key: cord-0044805-obx3pobo
authors: Çaylı, Gül Deniz
title: Construction of Nullnorms Based on Closure and Interior Operators on Bounded Lattices
date: 2020-05-15
journal: Information Processing and Management of Uncertainty in Knowledge-Based Systems
DOI: 10.1007/978-3-030-50143-3_37
sha: 44b99746ddc120b1c9a554033ffed63edf93fd53
doc_id: 44805
cord_uid: obx3pobo

In this paper, we introduce two rather effective methods for constructing new families of nullnorms with a zero element on the basis of the closure operators and interior operators on a bounded lattice under some additional conditions. Our constructions can be seen as a generalization of the ones in [28]. As a by-product, two types of idempotent nullnorms on bounded lattices are obtained. Several interesting examples are included to get a better understanding of the structure of new families of nullnorms.

The definitions of t-operators and nullnorms on the unit interval were introduced by Mas et al. [20] and Calvo et al. [4] , respectively. In [21] , it was shown that nullnorms coincide with t-operators on the unit interval [0, 1] since both of them have identical block structures on [0, 1] 2 . As a generalization of t-norms and tconorms on the unit interval, nullnorms have a zero element s derived from the whole domain, regardless of whether t-norms and t-conorms have zero elements 0 and 1, respectively. In particular, a nullnorm is a t-norm if s = 0 while it is a t-conorm if s = 1. These operators are effective in various fields of applications, such as expert systems, fuzzy decision making and fuzzy system modeling. They are interesting also from theoretical point of view. For more studies about tnorms, t-conorms, nullnorms and related operators on the unit interval, it can be referred to [5, 10, 14, 17, 19, [22] [23] [24] 26, 27, 29] .

In recent years, the study of nullnorms on bounded lattices was initiated by Karaçal et al. [18] . They demonstrated the presence of nullnorms with a zero element on the basis of a t-norm and a t-conorm on a bounded lattice. Notice that the families of nullnorms obtained in [18] are not idempotent, in general. For this reason, Ç aylı and Karaçal [6] introduced a method showing the presence idempotent nullnorms on bounded lattices such that there is only one element incomparable with the zero element. Moreover, they proposed that there does not need to exist an idempotent nullnorm on a bounded lattice. After then, Wang et al. [28] and Ç aylı [7] presented some methods for constructing idempotent nullnorms on a bounded lattice with some additional conditions on theirs zero element. Their methods can be viewed as a generalization of the proposed method in [6] . On the contrary to the approaches in [6, 7, 28] based on the only infimum t-norm on [s, 1] 2 and supremum t-conorm on [0, s] 2 , in [8, 9] by using an arbitrary t-norm on [s, 1] 2 and an arbitrary t-conorm on [0, s] 2 , it was described some construction methods for nullnorms on a bounded lattice L having a zero element s with some constraints. In general topology, by considering a nonempty set A and the set ℘ (A) of all subsets of A, the closure operator (resp. interior operator) on ℘ (A) is defined as an expansive, isotone and idempotent map cl : ℘ (A) → ℘ (A) (resp. a contractive, isotone and idempotent map int : ℘ (A) → ℘ (A)). Both of these operators can be used for constructing topologies on A in general topology [15] . More precisely, a one-to-one correspondence from the set of all topologies on A to the set of all closure (interior) operators on ℘ (A). That is, any topology on a nonempty set can induce the closure (interior) operator on its underlying powerset. It should be pointed out that closure and interior operators can be defined on a lattice (℘ (A) , ⊆) of all subsets of a set A with set union as the join and set intersection as the meet. Hence, Everett [16] extended the closure operator (resp. interior operator) on ℘ (A) to a general lattice L where the condition cl (∅) = ∅ (resp. int (A) = A) is omitted.

The main aim of this paper is to present some methods for yielding new families of nullnorms with a zero element by means of closure operators and interior operators on a bounded lattice. The remainder of this paper is organized as follows. In Sect. 2, we recall some preliminary details about bounded lattices and nullnorms, interior and closure operators on them. In Sect. 3, considering a bounded lattice L, we propose two new methods for generating nullnorms with a zero element based on the presence of closure operators cl : L → L and interior operators int : L → L. We note that our constructions are a generalization of the constructions in [28] . We also provide some corresponding examples showing that our constructions actually create new types of nullnorms on bounded lattices different from those in [28] . It should be pointed out that our methods need some sufficient and necessary conditions to generate a nullnorm on a bounded lattice. As a by product, two classes of idempotent nullnorms on bounded lattices are obtained when taking the closure operator cl : L → L as cl (x) = x for all x ∈ L and the interior operator int : L → L as int (x) = x for all x ∈ L. Furthermore, we exemplify that we cannot force new nullnorms to coincide with another predefined t-conorm [0, s] 2 

In this part, some basic results about bounded lattices and nullnorms, closure and interior operators on them are recalled. A lattice L is a nonempty set with the partial order ≤ where any two elements x, y ∈ L have a smallest upper bound (called join or supremum), denoted by x ∨ y and a greatest lower bound (called meet or infimum), denoted by x ∧ y. For a, b ∈ L, we use the notation a < b where a ≤ b and a = b. Moreover, we use the notation a b to denote that a and b are incomparable. For s ∈ L\{0, 1},

A bounded lattice (L, ≤, ∧, ∨) is a lattice having the top and bottom elements, which are written as 1 and 0, respectively, that is, there are two elements [1] ).

Let L be a bounded lattice. A binary operation F : L × L → L is called a nullnorm on L if, for any x, y, z ∈ L, it satisfies the following properties: F is called an idempotent nullnorm on L whenever F (x, x) = x for all x ∈ L. We note that a triangular norm T (t-norm for short) on L is a special case of nullnorm with s = 0 whereas a triangular conorm S (t-conorm for short) on L is a special case of nullnorm with s = 1 (see [2, 3] ).

Let L be a bounded lattice and F be a nullnorm on L with the zero element s ∈ L\{0, 1}. Then the following statements hold:

). Let L be a bounded lattice and F be an idempotent nullnorm on L with the zero element s ∈ L\{0, 1}. Then the following statements hold:

Let L be a lattice. A mapping cl : L → L is called a closure operator if, for any x, y ∈ L, it satisfies the following properties:

For a closure operator cl : L → L and for any x, y ∈ L, we obtain that cl(cl(x)) ≤ cl(x) and cl(x) ≤ cl(y) whenever x ≤ y.

Let L be a lattice. A mapping int : L → L is called an interior operator if, for any x, y ∈ L, it satisfies the following properties:

For an interior operator int : L → L and for any x, y ∈ L, we obtain that

In this section, considering a bounded lattice L, we introduce two methods to construct the classes of nullnorms F cl : L × L → L and F int : L × L → L with the zero element on the basis of the closure operator cl : L → L and interior operator int : L → L, respectively. We note that our constructions require some sufficient and necessary conditions on the bounded lattice and the closure (interior) operator. These conditions play an effective role in our constructions, and they yield a nullnorm on a bounded lattice in only particular cases. We also present some illustrative examples to have a better understanding of the structures of new constructions.

is a nullnorm on L with the zero element s.

for all a, b ∈ I s and cl : L → L be a closure operator. If the function F cl defined by the formula (1) is a nullnorm on L with the zero element s, then there holds p ∨ s = q ∨ s and cl (p) ∨ cl (q) ∈ I s for all p, q ∈ I s .

Proof. Let the function F cl defined by the formula (1) be a nullnorm on L with the zero element s.

Consider a bounded lattice L, s ∈ L\{0, 1} such that a ∧ s = b ∧ s for all a, b ∈ I s and a closure operator cl : L → L. We observe that the conditions p ∨ s = q ∨ s and cl (p) ∨ cl (q) ∈ I s for all p, q ∈ I s are both sufficient and necessary to yield a nullnorm on L with the zero element s of the function F cl defined by the formula (1). In this case, one can ask whether the condition a ∧ s = b ∧ s for all a, b ∈ I s is necessary to be a nullnorm on L with the zero element s of F cl . We firstly give an example to show that in Theorem 1, the condition a ∧ s = b ∧ s for all a, b ∈ I s cannot be omitted, in general. Notice that p ∨ s = q ∨ s and cl (p) ∨ cl (q) ∈ I s for all p, q ∈ I s but t ∧ s = v = 0 = r ∧ s for r, t ∈ I s . In this case, by using the approach in Theorem 1, we have F cl (0, F cl (u, r)) = F cl (0, cl (u) ∨ cl (r)) = F cl (0, t) = t ∧ s = v and F cl (F cl (0, u) , r) = F cl (u ∧ s, r) = F cl (0, r) = r ∧ s = 0. Then, F cl is not associative for the indicated closure operator on L 1 . Therefore, F cl is not a nullnorm on L 1 which does not satisfy the condition a ∧ s = b ∧ s for all a, b ∈ I s .

From Example 1, we observe that the condition a ∧ s = b ∧ s for all a, b ∈ I s is sufficient in Theorem 1. Taking into account the above mentioned question, we state that this condition is not necessary in Theorem 1. In order to show this fact, we provide an example of a bounded lattice violating this condition on which the function F cl defined by the formula (1) is a nullnorm with the zero element s.

Consider the lattice L 1 depicted in Example 1 and the closure operator cl : L 1 → L 1 defined by cl (x) = x for all x ∈ L 1 . If we apply the construction in Theorem 1, then we obtain the function F cl : L 1 ×L 1 → L 1 given as in Table 1 . It is easy to check that F cl is a nullnorm with the zero element s for the chosen closure operator on L 1 . In view of Theorems 1 and 2, when taking the closure operator cl : L → L as cl (x) = x for all x ∈ L, we have the following Corollary 1 which shows the presence of idempotent nullnorms on L with the zero element s ∈ L\{0, 1}.

Let L be a bounded lattice and s ∈ L\{0, 1} such that a ∧ s = b ∧ s for all a, b ∈ I s . Then, the following function F 1 : L × L → L defined by 

is a nullnorm on L with the zero element s. Theorem 4. Let L be a bounded lattice, s ∈ L\{0, 1} such that a ∨ s = b ∨ s for all a, b ∈ I s , int : L → L be an interior operator. If the function F int defined by the formula (2) is a nullnorm on L with the zero element s, then there holds p ∧ s = q ∧ s and int (p) ∧ int (q) ∈ I s for all p, q ∈ I s .

Proof. Let the function F int defined by the formula (2) be a nullnorm on L with the zero element s.

Given p, q ∈ I s , from the monotonicity of

In this case, we obtain q∧s ≤ p∧s and p∧s ≤ q∧s. So, it holds p ∧ s = q ∧ s for any p, q ∈ I s .

Assume that int (p)∧int (q) ∈ [s, 1] . Then we have s ≤ int (p)∧int (q) ≤ p∧q. That is, s ≤ p which is a contradiction. Hence, int (p) ∧ int (q) ∈ [s, 1] cannot hold. Suppose that int(p) ∧ int(q) ∈ [0, s[. Then, it is obtained that F int (1, F int (p, q)) = s and F int (F int (1, p) , q) = F int (p ∨ s, q) = (p ∨ s) ∧ (q ∨ s) = p ∨ s. Since F int is associative, we get s = p ∨ s, i.e., p ≤ s which is a contradiction. Hence, int(p) ∧ int(q) ∈ [0, s[ cannot hold. Therefore, it holds int (p) ∧ int (q) ∈ I s for any p, q ∈ I s .

Consider a bounded lattice L, s ∈ L\{0, 1} such that a ∨ s = b ∨ s for all a, b ∈ I s and an interior operator int : L → L. It should be pointed out that the conditions p ∧ s = q ∧ s and int (p) ∧ int (q) ∈ I s for all p, q ∈ I s are both sufficient and necessary to generate a nullnorm on L with the zero element s of the function F int defined by the formula (2) . Then a natural question arises: is it necessary the condition a ∨ s = b ∨ s for all a, b ∈ I s to be a nullnorm on L with the zero element s of F cl . At first, by the following example, we demonstrate that in Theorem 3, the condition a ∨ s = b ∨ s for all a, b ∈ I s cannot be omitted, in general. L 3 = {0, s, k, n, t, m, 1} with the lattice diagram shown in Fig. 3 . Define the interior operator int : L 3 → L 3 by int (x) = x for all x ∈ L 3 . It holds p ∧ s = q ∧ s and int (p) ∧ int (q) ∈ I s for all p, q ∈ I s , however, t ∨ s = m = 1 = n ∨ s for n, t ∈ I s . Then, by applying the method in Theorem 3, we obtain F cl (1, F cl (k, n) 

In that case, F int is not associative for the indicated interior operator on L 3 . Hence, F int is not a nullnorm on L 3 violating the condition a ∨ s = b ∨ s for all a, b ∈ I s .

By Example 4, we observe that the condition a ∨ s = b ∨ s for all a, b ∈ I s is sufficient in Theorem 3. Moreover, we answer the above question so that this is not a necessary condition in Theorem 3. In order to illustrative this observation, we provide an example of a bounded lattice violating this condition on which the function F int defined by the formula (2) is a nullnorm on L with the zero element s. Fig. 4 . Notice that t∨s = m = 1 = k∨s for k, t ∈ I s . Define the interior operator int : L 4 → L 4 by int (x) = x for all x ∈ L 4 . Then, by applying the construction in Theorem 3, we obtain the function F int : L 4 × L 4 → L 4 given as in Table 4 . It can be easily seen that F int is a nullnorm on L 4 with the zero element s. 

is an idempotent nullnorm on L with the zero element s if and only if p∧s = q ∧s and p ∧ q ∈ I s for all p, q ∈ I s .

Remark 2. Let L be a bounded lattice, s ∈ L\{0, 1}, a ∧ s = b ∧ s and a ∨ s = b ∨ s for all a, b ∈ I s . Consider an interior operator int : L → L such that int (p) ∧ int (q) ∈ I s for all p, q ∈ I s . We note that F int defined by the formula (2) in Theorem 3 can create new type of nullnorm different from V ∧ described in [28, Theorem 2] . In particular,

Both of them have same value on all remainder domains. By Corollary 2, when considering the interior operator int : L → L as int (x) = x for all x ∈ L, we observe that the nullnorm F int coincides with the nullnorm V ∧ . To be more precise, the class of the nullnorm F int is a generalization of the class of the nullnorm V ∧ . Moreover, the nullnorms F int and V ∧ do not need to coincide with each other. We provide the following example to demonstrate this observation.

Consider the lattice L 2 with the given order in Fig. 2 and the interior operator int : L 2 → L 2 defined by int(0) = 0, int (v) = v, int (p) = int (q) Tables 5 and  6 , respectively. These nullnorms are different from each other since F int (q, r) = t = q = V ∧ (q, r) for q, r ∈ L 2 . Table 8 . Then, by means of the construction approaches in Theorems 1 and 3, we have F cl (F cl (m, n) , k) = r (F int (F int (m, n) , k) = r) and F cl (m, F cl (n, k)) = t (F int (m, F int (n, k)) = t). Since F cl and F int do not satisfy associativity property, we cannot force F cl and F int to coincide with another prescribed t-norm except for the t-norm T ∧ on [s, 1] 2 .

Following the characterization of nullnorms on the real unit interval [0, 1], the structure of nullnorms concerning algebraic structures on bounded lattices has attracted researchers' attention. The definition of nullnorms was extended to bounded lattices by Karaçal et al. [18] . They also demonstrated the presence of nullnorms based on a t-norm and a t-conorm on bounded lattices. Some further methods for constructing nullnorms (in particular, idempotent nullnorms) on a bounded lattice were introduced in the papers [2, [6] [7] [8] [9] 28] . In this paper, we continued to investigate the methods for obtaining new classes of nullnorms on bounded lattices with the zero element different from the bottom and top elements. More particularly, by using the existence of closure operators and interior operators on a bounded lattice L, we proposed two different construction methods for nullnorms on L with the zero element s ∈ L\{0, 1} under some additional conditions. We also pointed out that our constructions encompass as a special case the ones in [28] . The proposed constructions for nullnorms both in this paper and in [28] 

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