, for any x∈ I,

    χ^L_{x}(t)=
    0,     t∈ [0, x),
    
    1,     t∈ [x, 1],

and

    χ^R_{x}(t)=
    1,     t∈ [0, x],
    
    0,     t∈ (x, 1].

This, together with Theorem <ref>, implies that

    χ_{x_1}⊑χ_{x_2}
       ⇔   χ^L_{x_2}≤χ^L_{x_1} and χ^R_{x_1}≤χ^R_{x_2}
       ⇔    x_1≤ x_2.




Let ⋆ be a continuous t-norm on I and ▵ be a binary operation on I.
Then,
for any x_1, x_2∈ I, χ_{x_1}⋏χ_{x_2}=χ_{x_1▵ x_2}.



Since ⋆ is a continuous t-norm, applying Lemma <ref> gives

  
  * for y, z∈ I, χ_{x_1}(y)⋆χ_{x_2}(z)∈{0, 1};
  
  * χ_{x_1}(y)⋆χ_{x_2}(z)=1 if and only if y=x_1 and z=x_2.

This, together with (χ_{x_1}⋏χ_{x_2})(x)
=sup{χ_{x_1}(y)⋆χ_{x_2}(z): y▵ z=x}, implies that

    χ_{x_1}⋏χ_{x_2}=χ_{x_1▵ x_2}.




Let ⋆ be a continuous t-norm on I and ▵ be a binary operation on I.
Then, 
  
  * ⋏ is commutative on 𝐋 if and only if ▵ is commutative;
  
  * If ⋏ is associative on 𝐋, then ▵ is
associative.



(1) The sufficiency holds trivially. It remains to check the necessity.

For x, y∈ I, since ⋏ is commutative, it follows from
Lemma <ref> that

    χ_{x▵ y}=χ_{x}⋏χ_{y}
    =χ_{y}⋏χ_{x}=χ_{y▵ x},

implying that

    x▵ y=y▵ x.

Thus,
▵ is commutative.

(2) For x, y, z∈ I, since ⋏ is associative, it follows from
Lemma <ref> that

    χ_(x▵ y)▵ z   =χ_{x▵ y}⋏χ_{z}=(χ_{x}⋏χ_{y})⋏χ_{z}
       =χ_{x}⋏ (χ_{y}⋏χ_{z})=χ_{x}⋏χ_{y▵ z}
       =χ_x▵ (y▵ z),

implying that

    (x▵ y)▵ x=x▵ (y▵ z).

Thus,
▵ is associative.



Let ⋆ be a continuous t-norm on I and ▵ be a binary operation on I.
If ⋏ is a t-norm on 𝐋, then
1▵ x=x▵ 1=x for all x∈ I.



For any x∈ I, since χ_{1} is an neutral element,
applying Lemma <ref> yields that

    χ_{1▵ x}=χ_{1}⋏χ_{x}=χ_{x}=χ_{x}⋏χ_{1}=χ_{x▵ 1}.

Thus,
1▵ x=x=x▵ 1.



Let ⋆ be a continuous t-norm on I and ▵ be a binary operation on I.
If ⋏ is a t-norm on 𝐋, then, for any y∈ (0, 1),
the functions ▵^r_y and ▵^l_y is increasing, where
▵^r_y(x)=x▵ y and ▵^l_y(x)=y▵ x for any x∈ I.



It follows from Lemma <ref> that ▵^r_y=▵^l_y.
So, it suffices to prove that ▵^r_y is increasing.

For 0≤ x_1≤ x_2≤ 1,
applying Lemma <ref> follows that

    χ_{x_1}⊑χ_{x_2}.

Since ⋏ is increasing in each argument, applying Lemma <ref> yields that

    χ_{x_1▵ y}=χ_{x_1}⋏χ_{y}⊑χ_{x_2}⋏χ_{y}
    =χ_{x_2▵ y}.

This, together with Lemma <ref>, implies that

    ▵^R_y(x_1)=x_1▵ y ≤ x_2▵ y=▵^R_y(x_2).

Therefore, ▵^R_y is increasing.




Combining together Lemmas <ref>–<ref> and <cit.>
immediately yields the following result.




Let ⋆ be a continuous t-norm on I and ▵ be
a continuous binary operation on I. Then, the following statements are equivalent:

  
  * ▵ is a t-norm on I;
  
  * ⋏ is a t_r-norm on 𝐋;
  
  * ⋏ is a t-norm on 𝐋.



Similarly, one can obtain the following result.

Let ⋆ be a continuous t-norm on I and ▵ be
a continuous binary operation on I. Then, the following statements are equivalent:

  
  * ▵ is a t-conorm on I;
  
  * ⋏ is a t_r-conorm on 𝐋;
  
  * ⋏ is a t-conorm on 𝐋.





§ CONCLUSION

This paper has further studied the binary operations ⋏ and ⋎ defined in (<ref>),
(<ref>) and (<ref>) on 𝐋. By introducing two special families of
functions 𝒲_x and 𝒱_x (x∈ I),
it first proves that, when the continuous t-norm ▵ or continuous
t-conorm ▿ is fixed, the following hold:

  
  * ⋏ is a continuous t_r-norm on 𝐋 if and only if ⋏ is a continuous t-norm on 𝐋
if and only if ⋆ is a continuous t-norm;
  
  * ⋎ is a continuous t_r-conorm on 𝐋 if and only if ⋎ is a continuous t-conorm on 𝐋
if and only if ⋆ is a continuous t-norm.

In particular, these results negatively answer Question <ref>. Similarly to
Question <ref>, the case that the binary operation ▵ is fixed (see (<ref>))
has been investigated and some analogous results were obtained.






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