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)
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
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
(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.