American Journal of Applied Mathematics
Volume 3, Issue 5, October 2015, Pages: 229-232

Investigation of Order among Some Known T-norms

Shohel Babu1, Fatema Tuj Johora1, Abdul Alim2

1Mathematics, IUBAT-International University of Business Agriculture and Technology, Dhaka, Bangladesh

2Mathematics, BGMEA University of Fashion and Technology, Dhaka, Bangladesh

Email address:

(S. Babu)
(F. T. Johora)
(A. Alim)

To cite this article:

Shohel Babu, Fatema Tuj Johora, Abdul Alim. Investigation of Order among Some Known T-norms.American Journal of Applied Mathematics.Vol.3, No. 5, 2015, pp. 229-232. doi: 10.11648/j.ajam.20150305.14

Abstract: In this paper, order among some known T-norms is investigated. Firstly, the T-norm which is the strongest or greatest and the T-norm which is the weakest is observed. Comparing two T-norms we establish the relation which is strong or weak. In addition, for parametric T-norms after changing the interval of their parameter a relation has established which is strong or weak. Finally, compared has done among three or more T-norms.

Keywords: T-norm Algebraic Product , T-norm Min  T-norm Drastic Product  T-norm Franks Product , T-norm Einstein Product , T-norm Hamacher Product , T-norm Dubois & Prade Product

1. Introduction

T-norms or triangular norms are generalization of the classical triangular inequality according to K. Menger in 1942. In 1960, B. Schweizer and A. Sklar after revision of this work redefined the concept of triangular norm as an associative and commutative binary operation. They play a fundamental role in probabilistic metric spaces, probabilistic norms and scalar products, multiple-valued logic, fuzzy sets theory.


A T-norm is a function: [0, 1] × [0, 1] → [0, 1] which satisfies the following properties:




[Boundary condition:

2. Ordering of T-norms

2.1. Proposition: Among all T-norm "Min" is the Strongest

Proof: Suppose T is any T-norm.


[by boundary condition]

 = [by boundary condition]


Again  [by commutativity]

  [by monotonicity]



Consequently,is a lower bound of .

Again  is the greatest lower bound(glb) of .

But I is in chain, so .

Hence "min" is the strongest or greatest t-norm.

2.2. Proposition: Among all T-norm "Drastic Product" is the Weakest

Proof: Suppose T is any T-norm.

I; there are two cases:

Case1: Without loss of generality suppose =1, then

and  [by boundary condition].

Consequently =1.

Case2: When 1 and 1, then

But in I; clearly, 0


Hence among all T-norm  is the weakest.

2.3. Proposition: Letkϵ]0,1[, Frank’S Product

And Algebraic Product


Proof: Let and

Then  and  which means that  is monotonically decreasing.

If we have  then by mean value theorem we get



Now let


Which means that  is monotonically decreasing and,



This means that

Again, we clam that



2.4. Proposition: Letkϵ]1, ∞[, Frank’s Product (x,y)k=logk (1+(kx-1)(ky-1)/k-1)

and Algebraic Product


Proof: Let and

Then  and  which means that  is monotonically decreasing.

Now we have


because  is monotonically decreasing and



Now let

We see that

According to (1) and  is monotonically decreasing.




From (2) we get

This means that




2.5. Proposition: Letkϵ]1, ∞[, Frank’s Product

And Boundary Product


Proof: We will distinguish two cases:

Case1: If  then

Now for  we get

Which is true, because  and

Case2: If  then


This is obvious.


2.6. Proposition: Letkϵ[0,1], Duboi’s & Prade Product


Proof: We will distinguish three cases, according to the maximum value of .

Caes1: If , then


Case2: If , then

Case3: If , then

Hence we can say from three cases

2.7. Proposition: Let Einstein Product TE:I×I→I


Proof: Now we will split the proof into two cases:

Case1: If , then

Which is true, because

Case2: If then

(i)If , then

In this case we get,



This is true

From above we may conclude that

2.8. Proposition: Let Hamacher Product TH:I×I→I


Proof: As above we will distinguish two cases:

Case1: In this case, it will prove that , then

This is obvious, since

Case2: Now we prove the relation between .

We may see that if then

Otherwise, if , and consider , then

This is true because


Hence we may conclude that

3. Conclusion

Form above discussions, it conclude that T-norm Min is the strongest and T-norm Drastic product is the weakest T-norms. Also, T-norm Algebraic product is stronger than T-norm Hamacher product and Boundery Product. Similarly, T-norm Dubois & Prade product  is stronger than T-norm Algebraic product. Since, T-norms are using for taking suitable decision from multi-valued logic. So, the ordering of T-norms, that’s means, this paper will help to take correct decision using sufficient T-norms according to their order.


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