Some notes on T-partial order
© Karaçal and Aşıcı; licensee Springer 2013
Received: 4 December 2012
Accepted: 18 April 2013
Published: 30 April 2013
In this study, by means of the T-partial order defined in Karaçal and Kesicioğlu (Kybernetika 47:300-314, 2011), an equivalence relation on the class of t-norms on is defined. Then, it is showed that the equivalence class of the weakest t-norm on contains a t-norm which is different from . Finally, defining the sets of some incomparable elements with any according to , these sets are studied.
Triangular norms were originally studied in the framework of probabilistic metric spaces [1–4] aiming at an extension of the triangle inequality and following some ideas of Menger . Later on, they turned out to be interpretations of the conjunction in many-valued logics [6–8], in particular in fuzzy logics, where the unit interval serves as a set of truth values.
where T is a t-norm. This order is called a t-partial order of T. Moreover, some connections between the natural order and the t-partial order were studied.
In , it was investigated that implies the natural order but its converse needs not be true. It was showed that a partially ordered set is not a lattice with respect to . Some sets which are lattices with respect to under some special conditions were determined. For more details on t-norms on bounded lattices, we refer to [11–17].
In the present paper, we introduce an equivalence on the class of t-norms on based on the equality of the sets of all incomparable elements with respect to . The paper is organized as follows. We shortly recall some basic notions in Section 2. In Section 3, we define an equivalence on the class of t-norms on , and we determine the equivalence class of the weakest t-norm on . Thus, we obtain that the equivalence class of the weakest t-norm contains a t-norm which is different from a t-norm . We obtain that for arbitrary , there exists an element such that or is not continuous.
2 Basic definitions and properties
Definition 2.1 
A triangular norm (t-norm for short) is a binary operation T on the unit interval , i.e., a function , such that for all the following four axioms are satisfied:
(T3) whenever (monotonicity);
(T4) (boundary condition).
Example 2.1 
Also, t-norms on a bounded lattice are defined in a similar way, and then extremal t-norms as well as on L are defined as and on .
Remark 2.1 
- (i)Directly from Definition 2.1, we can deduce that, for all , each t-norm T satisfies the following additional boundary conditions:
Therefore, all t-norms coincide on the boundary of the unit square .
- (ii)The monotonicity of a t-norm T in its second component described by (T3) is, together with the commutativity (T1), equivalent to the monotonicity in both components, i.e., to(2.1)
Definition 2.2 
Proposition 2.1 
A function which is non-decreasing, i.e., which satisfies (2.1), is continuous if and only if it is continuous in each component, i.e., if for all , both the vertical section and the horizontal section are continuous functions in one variable.
Proposition 2.2 
By the same token, the upper semicontinuity of a non-decreasing function is equivalent to its right-continuity in each component.
Proposition 2.3 
F is continuous;
- (ii)F is continuous in each variable, i.e., for any and any , the unary function
F has the intermediate value property: For any , with , and any , there exists , with , such that .
Definition 2.3 
Example 2.2 
Proposition 2.4 
Let L be a bounded lattice, let T be a t-norm on L. Then the binary relation is a partial order on L.
Definition 2.4 
This partial order is called a T-partial order on L.
We will use the notation to denote the set of all incomparable elements with respect to .
3 The equivalence of any two t-norms
Let L be a lattice and let T be any t-norm on L. In , a partial order for a t-norm T on L was defined. In this section, we define an equivalence relation with the help of the sets of all incomparable elements with respect to . The above introduced T-partial order allows us to introduce the next equivalence relation on the class of all t-norms on .
Proposition 3.1 The relation ∼ given in Definition 3.1 is an equivalence relation.
Proof Let , and be t-norms on . Since , it is obtained that . Thus, the reflexivity is satisfied.
Let . Then we have that , and since , it is obtained that . Thus, the symmetry is satisfied. Let and . Then we have and . Since , it is obtained that . This means that the relation ∼ satisfies the transitivity. So, we have that ∼ is an equivalence relation. □
Proposition 3.2 shows that the equivalence class of the t-norm contains a t-norm which is different from .
Proposition 3.2 Let the t-norm be on . Then .
We give a contrary example as follows for the proof of Proposition 3.2.
Then . Firstly, let us show that . Let and . Then , but . Suppose that . Then, for some ℓ, . Since , it is not possible . Then , whence it is obtained that , a contradiction. Since for any there exists an element such that but , . Conversely, for any t-norm T, it is clear that . So, it is obtained that .
Now, we will show that . Let . For any with , it is obvious that . Otherwise, it would be for some ℓ. Since , . Thus, it must be for , a contradiction. Since there is an element y with such that , . This shows that . So, it is obtained that .
Lemma 3.1 Let T be a t-norm on and be arbitrarily chosen. If T is continuous at for all , then .
Proof Let T be a t-norm on and let be arbitrarily chosen. Suppose that . Then there exists an element such that , but . Since the t-norm is continuous, there exists an element such that for by Proposition 2.3. So, it is obtained that , a contradiction. Therefore we have that . □
Lemma 3.2 Let T be a t-norm on and the function be continuous. Then, for all with , we have that .
Proof Let T be a t-norm on and the function be continuous. Suppose that there exists an element such that and . Since is continuous, there exists an element such that for by Proposition 2.3. Thus, it is obtained that , a contradiction. Therefore, for all with , we have that . □
Theorem 3.1 Let T be a t-norm on and . Then, for arbitrary , there exists an element such that or is not continuous.
Proof Let T be a t-norm on and . Suppose that is continuous for . Choose arbitrarily. Then there exists an element such that but , or but . Let but . Since is continuous, then it is obtained that by Lemma 3.2, a contradiction. Let but . Since is continuous, then it is obtained that by Lemma 3.2, a contradiction. Therefore, for arbitrary , there exists an element such that or is not continuous. □
Dedicated to Professor Hari M Srivastava.
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