1. Introduction
The notions of triangular norms (t-norms for short) and triangular conorms (t-conorms for short) were introduced by Schweizer and Sklar [1]. Nullnorms are generalizations of triangular norms and triangular conorms with a zero element in the interior of the unit interval, and have to satisfy some additional constraints. Nullnorms are important from a theoretical viewpoint but also because of their numerous potential applications, such as expert systems, fuzzy quantifiers, neural networks, fuzzy logic [2]. The constructions of nullnorms were first studied on the unit interval [2] - [9]. In the subsequent studies, the interval has extended to bounded lattices [10] [11] [12].
Some constructions of nullnorms on bounded lattices were demonstrated in previously papers. Based on the existence of t-norms and t-conorms on an arbitrary bounded lattice, Karaçal et al. [10] proposed three construction methods of nullnorms on bounded lattices with an arbitrary zero element . Subsequently, Ümit Ertuğrul [11] proposed two construction methods of nullnorms on bounded lattices, which can be recognized as generalizations of two construction methods proposed in [10].
In this paper, we propose two more general construction methods of nullnorms on an arbitrary bounded lattice. The present study is organized as follows: In Section 2, we recall some basic concepts and show some existing constructions of nullnorms on an arbitrary bounded lattice. In Section 3, we introduce the notions of t-subnorm and t-subcnonorm. By using these operations, we propose new methods to obtain nullnorms on L under some additional constraints and their characteristics are examined. Finally, this summarization can be found in Section 4.
2. Preliminaries
A lattice is a partially ordered set in which each two-element subset has an infimum, denoted as , and a supremum, denoted as . A bounded lattice is a lattice that has the bottom and top elements written as 0 and 1, respectively. We denote simply by L in this article.
Let be a bounded lattice and be two binary operations on L, we can define a partial order:
Given a bounded lattice and , , a subset of L is defined as . Similarly, denote , and . If a and b are incomparable, we use the notation . The set of all elements which are incomparable with a are denoted by .
Definition 2.1. ( [13] [14] ) Let be a bounded lattice. An operation is called a triangular norm (t-norm for short) if it is commutative, associative, increasing with respect to both variables and has the neutral element such that for all .
Definition 2.2. ( [13] [14] ) Let be a bounded lattice. An operation is called a triangular conorm (t-conorm for short) if it is commutative, associative, increasing with respect to both variables and has the neutral element such that for all .
Definition 2.3. ( [15] ) Let be a bounded lattice. An operation is called a t-subnorm on L if it is commutative, associative, increasing with respect to both variables and for all .
Definition 2.4. ( [15] ) Let be a bounded lattice. An operation is called a t-subconorm on L if it is commutative, associative, increasing with respect to both variables and both for all .
Proposition 2.5. ( [15] ) If is a t-subnorm on a bounded lattice L, then defined by
(1)
is a t-norm on L.
Dually, if is a t-subconorm on a bounded lattice L, then defined by
(2)
is a t-conorm on L.
Definition 2.6. ( [10] ) Let be a bounded lattice. A commutative, associative, non-decreasing in each variable function is called a nullnorm if an element exists such that for all and for all .
It is easy to see that for all , and thus a is the zero element for V [10].
Proposition 2.7. ( [16] ) Let be a bounded lattice and be a nullnorm on L with the zero element a. Then, [(i)]
(i) is a t-conorm on ;
(ii) is a t-norm on .
Let be a bounded lattice and . Let be a t-norm on and be a t-conorm on . Based on the knowledge of the existence of t-norms and t-conorms on an arbitrary given bounded lattice, many construction methods of nullnorms were presented in previous papers. Generally speaking, these construction methods on an arbitrary bounded lattice under no additional constraints can be divided into two groups. One is proposed by Karaçal et al. in [10], which is defined as
(3)
The structures of is shown in Figure 1.
The other group is and its dual, i.e., , which are proposed by Ümit Ertuğrul [11] and defined as
(4)
Figure 1. The frame of .
and
(5)
The structures of and are shown in Figure 2 and Figure 3, respectively. In these figures, we denote and .
3. New Methods for Constructing Nullnorms on Bounded Lattices
In order to reduce the complexity in the proof of associativity, we introduce the following proposition.
Proposition 3.1. ( [17] ) Let S be a nonempty set and be subsets of S. Let H be a commutative binary operation on S. Then H is associative on if both of the following statements hold:
1) for all ;
2) for all .
Now, we introduce two construction methods which can be regard as generalizations of existing methods.
Theorem 3.2. Let be a bounded lattice and . Let
Figure 2. The frame of .
Figure 3. The frame of .
be a t-norm on , be a t-conorm on and be a t-subconorm on . If and
(6)
then is a nullnorm on L with the zero element a, where
(7)
Proof. The commutativity of can be proven directly based on its description. Similarly, we can express for all and for all .
Monotonicity: Let us prove that if , then for all . If , or , or , then it is clear that because and are in the same piece of U and U is monotonic in each piece. Moreover, contradicts the assumption that . Therefore, there are only three cases left to consider, namely, , , and .
(I) Assume that and .
(i) If , then and . As , we have .
(ii) If , then and . As , we have .
(iii) If , then and . As , we have .
Therefore, holds for .
(II) Assume that and such that .
(i) If , then and . As , we have .
(ii) If , then and , and thus .
(iii) If , then and . As , we have .
Therefore, holds for .
(III) Assume that and such that .
(i) If , then and . As , we have .
(ii) If , then and . As , we have .
(iii) If , then and . As , we have .
Therefore, holds for .
Combining the above cases, we obtain that holds for such that . Therefore, is monotonic.
Associativity: It can be shown that for all . By Proposition 3.1, We only need to consider the following cases:
(i) If , then sinceS is associative, we have .
(ii) If , then since T is associative, we have .
(iii) If , then , . As R is an associative function on , we have .
(iv) If and , then and , and thus .
(v) If and , then and . Thus .
(vi) If and , then and . It follows from (6) that .
(vii) If and , then and . It follows from (6) that .
(viii) If and , then and . Thus .
(ix) If and , then and . Thus .
(x) If , , , then , and . Thus .
From (i) to (x), we obtain that for all by Proposition 3.1. Therefore, is a nullnorm on L with the zero element a.¨
Theorem 3.3. Let be a bounded lattice and . Let be a t-norm on , be a t-subnorm on and be a t-conorm on . If and for all , then is a nullnorm on L with the zero element a, where
(8)
Proof. This can be proved similarly as Theorem 3.2.¨
The structures of and from Formula (7) and Formula (8) are shown in Figure 4 and Figure 5, respectively. We denote and in these figures.
Let be a bounded lattice and . Let be a t-norm on , be a t-conorm on . Taking in Formula (7), we obtain that
Figure 4. The frame of .
Figure 5. The frame of .
(9)
which is equal to given by Formula (4).
Dually, taking in Formula (8), we obtain that
(10)
which is equal to given by Formula (5).
Taking for all in Formula (7), then
(11)
which is equal to given by Formula (3).
Taking for all in Formula (8), then it is clear that also coincides with , which is given by Formula (3). Therefore, the two methods proposed in this study are more generalized than the methods proposed previously by [10] [11]. Now we give an example to show that we can obtain new nullnorms by the construction methods proposed in this paper.
Example 3.4. Let be a bounded lattice and let .
(i) Let be a t-norm on and be such that . Let and be two functions on defined by
(12)
and
(13)
Then S is a t-conorm and R is a t-subconorm on . It is easy to verify and the condition (6) holds. Therefore,
(14)
is a nullnorm on L with the zero element a by Theorem 3.2.
(ii) Dually, let be a t-conorm on and be such that . Then is a nullnorm on L with the zero element a by Theorem 3.3, where
(15)
4. Conclusion
In this study, based on the existing constructions of nullnorms on L, we continue to study construction methods of nullnorms on bounded lattices. Two methods for obtaining nullnorms on L are presented in this paper. Some examples were provided to show that the construction methods proposed in this paper generalized the methods presented in previous studies.
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