On Generating -Norms (-Conorms) on Some Special Classes of Bounded Lattices

Emel A<a href=

doi:10.5391/IJFIS.2022.22.1.48

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International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(1): 48-58

Published online March 25, 2022

https://doi.org/10.5391/IJFIS.2022.22.1.48

On Generating -Norms (-Conorms) on Some Special Classes of Bounded Lattices

Emel A?ıcı

Department of Software Engineering, Faculty of Technology, Karadeniz Technical University, Trabzon, Turkey

Correspondence to :
Emel A?ıcı (emelkalin@hotmail.com)

Received: June 29, 2021; Revised: August 7, 2021; Accepted: September 2, 2021

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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Abstract
1. Introduction
2. Preliminaries
3. New Methods to Construct t-Norms and t-Conorms on Appropriate Bounded Lattices
4. Modified Ordinal Sum Constructions of t-Norms and t-Conorms on Appropriate Bounded Lattices
5. Conclusion
Conflict of Interest
Figures
Tables
References
Biography

In recent years, construction methods for triangular norms (t-norms) and triangular conorms (t-conorms) on bounded lattices have been studied extensively. This paper presents the continued study of the construction of t-norms and t-conorms on bounded lattices. We introduce new methods for constructing t-norms and t-conorms on an arbitrary bounded lattice. Furthermore, we provide illustrative examples for clarity. Subsequently, we demonstrate how our new construction methods differ from certain existing methods for the construction of t-norms and t-conorms on bounded lattices. Finally, we reveal that new construction methods can be generalized by induction to a modified ordinal sum for t-norms and t-conorms on an arbitrary bounded lattice.

Keywords: t-norm, t-conorm, Bounded lattice

1. Introduction

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Abstract
1. Introduction
2. Preliminaries
3. New Methods to Construct t-Norms and t-Conorms on Appropriate Bounded Lattices
4. Modified Ordinal Sum Constructions of t-Norms and t-Conorms on Appropriate Bounded Lattices
5. Conclusion
Conflict of Interest
Figures
Tables
References
Biography

In 1960, Schweizer and Sklar [1] presented t-norms and t-conorms on the unit interval [0, 1]. Following the introduction, they identified many applications, such as fuzzy sets and fuzzy systems modeling. The t-norm and t-conorm are binary operations that are used in the framework of probabilistic metric spaces and multi-valued logic, multi-criteria decision-making problems, specifically in fuzzy logic [2,3]. The t-norm and t-conorm generalize the intersection in a lattice and conjunction in logic. The names t-norm and t-conorm refer to the fact that in the framework of probabilistic metric spaces, t-norms are used to generalize the triangle inequality of ordinary metric spaces. As a bounded lattice is more general, namely [0, 1], several attempts have been made to characterize t-norms and t-conorms on bounded lattices in various studies. Moreover, in 2000, Klement et al. [4] researched t-norms as ordinal sums of semigroups using the method of Clifford [5]. Saminger [6] initially introduced the ordinal sum of t-norms on bounded lattices in 2006. The author also concentrated on the ordinal sums of t-norms (t-conorms) acting on bounded chains or ordinal sums of posets. Saminger-Platz et al. [7] presented the expansion of t-norms (t-conorms) on bounded lattices in 2008. Subsequently, in 2012, Medina [8] characterized ordinal sums as t-norms (t-conorms) on bounded lattices. In 2015, Ertu?rul et al. [9] proposed a modification of the ordinal sums of t-norms (t-conorms), resulting in a t-norm (t-conorm) on an arbitrary bounded lattice.

The above authors presented a new method for constructing the t-norms and t-conorms on special bounded lattices L using the existence of t-norms on a sublattice [a, 1] and t-conorms on a sublattice [0, a], where a ∈ L \ {0, 1}. Further modifications were proposed in [10–14].

Moreover, when considering the existence of t-norms on a sublattice [0, a] and t-conorms on a sublattice [a, 1], very few investigations have been carried out on the ordinal sums construction of t-norms and t-conorms on bounded lattices.

In this paper, we introduce a new ordinal sum construction of t-norms and t-conorms on an arbitrary bounded lattice satisfying certain constraints for a fixed element a ∈ L \ {0, 1}, using the existence of t-norms on the sublattice [0, a] and t-conorms on the sublattice [a, 1]. We present construction methods for t-norms and t-conorms on an appropriate bounded lattice that has certain restrictions for a fixed element a ∈ L \ {0, 1}. The remainder of this paper is organized as follows. In Section 2, we provide the basic definitions for our main study. In Section 3, we introduce a new ordinal sum construction of t-norms and t-conorms on an arbitrary bounded lattice with a fixed element a ∈ L \ {0, 1} based on the existence of a t-norm V acting on [0, a] and a t-conorm W acting on [a, 1], where several additional conditions on its a ∈ L\{0, 1} are required. Furthermore, the role of these conditions is stressed by providing examples, following which our constructions yield a t-norm and t-conorm on a bounded lattice in particular cases. We also provide illustrative examples for clarity. Thereafter, we present remarks and examples to examine the connection and difference between our methods and other methods outlined by Ertu?rul et al. [9]. In Section 4, we present the modified ordinal sum constructions in full generality. Finally, concluding remarks are provided.

2. Preliminaries

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Abstract
1. Introduction
2. Preliminaries
3. New Methods to Construct t-Norms and t-Conorms on Appropriate Bounded Lattices
4. Modified Ordinal Sum Constructions of t-Norms and t-Conorms on Appropriate Bounded Lattices
5. Conclusion
Conflict of Interest
Figures
Tables
References
Biography

Definition 1 [15]

A lattice is a partially ordered set (L,≤) in which each two-element subset {x, y} has an infimum, denoted as x ∧ y, and a supremum, denoted as x ∨ y. A bounded lattice (L,≤, 0, 1) is a lattice in which the bottom and top elements are written as 0 and 1, respectively.

Definition 2 [15–17]

Let (L,≤, 0, 1) and a, b ∈ L. If a and b are incomparable, we use the notation a || b. We denote the set of elements that are incomparable with a as I_a. Thus, I_a = {x ∈ L | x || a}.

Definition 3 [15,18–20]

Given a bounded lattice (L,≤, 0, 1) and a, b ∈ L, a ≤ b, a subinterval [a, b] of L is defined as

[a, b] = {x \in L ? a \leq x \leq b} .

Similarly, [a, b) = {x ∈ L | a ≤ x < b}, (a, b] = {x ∈ L | a < x ≤ b}, and (a, b) = {x ∈ L | a < x < b}.

Definition 4 [4,6]

Let (L,≤, 0, 1) be a bounded lattice. A t-norm T (a t-conorm S) is a binary operation on L that is commutative, associative, monotonic, and satisfies the neutral element 1 (0); that is, T(x, 1) = x (S(x, 0) = x) for all x ∈ L.

Theorem 1 [9]

Let (L,≤, 0, 1) be a bounded lattice and a ∈ L \ {0, 1}. If V_T is a t-norm on [a, 1] and W_S is a t-conorm on [0, a], the operations T* and S* are a t-norm and t-conorm on L, respectively, where

\begin{array}{l} T^{*} (x, y) = {\begin{array}{l} x \land y, & if x = 1 or y = 1, \\ V_{T} (x, y), & if x, y \in [a, 1), \\ x \land y \land a, & otherwise. \end{array} \\ S^{*} (x, y) = {\begin{array}{l} x \lor y, & if x = 0 or y = 0, \\ W_{S} (x, y), & if x, y \in (0, a], \\ x \lor y \lor a, & otherwise. \end{array} \end{array}

3. New Methods to Construct t-Norms and t-Conorms on Appropriate Bounded Lattices

Go to

Abstract
1. Introduction
2. Preliminaries
3. New Methods to Construct t-Norms and t-Conorms on Appropriate Bounded Lattices
4. Modified Ordinal Sum Constructions of t-Norms and t-Conorms on Appropriate Bounded Lattices
5. Conclusion
Conflict of Interest
Figures
Tables
References
Biography

In this section, we present a new means of constructing t-norms and t-conorms on an arbitrary bounded lattice in Theorems 2 and 3, respectively, where a ∈ L \ {0, 1}, V is a t-norm on [0, a], and W is a t-conorm on [a, 1]. Furthermore, we provide several examples to discuss the method introduced in Theorem 2. Thereafter, we investigate the relation between the introduced method presented in Theorem 2 and other methods proposed in [9]. We also provide illustrative examples to facilitate the understanding of our approach. In Example 2, we obtain a t-norm on L₂ using the method introduced in Theorem 2. In Examples 3 and 4, we cannot ignore the constraints of Theorem 2. In Examples 5 and 6, we demonstrate that the t-norms T and T* on L₂ and L₅ are comparable. In Example 7, we show that the t-norms T and T* on L₇ are not comparable.

Definition 5 [6]

Consider (L,≤, 0, 1) as a bounded lattice and fix a subinterval [a, b] of L. Let V be a t-norm on [a, b]. Then, T_V : L² → L, which is defined by

(1)

T_{V} (x, y) = {\begin{array}{l} V (x, y), & if (x, y) \in {[a, b]}^{2}, \\ x \land y, & otherwise. \end{array}

is an ordinal sum (< a,b,V >) of V on L.

Definition 6 [6]

Consider (L,≤, 0, 1) as a bounded lattice and fix a subinterval [a, b] of L. Let S_W be a t-conorm on [a, b]. Then, S_W : L² → L, which is defined by

(2)

S_{W} (x, y) = {\begin{array}{l} W (x, y), & if (x, y) \in {[a, b]}^{2}, \\ x \lor y, & otherwise. \end{array}

is an ordinal sum (< a,b,W >) of W on L.

In Formula (1), function T_V does not need to be a t-norm (resp. t-conorm). We present an example only for t-norms because the example can be obtained for t-conorms from duality.

Example 1

Consider the bounded lattice (L₁ = {0_L_₁, b, c, d, e, a, 1 _L_₁}, ≤, 0 _L_₁, 1 _L_₁) that is illustrated in Figure 1. Consider the t-norm V on [0_L_₁, a] as follows:

V (x, y) = {\begin{array}{l} x \land y, & if a \in {x, y}, \\ 0_{L_{1}}, & otherwise. \end{array}

We can observe that the function T_V is not a t-norm on L₁ (see Table 1). Clearly, T_V (a, T_V (e, b)) = T_V (a, b) = b ≠ 0_L_₁ = T_V (b, b) = T_V (T_V (a, e), b). Therefore, it does not hold associativity.

Theorem 2

Let (L,≤, 0, 1) be a bounded lattice and a ∈ L \ {0, 1}. If, for all x ∈ I_a and y ∈ (0, a], and for all x ∈ I_a and y ∈ [a, 1), x || y holds, the operation T that is defined as follows is a t-norm on L, where V is a t-norm on [0, a].

T (x, y) = {\begin{array}{l} V (x, y) & if (x, y) \in {[0, a]}^{2}, \\ 0, & if (x, y) \in [0, a] \times I_{a} \cup I_{a} \times [0, a) \cup [a, 1) \times I_{a}, \\ x \land y, & otherwise. \end{array}

Proof

It can easily be observed that T(x, y) = T(y, x) for all x, y ∈ L. Thus, the commutativity of T holds. Moreover, we obtain T(x, 1) = x ∧ 1 = x for all x ∈ L. Thus, the fact that 1 ∈ L is a neutral element of T.

i) Monotonicity

We prove that if x ≤ y, T(x, z) ≤ T(y, z) for all z ∈ L. Subsequently, we demonstrate that T(x, z) ≤ T(y, z) for all z ∈ L. If z = 1, x = 0, y = 0, and z = 0, the monotonicity is clearly held. The proof can be divided into all possible cases.

x ∈ (0, a).
- 1.1 y ∈ (0, a).
  - 1.1.1. z ∈ (0, a).
    
    $T (x, z) = V (x, z) \leq V (y, z) = T (y, z) .$
  - 1.1.2. z ∈ [a, 1).
    
    $T (x, z) = x \land z \leq y \land z = T (y, z) .$
  - 1.1.3. z ∈ I_a.
    
    $T (x, z) = 0 = T (y, z) .$
- 1.2. y ∈ [a, 1).
  - 1.2.1. z ∈ (0, a).
    
    $T (x, z) = V (x, z) \leq z = T (y, z) .$
  - 1.2.2. z ∈ [a, 1).
    
    $T (x, z) = x \land z \leq y \land z = T (y, z) .$
  - 1.2.3. z ∈ I_a.
    
    $T (x, z) = 0 = T (y, z) .$
- 1.3. y ∈ I_a
  
  Because x ∈ (0, a) and y ∈ I_a, according to our constraint, x || y must hold. Therefore, we cannot consider the case of y ∈ I_a.
- 1.4. y = 1.
  - 1.4.1. z ∈ (0, a).
    
    $T (x, z) = V (x, z) \leq z = T (1, z) .$
  - 1.4.2. z ∈ [a, 1).
    
    $T (x, z) = x < a \leq z = T (1, z) .$
  - 1.4.3. z ∈ I_a.
    
    $T (x, z) = 0 \leq z = T (1, z) .$
x ∈ [a, 1).

Thus, it must be the case that y ∈ [a, 1].
- 2.1 y ∈ [a, 1).
  - 2.1.1. z ∈ (0, a).
    
    $T (x, z) = z = T (y, z) .$
  - 2.1.2. z ∈ [a, 1).
    
    $T (x, z) = x \land z \leq y \land z = T (y, z) .$
  - 2.1.3. z ∈ I_a.
    
    $T (x, z) = 0 = T (y, z) .$
- 2.1 y = 1.
  - 2.1.1. z ∈ (0, a).
    
    $T (x, z) = z = T (1, z) .$
  - 2.1.2. z ∈ [a, 1).
    
    $T (x, z) = x \land z \leq z = T (1, z) .$
  - 2.1.3. z ∈ I_a.
    
    $T (x, z) = 0 \leq z = T (1, z) .$
x ∈ I_a.

Therefore, it must be the case that y ∈ I_a or y ∈ (a, 1].
- 3.1 y ∈ I_a.
  - 3.1.1. z ∈ (0, a) or z ∈ [a, 1).
    
    $T (x, z) = 0 = T (y, z) .$
  - 3.1.2. z ∈ I_a.
    
    $T (x, z) = x \land z \leq y \land z = T (y, z) .$
- 3.2. y ∈ (a, 1).
  
  As x ∈ I_a and y ∈ (a, 1), according to our constraint, x || y must hold. The case of y ∈ (a, 1) is not possible.
- 3.3 y = 1.
  - 3.3.1. z ∈ (0, a) or z ∈ [a, 1).
    
    $T (x, z) = 0 \leq z = T (1, z) .$
  - 3.3.2. z ∈ I_a.
    
    $T (x, z) = x \land z \leq z = T (1, z) .$
x = 1. Subsequently, because y = 1, it is clear that T(x, z) = T(y, z) for all z ∈ L.

ii) Associativity

If at least one of x, y, z in L is 1, the associativity is satisfied.

x ∈ [0, a).
- 1.1 y ∈ [0, a).
  - 1.1.1. z ∈ [0, a).
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, V (y, z)) \\ = V (x, V (y, z)) \\ = V (V (x, y), z) \\ = T (T (x, y), z) . \end{array}$
  - 1.1.2. z ∈ [a, 1).
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, y) = V (x, y) \\ = T (V (x, y), z) \\ = T (T (x, y), z) . \end{array}$
  - 1.1.3. z ∈ I_a.
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, 0) = 0 \\ = T (V (x, y), z) \\ = T (T (x, y), z) . \end{array}$
- 1.2. y ∈ [a, 1).
  - 1.2.1. z ∈ [0, a).
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, z) \\ = T (T (x, y), z) . \end{array}$
  - 1.2.2. z ∈ [a, 1).
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, y \land z) = x \\ = T (x, z) \\ = T (T (x, y), z) . \end{array}$
  - 1.2.3. z ∈ I_a.
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, 0) = 0 = T (x, z) \\ = T (T (x, y), z) . \end{array}$
- 1.3. y ∈ I_a.
  - 1.3.1. z ∈ [0, a) or z ∈ [a, 1).
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, 0) = 0 = T (0, z) \\ = T (T (x, y), z) . \end{array}$
  - 1.3.2. z ∈ I_a.
    
    It holds that y ∧z ∈ I_a or y ∧ z = 0. Otherwise, if y ∧ z ∈ (0, a], because y || a, we obtain y || y ∧ z. Thus, there is a contradiction.
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, y \land z) = 0 \\ = T (0, z) = T (T (x, y), z) . \end{array}$
x ∈ [a, 1).
- 2.1 y ∈ [0, a).
  - 2.1.1. z ∈ [0, a).
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, V (y, z)) = V (y, z) \\ = T (y, z) = T (T (x, y), z) . \end{array}$
  - 2.1.2. z ∈ [a, 1).
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, y) = y \\ = T (y, z) = T (T (x, y), z) . \end{array}$
  - 2.1.3. z ∈ I_a.
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, 0) = 0 = T (y, z) \\ = T (T (x, y), z) . \end{array}$
- 2.2. y ∈ [a, 1).
  - 2.2.1. z ∈ [0, a).
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, z) = z = T (x \land y, z) \\ = T (T (x, y), z) . \end{array}$
  - 2.2.2. z ∈ [a, 1).
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, y \land z) = x \land y \land z \\ = T (x \land y, z) = T (T (x, y), z) . \end{array}$
  - 2.2.3. z ∈ I_a.
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, 0) = 0 = T (x \land y, z) \\ = T (T (x, y), z) . \end{array}$
- 2.3. y ∈ I_a.
  - 2.3.1. z ∈ [0, a) or z ∈ [a, 1).
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, 0) = 0 = T (0, z) \\ = T (T (x, y), z) . \end{array}$
  - 2.3.2. z ∈ I_a.
    
    Then, y ∧ z ∈ I_a or y ∧ z = 0 must be true.
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, y \land z) = 0 \\ = T (0, z) = T (T (x, y), z) . \end{array}$
x ∈ I_a.
- 3.1 y ∈ [0, a).
  - 3.1.1. z ∈ [0, a).
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, V (y, z)) = 0 = T (0, z) \\ = T (T (x, y), z) . \end{array}$
  - 3.1.2. z ∈ [a, 1).
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, y) = 0 = 0 T (0, z) \\ = T (T (x, y), z) . \end{array}$
  - 3.1.3. z ∈ I_a.
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, 0) = 0 = T (0, z) \\ = T (T (x, y), z) . \end{array}$
- 3.2. y ∈ [a, 1).
  - 3.2.1. z ∈ [0, a).
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, z) = 0 = T (0, z) \\ = T (T (x, y), z) . \end{array}$
  - 3.2.2. z ∈ [a, 1).
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, y \land z) = 0 = T (0, z) \\ = T (T (x, y), z) . \end{array}$
  - 3.2.3. z ∈ I_a.
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, 0) = 0 = T (0, z) \\ = T (T (x, y), z) . \end{array}$
- 3.3. y ∈ I_a.
  - 3.3.1. z ∈ [0, a) or z ∈ [a, 1).
    
    Thus, by assumption, x ∧ y ∈ I_a or x ∧ y = 0 must hold.
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, 0) = 0 \\ = T (x \land y, z) = T (T (x, y), z) . \end{array}$
  - 3.3.2. z ∈ I_a.
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, y \land z) \\ = x \land y \land z = T (x \land y, z) \\ = T (T (x, y), z) . \end{array}$

Corollary 1

If we use V = T_∧ on [0, a] provided in Theorem 2, we obtain the following t-norm on L:

T (x, y) = {\begin{array}{l} 0, & if (x, y) \in [0, a) \times I_{a} \cup I_{a} \times [0, a) \cup [a, 1) \times I_{a}, \cup I_{a} \times [a, 1), \\ x \land y, & otherwise. \end{array}

Example 2

Consider the bounded lattice (L₂ = {0_L_₂, b, c, e, d, a, p, 1_L_₂}, ≤, 0_L_₂, 1_L_₂) illustrated in Figure 2, which satisfies the constraints of Theorem 2.

V (x, y) = {\begin{array}{l} x \land y, & if a \in {x, y}, \\ 0_{L_{2}}, & otherwise. \end{array}

Thus, the function T on L₂ defined in Table 2 is a t-norm.

Remark 1

Let (L,≤, 0, 1) be a bounded lattice and a ∈ L \ {0, 1}. In Theorem 2, it can be observed that the condition for all x ∈ I_a and y ∈ (0, a] holds; thus, x || y cannot be ignored. The following example illustrates that the function T : L² → L defined by Theorem 2 is not a t-norm.

Example 3

Consider the bounded lattice (L₃ = {0_L_₃, b, c, e, d, a, p, 1_L_₃}, ≤, 0_L_₃, 1_L_₃) described in Figure 3, which does not satisfy one of the constraints of Theorem 2; that is, there exists an element b ∈ L₃, where b < e for e ∈ I_a and b ∈ (0_L_₃, a). Consider the t-norm V on [0_L_₃, a] such that V (x, y) = x ∧ y.

Thus, the function T on L₃ defined in Table 3 is not a t-norm. Indeed, it does not satisfy monotonicity. Clearly, b < e and T(b, b) = b ? 0_L_₃ = T(b, e).

Remark 2

Let (L,≤, 0, 1) be a bounded lattice and a ∈ L \ {0, 1}. In Theorem 2, it can be observed that the condition for all x ∈ I_a and y ∈ [a, 1) holds, and that x || y cannot be omitted. The following example illustrates that the function T : L² → L defined by Theorem 2 is not a t-norm.

Example 4

Consider the bounded lattice (L₄ = {0_L_₄, b, c, e, d, a, p, 1_L_₄}, ≤, 0_L_₄, 1_L_₅) described in Figure 4, which does not satisfy one of the constraints of Theorem 2; that is, there is an element e ∈ L₄, where e < p for e ∈ I_a and p ∈ (a, 1_L_₄). Consider the t-norm V on [0_L_₄, a] such that V (x, y) = x ∧ y.

Therefore, the function T on L₄ that is defined in Table 4 is not a t-norm. Again, it does not satisfy monotonicity. Clearly, e < p and T(e, e) = e ? 0_L_₄= T(e, p).

Remark 3

We may question whether the t-norms T and T* defined in Theorems 2 and 1, respectively, can be comparable on any bounded lattice; that is, T ≤ T* or T* ≤ T on any bounded lattice. We demonstrate these arguments using the following examples.

Example 5

Consider the bounded lattice (L₂ = {0_L_₂, b, c, e, d, a, p, 1_L_₂},≤, 0_L_₂, 1_L_₂) described in Figure 2 and the t-norm V on [0_L_₂, a] such that V (x, y) = x ∧ y, and consider the t-norm V_T on [a, 1_L_₂], V_T (x, y) = x ∧ y. By using the construction approaches introduced in Theorems 1 and 2, we define the t-norm T* in Tables 5 and 6, respectively. Subsequently, T* ≤ T.

Example 6

Consider the bounded lattice (L₅ = {0_L_₅, p, t, s, m, a, 1_L_₅},≤, 0_L_₅, 1_L_₅) described in Figure 5, and consider the t-norm V on [0_L_₅, a] as follows:

V (x, y) = {\begin{array}{l} x \land y, & if a \in {x, y}, \\ 0_{L_{5}}, & otherwise. \end{array}

Moreover, consider any t-norm V_T on [a, 1_L_₅]. By using the construction approaches introduced in Theorems 2 and 1, we define the t-norms T and T*in Tables 7 and 8, respectively. Subsequently, T ≤ T*.

Remark 4

In Example 6, if we take the t-norm V on [0_L₅, a] such that V (x, y) = x ∧ y, we obtain T = T*.

Example 7

Consider the bounded lattice (L₆ = {0_L₆, m, n, k, a, 1_L_₆}, ≤, 0_L_₆, 1_L_₆) described in Figure 6, and consider the t-norm V on [0_L_₆, a] as follows:

V (x, y) = {\begin{array}{l} x \land y, & if a \in {x, y}, \\ 0_{L_{6}}, & otherwise. \end{array}

Furthermore, consider any t-norm V_T on [a, 1_L_₆]. By using the construction approaches in Theorems 2 and 1, we define the t-norms T and T*in Tables 9 and 10, respectively. Thus, neither T ≤ T*nor T* ≤ T.

We present the same arguments for t-conorms. We omit the proof of Theorem and its examples, owing to its similarity to the proof of Theorem 2.

Theorem 3

Let (L,≤, 0, 1) be a bounded lattice and a ∈ L \ {0, 1}. If, for all x ∈ I_a and y ∈ [a, 1), and for all x ∈ I_a and y ∈ (0, a], x || y holds, the operation S, which is defined as follows, is a t-conorm on L, such that W is a t-conorm on [a, 1].

S (x, y) = {\begin{array}{l} W (x, y), & if (x, y) \in {[a, 1]}^{2}, \\ 1, & if (x, y) \in (a, 1] \times I_{a} \cup I_{a} \times (a, 1] \cup (0, a] \times I_{a} \cup I_{a} \times (0, a], \\ x \lor y, & otherwise. \end{array}

Corollary 2

In Theorem 3, if we use W = S_∨ on [a, 1], we obtain the following t-conorm on L:

S (x, y) = {\begin{array}{l} 1, & if (x, y) \in (a, 1] \times I_{a} \cup I_{a} \times (a, 1] \cup (0, a] \times I_{a} \cup I_{a} \times (0, a], \\ x \lor y, & otherwise. \end{array}

4. Modified Ordinal Sum Constructions of t-Norms and t-Conorms on Appropriate Bounded Lattices

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Abstract
1. Introduction
2. Preliminaries
3. New Methods to Construct t-Norms and t-Conorms on Appropriate Bounded Lattices
4. Modified Ordinal Sum Constructions of t-Norms and t-Conorms on Appropriate Bounded Lattices
5. Conclusion
Conflict of Interest
Figures
Tables
References
Biography

In this section, based on the proposed approaches for constructing t-norms and t-conorms in Section 3, we introduce new construction methods for t-norms and t-conorms on an appropriate bounded lattice L using recursion.

Theorem 4

Let (L,≤, 0, 1) be a bounded lattice and{a₀, a₁, a₂, ···, a_n} be a finite chain in L such that 0 = a₀ < a₁ < a₂ < ··· < a_n = 1. Furthermore, let x || y for all x ∈ I_a_{_i} and y ∈ (0, a_i], and for all x ∈ I_a_{_i}and y ∈ [a_i, 1). We use the t-norm V on [0, a₁]. Thus, the operation T_n on L, which is defined as follows, is a t-norm, such that T₁ = V, and for i ∈ {2, ···, n}, the function T_i on [0, a_i] is expressed by

(3)

T_{i} (x, y) = {\begin{array}{l} T_{i - 1} (x, y) & i f (x, y) \in {[0, a_{i - 1}]}^{2}, \\ 0, & if (x, y) \in [0, a_{i - 1}) \times I_{a_{i - 1}} \cup I_{a_{i - 1}} \times [0, a_{i - 1}) \cup [a_{i - 1}, a_{i}) \times I_{a_{i - 1}} \cup I_{a_{i - 1}} \times [a_{i - 1}, a_{i}), \\ x \land y, & otherwise. \end{array}

Theorem 5

Let (L,≤, 0, 1) be a bounded lattice and {a₀, a₁, a₂, …, a_n} be a finite chain in L such that 1 = a₀ > a₁ > a₂ > … > a_n = 0. Let x || y for all x ∈ I_a_{_i} and y ∈ [a_{_i}, 1), and for all x ∈ I_a_{_i} and y ∈ (0, a_i]. We place the t-conorm S₀ on [a₁, 1]. Therefore, the operation S_n : L² → L, which is defined as follows, is a t-conorm, such that S₁ = S₀, and for i ∈ {2, …, n}, the function S_i : [a_i, 1]² → [a_i, 1] is expressed by

(4)

S_{i} (x, y), = {\begin{array}{l} S_{i - 1} (x, y) & if (x, y) \in {[a_{i - 1}, 1]}^{2}, \\ 1, & if (x, y) \in (a_{i - 1}, 1] \times I_{a_{i - 1}} \cup I_{a_{i - 1}} \times (a_{i - 1}, 1] \cup (a_{i}, a_{i - 1}] \times I_{a_{i - 1}} \cup I_{a_{i - 1}} \times (a_{i}, a_{i - 1}], \\ x \land y, & otherwise. \end{array}

5. Conclusion

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Abstract
1. Introduction
2. Preliminaries
3. New Methods to Construct t-Norms and t-Conorms on Appropriate Bounded Lattices
4. Modified Ordinal Sum Constructions of t-Norms and t-Conorms on Appropriate Bounded Lattices
5. Conclusion
Conflict of Interest
Figures
Tables
References
Biography

In this study, we have investigated and introduced new construction methods for building t-norms and t-conorms on appropriate bounded lattices with a constraint concerning the specific splitting point a. Based on this ordinal sum method, we introduced a new class of t-norms T and t-conorms S on an arbitrary bounded lattice by using the existence of a t-norm V on a sublattice [0, a] and a t-conorm W on a sublattice [a, 1] in Theorems 2 and 3, respectively. We have provided several illustrative examples to facilitate better understanding of the constructed t-norm T and t-conorm S. Furthermore, we investigated the relationship between the introduced methods and certain other approaches. Finally, we demonstrated that the new construction methods can be generalized by induction to a modified ordinal sum for t-norms and t-conorms on an arbitrary bounded lattice.

Conflict of Interest

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Abstract
1. Introduction
2. Preliminaries
3. New Methods to Construct t-Norms and t-Conorms on Appropriate Bounded Lattices
4. Modified Ordinal Sum Constructions of t-Norms and t-Conorms on Appropriate Bounded Lattices
5. Conclusion
Conflict of Interest
Figures
Tables
References
Biography

No potential conflict of interest relevant to this article is reported.

Figures

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Abstract
1. Introduction
2. Preliminaries
3. New Methods to Construct t-Norms and t-Conorms on Appropriate Bounded Lattices
4. Modified Ordinal Sum Constructions of t-Norms and t-Conorms on Appropriate Bounded Lattices
5. Conclusion
Conflict of Interest
Figures
Tables
References
Biography

Fig. 1.

The lattice L₁.

Fig. 2.

The lattice L₂.

Fig. 3.

The lattice L₃.

Fig. 4.

The lattice L₄.

Fig. 5.

The lattice L₅.

Fig. 6.

The lattice L₆.

Tables

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Abstract
1. Introduction
2. Preliminaries
3. New Methods to Construct t-Norms and t-Conorms on Appropriate Bounded Lattices
4. Modified Ordinal Sum Constructions of t-Norms and t-Conorms on Appropriate Bounded Lattices
5. Conclusion
Conflict of Interest
Figures
Tables
References
Biography

Table. 1.

Table 1. The function T_V on L₁.

T_V	0_L_₁	b	c	d	e	a	1_L_₁
0_L_₁	0_L_₁	0_L_₁	0_L_₁	0_L_₁	0_L_₁	0_L_₁	0_L_₁
b	0_L_₁	0_L_₁	0_L_₁	0_L_₁	b	b	b
c	0_L_₁	0_L_₁	0_L_₁	0_L_₁	b	c	c
d	0_L_₁	0_L_₁	0_L_₁	d	b	0_L_₁	d
e	0_L_₁	b	b	b	e	b	e
a	0_L_₁	b	c	b	b	a	a
1_L_₁	0_L_₁	b	c	d	e	a	1_L_₁

Table. 2.

Table 2. The t-norm T on L₂.

T	0_L_₂	b	c	e	d	a	p	1_L_₂
0_L_₂	0_L_₂	0_L_₂	0_L_₂	0_L_₂	0_L_₂	0_L_₂	0_L_₂	0_L_₂
b	0_L_₂	0_L_₂	0_L_₂	0_L_₂	0_L_₂	b	b	b
c	0_L_₂	0_L_₂	c	0_L_₂	c	0_L_₂	0_L_₂	c
e	0_L_₂	0_L_₂	0_L_₂	e	e	0_L_₂	0_L_₂	e
d	0_L_₂	0_L_₂	c	e	d	0_L_₂	0_L_₂	d
a	0_L_₂	b	0_L_₂	0_L_₂	0_L_₂	a	a	a
p	0_L_₂	b	0_L_₂	0_L_₂	0_L_₂	a	p	p
1_L_₂	0_L_₂	b	c	e	d	a	p	1_L_₂

Table. 3.

Table 3. The function T on L₃.

T	0_L_₃	b	c	e	d	a	p	1_L_₃
0_L_₃	0_L_₃	0_L_₃	0_L_₃	0_L_₃	0_L_₃	0_L_₃	0_L_₃	0_L_₃
b	0_L_₃	b	0_L_₃	0_L_₃	0_L_₃	b	b	b
c	0_L_₃	0_L_₃	c	b	c	0_L_₃	0_L_₃	c
e	0_L_₃	0_L_₃	b	e	b	0_L_₃	0_L_₃	e
d	0_L_₃	0_L_₃	c	b	d	0_L_₃	0_L_₃	d
a	0_L_₃	b	0_L_₃	0_L_₃	0_L_₃	a	a	a
p	0_L_₃	b	0_L_₃	0_L_₃	0_L_₃	a	p	p
1_L_₃	0_L_₃	b	c	e	d	a	p	1_L_₃

Table. 4.

Table 4. The function T on L₄.

T	0_L_₄	b	c	e	d	a	p	1_L_₄
0_L_₄	0_L_₄	0_L_₄	0_L_₄	0_L_₄	0_L_₄	0_L_₄	0_L_₄	0_L_₄
b	0_L_₄	b	0_L_₄	0_L_₄	0_L_₄	b	b	b
c	0_L_₄	0_L_₄	c	0_L_₄	c	0_L_₄	0_L_₄	c
e	0_L_₄	0_L_₄	0_L_₄	e	e	0_L_₄	0_L_₄	e
d	0_L_₄	0_L_₄	c	e	d	0_L_₄	0_L_₄	d
a	0_L_₄	b	0_L_₄	0_L_₄	0_L_₄	a	a	a
p	0_L_₄	b	0_L_₄	0_L_₄	0_L_₄	a	p	p
1_L_₄	0_L_₄	b	c	e	d	a	p	1_L_₄

Table. 5.

Table 5. The t-norm T* on L₂.

T*	0_L_₂	b	c	e	d	a	p	1_L_₂
0_L_₂	0_L_₂	0_L_₂	0_L_₂	0_L_₂	0_L_₂	0_L_₂	0_L_₂	0_L_₂
b	0_L_₂	b	0_L_₂	0_L_₂	0_L_₂	b	b	b
c	0_L_₂	0_L_₂	0_L_₂	0_L_₂	0_L_₂	0_L_₂	0_L_₂	c
e	0_L_₂	0_L_₂	0_L_₂	0_L_₂	0_L_₂	0_L_₂	0_L_₂	e
d	0_L_₂	0_L_₂	0_L_₂	0_L_₂	0_L_₂	0_L_₂	0_L_₂	d
a	0_L_₂	b	0_L_₂	0_L_₂	0_L_₂	a	a	a
p	0_L_₂	b	0_L_₂	0_L_₂	0_L_₂	a	p	p
1_L_₂	0_L_₂	b	c	e	d	a	p	1_L_₂

Table. 6.

Table 6. The t-norm T on L₂.

T	0_L_₂	b	c	e	d	a	p	1_L_₂
0_L_₂	0_L_₂	0_L_₂	0_L_₂	0_L_₂	0_L_₂	0_L_₂	0_L_₂	0_L_₂
b	0_L_₂	b	0_L_₂	0_L_₂	0_L_₂	b	b	b
c	0_L_₂	0_L_₂	c	0_L_₂	c	0_L_₂	0_L_₂	c
e	0_L_₂	0_L_₂	0_L_₂	e	e	0_L_₂	0_L_₂	e
d	0_L_₂	0_L_₂	c	e	d	0_L_₂	0_L_₂	d
a	0_L_₂	b	0_L_₂	0_L_₂	0_L_₂	a	a	a
p	0_L_₂	b	0_L_₂	0_L_₂	0_L_₂	a	p	p
1_L_₂	0_L_₂	b	c	e	d	a	p	1_L_₂

Table. 7.

Table 7. The t-norm T on L₅.

T	0_L_₅	p	t	s	m	a	1_L_₅
0_L_₅	0_L_₅	0_L_₅	0_L_₅	0_L_₅	0_L_₅	0_L_₅	0_L_₅
p	0_L_₅	0_L_₅	0_L_₅	0_L_₅	0_L_₅	p	p
t	0_L_₅	0_L_₅	0_L_₅	0_L_₅	0_L_₅	t	t
s	0_L_₅	0_L_₅	0_L_₅	0_L_₅	0_L_₅	s	s
m	0_L_₅	0_L_₅	0_L_₅	0_L_₅	0_L_₅	m	m
a	0_L_₅	p	t	s	m	a	a
1_L_₅	0_L_₅	p	t	s	m	a	1_L_₅

Table. 8.

Table 8. The t-norm T* on L₅.

T*	0_L_₅	p	t	s	m	a	1_L_₅
0_L_₅	0_L_₅	0_L_₅	0_L_₅	0_L_₅	0_L_₅	0_L_₅	0_L_₅
p	0_L_₅	p	p	0_L_₅	0_L_₅	p	p
t	0_L_₅	p	t	0_L_₅	0_L_₅	t	t
s	0_L_₅	0_L_₅	0_L_₅	s	0_L_₅	s	s
m	0_L_₅	0_L_₅	0_L_₅	0_L_₅	m	m	m
a	0_L_₅	p	t	s	m	a	a
1_L_₅	0_L_₅	p	t	s	m	a	1_L_₅

Table. 9.

Table 9. The t-norm T on L₆.

T	0_L_₆	m	n	k	a	1_L_₆
0_L_₆	0_L_₆	0_L_₆	0_L_₆	0_L_₆	0_L_₆	0_L_₆
m	0_L_₆	m	0_L_₆	0_L_₆	0_L_₆	m
n	0_L_₆	0_L_₆	0_L_₆	0_L_₆	n	n
k	0_L_₆	0_L_₆	0_L_₆	k	0_L_₆	k
a	0_L_₆	0_L_₆	n	0_L_₆	a	a
1_L_₆	0_L_₆	m	n	k	a	1_L_₆

Table. 10.

Table 10. The t-norm T* on L₆.

T*	0_L_₆	m	n	k	a	1_L_₆
0_L_₆	0_L_₆	0_L_₆	0_L_₆	0_L_₆	0_L_₆	0_L_₆
m	0_L_₆	0_L_₆	0_L_₆	0_L_₆	0_L_₆	m
n	0_L_₆	0_L_₆	n	0_L_₆	n	n
k	0_L_₆	0_L_₆	0_L_₆	0_L_₆	0_L_₆	k
a	0_L_₆	0_L_₆	n	0_L_₆	a	a
1_L_₆	0_L_₆	m	n	k	a	1_L_₆

References

Go to

Abstract
1. Introduction
2. Preliminaries
3. New Methods to Construct t-Norms and t-Conorms on Appropriate Bounded Lattices
4. Modified Ordinal Sum Constructions of t-Norms and t-Conorms on Appropriate Bounded Lattices
5. Conclusion
Conflict of Interest
Figures
Tables
References
Biography

Schweizer, B, and Sklar, A (1960). Statistical metric spaces. Pacific Journal of Mathematics. 10, 313-334.
Agarwal, N (2021). Parametric directed divergence measure for Pythagorean fuzzy set and their applications to multi-criteria decision-making. Pythagorean Fuzzy Sets. Singapore: Springer, pp. 65-81 https://doi.org/10.1007/978-981-16-1989-2_3
Garg, H, Agarwal, N, and Tripathi, A (2015). Entropy based multi-criteria decision making method under fuzzy environment and unknown attribute weights. Global Journal of Technology and Optimization. 6, 13-20. https://doi.org/10.4172/2229-8711.1000182
Klement, EP, Mesiar, R, and Pap, E (2000). Triangular Norms. Dordrecht, Netherlands: Kluwer Academic Publishers
Clifford, AH (1954). Naturally totally ordered commutative semigroups. American Journal of Mathematics. 76, 631-646. https://doi.org/10.2307/2372706
Saminger, S (2006). On ordinal sums of triangular norms on bounded lattices. Fuzzy Sets and Systems. 157, 1403-1416. https://doi.org/10.1016/j.fss.2005.12.021
Saminger-Platz, S, Klement, EP, and Mesiar, R (2008). On extensions of triangular norms on bounded lattices. Indagationes Mathematicae. 19, 135-150. https://doi.org/10.1016/S0019-3577(08)80019-5
Medina, J (2012). Characterizing when an ordinal sum of tnorms is a t-norm on bounded lattices. Fuzzy Sets and Systems. 202, 75-88. https://doi.org/10.1016/j.fss.2012.03.002
Ertugrul, U, Karacal, F, and Mesiar, R (2015). Modified ordinal sums of triangular norms and triangular conorms on bounded lattices. International Journal of Intelligent Systems. 30, 807-817. https://doi.org/10.1002/int.21713
Asici, E (2021). Construction methods for triangular norms and triangular conorms on appropriate bounded lattices. Iranian Journal of Fuzzy Systems. 18, 81-98. https://doi.org/10.22111/ijfs.2021.6083
Asici, E, and Mesiar, R (2020). Alternative approaches to obtain t-norms and t-conorms on bounded lattices. Iranian Journal of Fuzzy Systems. 17, 121-138. https://doi.org/10.22111/ijfs.2020.5410
Asici, E, and Mesiar, R (2020). New constructions of triangular norms and triangular conorms on an arbitrary bounded lattice. International Journal of General Systems. 49, 143-160. https://doi.org/10.1080/03081079.2019.1668385
Cayli, GD (2019). Some methods to obtain t-norms and t-conorms on bounded lattices. Kybernetika. 55, 273-294. https://doi.org/10.14736/kyb-2019-2-0273
Cayli, GD (2018). On a new class of t-norms and t-conorms on bounded lattices. Fuzzy Sets and Systems. 332, 129-143. https://doi.org/10.1016/j.fss.2017.07.015
Birkhoff, G (1967). Lattice Theory. Providence, RI: American Mathematical Society
Asici, E, and Mesiar, R (2021). On generating uninorms on some special classes of bounded lattices. Fuzzy Sets and Systems. https://doi.org/10.1016/j.fss.2021.06.010
Asici, E (2019). The equivalence of uninorms induced by the U-partial order. Hacettepe Journal of Mathematics and Statistics. 48, 439-450. https://doi.org/10.15672/hjms.2019.662
Asici, E (2019). Equivalence classes of uninorms. Filomat. 33, 571-582. https://doi.org/10.2298/fil1902571a
Ince, MA, Karacal, F, and Mesiar, R (2016). Medians and nullnorms on bounded lattices. Fuzzy Sets and Systems. 289, 74-81. https://doi.org/10.1016/j.fss.2015.05.015
Karacal, F, Ince, MA, and Mesiar, R (2015). Nullnorms on bounded lattices. Information Sciences. 325, 227-236. https://doi.org/10.1016/j.ins.2015.06.05257

Biography

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Abstract
1. Introduction
2. Preliminaries
3. New Methods to Construct t-Norms and t-Conorms on Appropriate Bounded Lattices
4. Modified Ordinal Sum Constructions of t-Norms and t-Conorms on Appropriate Bounded Lattices
5. Conclusion
Conflict of Interest
Figures
Tables
References
Biography

Emel Asšıcı has been working as an associate professor at Karadeniz Technical University since 2018. She obtained her bachelor’s degree in the Department of Mathematics (2005) and Ph.D. degree in the Department of Mathematics (2013) at Karadeniz Technical University. She has over 10 publications in international peer-reviewed journals. Her research interests are fuzzy logic, triangular norms and conorms, uninorms, nullnorms, and aggregation functions.

E-mail: emelkalin@hotmail.com

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(1): 48-58

Published online March 25, 2022 https://doi.org/10.5391/IJFIS.2022.22.1.48

On Generating -Norms (-Conorms) on Some Special Classes of Bounded Lattices

Emel A?ıcı

Department of Software Engineering, Faculty of Technology, Karadeniz Technical University, Trabzon, Turkey

Correspondence to:Emel A?ıcı (emelkalin@hotmail.com)

Received: June 29, 2021; Revised: August 7, 2021; Accepted: September 2, 2021

Abstract

Keywords: t-norm, t-conorm, Bounded lattice

1. Introduction

2. Preliminaries

Definition 1 [15]

Definition 2 [15–17]

Let (L,≤, 0, 1) and a, b ∈ L. If a and b are incomparable, we use the notation a || b. We denote the set of elements that are incomparable with a as I_a. Thus, I_a = {x ∈ L | x || a}.

Definition 3 [15,18–20]

Given a bounded lattice (L,≤, 0, 1) and a, b ∈ L, a ≤ b, a subinterval [a, b] of L is defined as

[a, b] = {x \in L ? a \leq x \leq b} .

Similarly, [a, b) = {x ∈ L | a ≤ x < b}, (a, b] = {x ∈ L | a < x ≤ b}, and (a, b) = {x ∈ L | a < x < b}.

Definition 4 [4,6]

Theorem 1 [9]

Let (L,≤, 0, 1) be a bounded lattice and a ∈ L \ {0, 1}. If V_T is a t-norm on [a, 1] and W_S is a t-conorm on [0, a], the operations T* and S* are a t-norm and t-conorm on L, respectively, where

\begin{array}{l} T^{*} (x, y) = {\begin{array}{l} x \land y, & if x = 1 or y = 1, \\ V_{T} (x, y), & if x, y \in [a, 1), \\ x \land y \land a, & otherwise. \end{array} \\ S^{*} (x, y) = {\begin{array}{l} x \lor y, & if x = 0 or y = 0, \\ W_{S} (x, y), & if x, y \in (0, a], \\ x \lor y \lor a, & otherwise. \end{array} \end{array}

3. New Methods to Construct t-Norms and t-Conorms on Appropriate Bounded Lattices

Definition 5 [6]

Consider (L,≤, 0, 1) as a bounded lattice and fix a subinterval [a, b] of L. Let V be a t-norm on [a, b]. Then, T_V : L² → L, which is defined by

(1)

T_{V} (x, y) = {\begin{array}{l} V (x, y), & if (x, y) \in {[a, b]}^{2}, \\ x \land y, & otherwise. \end{array}

is an ordinal sum (< a,b,V >) of V on L.

Definition 6 [6]

Consider (L,≤, 0, 1) as a bounded lattice and fix a subinterval [a, b] of L. Let S_W be a t-conorm on [a, b]. Then, S_W : L² → L, which is defined by

(2)

S_{W} (x, y) = {\begin{array}{l} W (x, y), & if (x, y) \in {[a, b]}^{2}, \\ x \lor y, & otherwise. \end{array}

is an ordinal sum (< a,b,W >) of W on L.

In Formula (1), function T_V does not need to be a t-norm (resp. t-conorm). We present an example only for t-norms because the example can be obtained for t-conorms from duality.

Example 1

Consider the bounded lattice (L₁ = {0_L_₁, b, c, d, e, a, 1 _L_₁}, ≤, 0 _L_₁, 1 _L_₁) that is illustrated in Figure 1. Consider the t-norm V on [0_L_₁, a] as follows:

V (x, y) = {\begin{array}{l} x \land y, & if a \in {x, y}, \\ 0_{L_{1}}, & otherwise. \end{array}

Theorem 2

T (x, y) = {\begin{array}{l} V (x, y) & if (x, y) \in {[0, a]}^{2}, \\ 0, & if (x, y) \in [0, a] \times I_{a} \cup I_{a} \times [0, a) \cup [a, 1) \times I_{a}, \\ x \land y, & otherwise. \end{array}

Proof

i) Monotonicity

x ∈ (0, a).
- 1.1 y ∈ (0, a).
  - 1.1.1. z ∈ (0, a).
    
    $T (x, z) = V (x, z) \leq V (y, z) = T (y, z) .$
  - 1.1.2. z ∈ [a, 1).
    
    $T (x, z) = x \land z \leq y \land z = T (y, z) .$
  - 1.1.3. z ∈ I_a.
    
    $T (x, z) = 0 = T (y, z) .$
- 1.2. y ∈ [a, 1).
  - 1.2.1. z ∈ (0, a).
    
    $T (x, z) = V (x, z) \leq z = T (y, z) .$
  - 1.2.2. z ∈ [a, 1).
    
    $T (x, z) = x \land z \leq y \land z = T (y, z) .$
  - 1.2.3. z ∈ I_a.
    
    $T (x, z) = 0 = T (y, z) .$
- 1.3. y ∈ I_a
  
  Because x ∈ (0, a) and y ∈ I_a, according to our constraint, x || y must hold. Therefore, we cannot consider the case of y ∈ I_a.
- 1.4. y = 1.
  - 1.4.1. z ∈ (0, a).
    
    $T (x, z) = V (x, z) \leq z = T (1, z) .$
  - 1.4.2. z ∈ [a, 1).
    
    $T (x, z) = x < a \leq z = T (1, z) .$
  - 1.4.3. z ∈ I_a.
    
    $T (x, z) = 0 \leq z = T (1, z) .$
x ∈ [a, 1).

Thus, it must be the case that y ∈ [a, 1].
- 2.1 y ∈ [a, 1).
  - 2.1.1. z ∈ (0, a).
    
    $T (x, z) = z = T (y, z) .$
  - 2.1.2. z ∈ [a, 1).
    
    $T (x, z) = x \land z \leq y \land z = T (y, z) .$
  - 2.1.3. z ∈ I_a.
    
    $T (x, z) = 0 = T (y, z) .$
- 2.1 y = 1.
  - 2.1.1. z ∈ (0, a).
    
    $T (x, z) = z = T (1, z) .$
  - 2.1.2. z ∈ [a, 1).
    
    $T (x, z) = x \land z \leq z = T (1, z) .$
  - 2.1.3. z ∈ I_a.
    
    $T (x, z) = 0 \leq z = T (1, z) .$
x ∈ I_a.

Therefore, it must be the case that y ∈ I_a or y ∈ (a, 1].
- 3.1 y ∈ I_a.
  - 3.1.1. z ∈ (0, a) or z ∈ [a, 1).
    
    $T (x, z) = 0 = T (y, z) .$
  - 3.1.2. z ∈ I_a.
    
    $T (x, z) = x \land z \leq y \land z = T (y, z) .$
- 3.2. y ∈ (a, 1).
  
  As x ∈ I_a and y ∈ (a, 1), according to our constraint, x || y must hold. The case of y ∈ (a, 1) is not possible.
- 3.3 y = 1.
  - 3.3.1. z ∈ (0, a) or z ∈ [a, 1).
    
    $T (x, z) = 0 \leq z = T (1, z) .$
  - 3.3.2. z ∈ I_a.
    
    $T (x, z) = x \land z \leq z = T (1, z) .$
x = 1. Subsequently, because y = 1, it is clear that T(x, z) = T(y, z) for all z ∈ L.

ii) Associativity

If at least one of x, y, z in L is 1, the associativity is satisfied.

x ∈ [0, a).
- 1.1 y ∈ [0, a).
  - 1.1.1. z ∈ [0, a).
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, V (y, z)) \\ = V (x, V (y, z)) \\ = V (V (x, y), z) \\ = T (T (x, y), z) . \end{array}$
  - 1.1.2. z ∈ [a, 1).
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, y) = V (x, y) \\ = T (V (x, y), z) \\ = T (T (x, y), z) . \end{array}$
  - 1.1.3. z ∈ I_a.
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, 0) = 0 \\ = T (V (x, y), z) \\ = T (T (x, y), z) . \end{array}$
- 1.2. y ∈ [a, 1).
  - 1.2.1. z ∈ [0, a).
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, z) \\ = T (T (x, y), z) . \end{array}$
  - 1.2.2. z ∈ [a, 1).
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, y \land z) = x \\ = T (x, z) \\ = T (T (x, y), z) . \end{array}$
  - 1.2.3. z ∈ I_a.
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, 0) = 0 = T (x, z) \\ = T (T (x, y), z) . \end{array}$
- 1.3. y ∈ I_a.
  - 1.3.1. z ∈ [0, a) or z ∈ [a, 1).
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, 0) = 0 = T (0, z) \\ = T (T (x, y), z) . \end{array}$
  - 1.3.2. z ∈ I_a.
    
    It holds that y ∧z ∈ I_a or y ∧ z = 0. Otherwise, if y ∧ z ∈ (0, a], because y || a, we obtain y || y ∧ z. Thus, there is a contradiction.
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, y \land z) = 0 \\ = T (0, z) = T (T (x, y), z) . \end{array}$
x ∈ [a, 1).
- 2.1 y ∈ [0, a).
  - 2.1.1. z ∈ [0, a).
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, V (y, z)) = V (y, z) \\ = T (y, z) = T (T (x, y), z) . \end{array}$
  - 2.1.2. z ∈ [a, 1).
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, y) = y \\ = T (y, z) = T (T (x, y), z) . \end{array}$
  - 2.1.3. z ∈ I_a.
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, 0) = 0 = T (y, z) \\ = T (T (x, y), z) . \end{array}$
- 2.2. y ∈ [a, 1).
  - 2.2.1. z ∈ [0, a).
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, z) = z = T (x \land y, z) \\ = T (T (x, y), z) . \end{array}$
  - 2.2.2. z ∈ [a, 1).
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, y \land z) = x \land y \land z \\ = T (x \land y, z) = T (T (x, y), z) . \end{array}$
  - 2.2.3. z ∈ I_a.
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, 0) = 0 = T (x \land y, z) \\ = T (T (x, y), z) . \end{array}$
- 2.3. y ∈ I_a.
  - 2.3.1. z ∈ [0, a) or z ∈ [a, 1).
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, 0) = 0 = T (0, z) \\ = T (T (x, y), z) . \end{array}$
  - 2.3.2. z ∈ I_a.
    
    Then, y ∧ z ∈ I_a or y ∧ z = 0 must be true.
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, y \land z) = 0 \\ = T (0, z) = T (T (x, y), z) . \end{array}$
x ∈ I_a.
- 3.1 y ∈ [0, a).
  - 3.1.1. z ∈ [0, a).
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, V (y, z)) = 0 = T (0, z) \\ = T (T (x, y), z) . \end{array}$
  - 3.1.2. z ∈ [a, 1).
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, y) = 0 = 0 T (0, z) \\ = T (T (x, y), z) . \end{array}$
  - 3.1.3. z ∈ I_a.
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, 0) = 0 = T (0, z) \\ = T (T (x, y), z) . \end{array}$
- 3.2. y ∈ [a, 1).
  - 3.2.1. z ∈ [0, a).
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, z) = 0 = T (0, z) \\ = T (T (x, y), z) . \end{array}$
  - 3.2.2. z ∈ [a, 1).
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, y \land z) = 0 = T (0, z) \\ = T (T (x, y), z) . \end{array}$
  - 3.2.3. z ∈ I_a.
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, 0) = 0 = T (0, z) \\ = T (T (x, y), z) . \end{array}$
- 3.3. y ∈ I_a.
  - 3.3.1. z ∈ [0, a) or z ∈ [a, 1).
    
    Thus, by assumption, x ∧ y ∈ I_a or x ∧ y = 0 must hold.
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, 0) = 0 \\ = T (x \land y, z) = T (T (x, y), z) . \end{array}$
  - 3.3.2. z ∈ I_a.
    
    $\begin{array}{l} T (x, T (y, z)) = T (x, y \land z) \\ = x \land y \land z = T (x \land y, z) \\ = T (T (x, y), z) . \end{array}$

Corollary 1

If we use V = T_∧ on [0, a] provided in Theorem 2, we obtain the following t-norm on L:

T (x, y) = {\begin{array}{l} 0, & if (x, y) \in [0, a) \times I_{a} \cup I_{a} \times [0, a) \cup [a, 1) \times I_{a}, \cup I_{a} \times [a, 1), \\ x \land y, & otherwise. \end{array}

Example 2

Consider the bounded lattice (L₂ = {0_L_₂, b, c, e, d, a, p, 1_L_₂}, ≤, 0_L_₂, 1_L_₂) illustrated in Figure 2, which satisfies the constraints of Theorem 2.

V (x, y) = {\begin{array}{l} x \land y, & if a \in {x, y}, \\ 0_{L_{2}}, & otherwise. \end{array}

Thus, the function T on L₂ defined in Table 2 is a t-norm.

Remark 1

Example 3

Thus, the function T on L₃ defined in Table 3 is not a t-norm. Indeed, it does not satisfy monotonicity. Clearly, b < e and T(b, b) = b ? 0_L_₃ = T(b, e).

Remark 2

Example 4

Therefore, the function T on L₄ that is defined in Table 4 is not a t-norm. Again, it does not satisfy monotonicity. Clearly, e < p and T(e, e) = e ? 0_L_₄= T(e, p).

Remark 3

Example 5

Example 6

Consider the bounded lattice (L₅ = {0_L_₅, p, t, s, m, a, 1_L_₅},≤, 0_L_₅, 1_L_₅) described in Figure 5, and consider the t-norm V on [0_L_₅, a] as follows:

V (x, y) = {\begin{array}{l} x \land y, & if a \in {x, y}, \\ 0_{L_{5}}, & otherwise. \end{array}

Remark 4

In Example 6, if we take the t-norm V on [0_L₅, a] such that V (x, y) = x ∧ y, we obtain T = T*.

Example 7

Consider the bounded lattice (L₆ = {0_L₆, m, n, k, a, 1_L_₆}, ≤, 0_L_₆, 1_L_₆) described in Figure 6, and consider the t-norm V on [0_L_₆, a] as follows:

V (x, y) = {\begin{array}{l} x \land y, & if a \in {x, y}, \\ 0_{L_{6}}, & otherwise. \end{array}

We present the same arguments for t-conorms. We omit the proof of Theorem and its examples, owing to its similarity to the proof of Theorem 2.

Theorem 3

S (x, y) = {\begin{array}{l} W (x, y), & if (x, y) \in {[a, 1]}^{2}, \\ 1, & if (x, y) \in (a, 1] \times I_{a} \cup I_{a} \times (a, 1] \cup (0, a] \times I_{a} \cup I_{a} \times (0, a], \\ x \lor y, & otherwise. \end{array}

Corollary 2

In Theorem 3, if we use W = S_∨ on [a, 1], we obtain the following t-conorm on L:

S (x, y) = {\begin{array}{l} 1, & if (x, y) \in (a, 1] \times I_{a} \cup I_{a} \times (a, 1] \cup (0, a] \times I_{a} \cup I_{a} \times (0, a], \\ x \lor y, & otherwise. \end{array}

4. Modified Ordinal Sum Constructions of t-Norms and t-Conorms on Appropriate Bounded Lattices

Theorem 4

(3)

T_{i} (x, y) = {\begin{array}{l} T_{i - 1} (x, y) & i f (x, y) \in {[0, a_{i - 1}]}^{2}, \\ 0, & if (x, y) \in [0, a_{i - 1}) \times I_{a_{i - 1}} \cup I_{a_{i - 1}} \times [0, a_{i - 1}) \cup [a_{i - 1}, a_{i}) \times I_{a_{i - 1}} \cup I_{a_{i - 1}} \times [a_{i - 1}, a_{i}), \\ x \land y, & otherwise. \end{array}

Theorem 5

(4)

S_{i} (x, y), = {\begin{array}{l} S_{i - 1} (x, y) & if (x, y) \in {[a_{i - 1}, 1]}^{2}, \\ 1, & if (x, y) \in (a_{i - 1}, 1] \times I_{a_{i - 1}} \cup I_{a_{i - 1}} \times (a_{i - 1}, 1] \cup (a_{i}, a_{i - 1}] \times I_{a_{i - 1}} \cup I_{a_{i - 1}} \times (a_{i}, a_{i - 1}], \\ x \land y, & otherwise. \end{array}

5. Conclusion

Abstract
1. Introduction
2. Preliminaries
3. New Methods to Construct t-Norms and t-Conorms on Appropriate Bounded Lattices
4. Modified Ordinal Sum Constructions of t-Norms and t-Conorms on Appropriate Bounded Lattices
5. Conclusion

Fig 1.

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