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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

© The Korean Institute of Intelligent Systems

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.

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

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 aL \ {0, 1}. Further modifications were proposed in [1014].

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 aL \ {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 aL \ {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 aL \ {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 aL\{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.

Definition 1 [15]

A lattice is a partially ordered set (L,≤) in which each two-element subset {x, y} has an infimum, denoted as xy, and a supremum, denoted as xy. 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 [1517]

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

Definition 3 [15,1820]

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

[a,b]={xL?axb}.

Similarly, [a, b) = {xL | ax < b}, (a, b] = {xL | a < xb}, and (a, b) = {xL | 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 xL.

Theorem 1 [9]

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

T*(x,y)={xy,if x=1or y=1,VT(x,y),if x,y[a,1),xya,otherwise.S*(x,y)={xy,if x=0or y=0,WS(x,y),if x,y(0,a],xya,otherwise.

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 aL \ {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, TV : L2L, which is defined by

TV(x,y)={V(x,y),if (x,y)[a,b]2,xy,otherwise.

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 SW be a t-conorm on [a, b]. Then, SW : L2L, which is defined by

SW(x,y)={W(x,y),if (x,y)[a,b]2,xy,otherwise.

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

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

V(x,y)={xy,if a{x,y},0L1,otherwise.

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

Theorem 2

Let (L,≤, 0, 1) be a bounded lattice and aL \ {0, 1}. If, for all xIa and y ∈ (0, a], and for all xIa 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)={V(x,y)if (x,y)[0,a]2,0,if (x,y)[0,a]×IaIa×[0,a)[a,1)×Ia,xy,otherwise.
Proof

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

i) Monotonicity

We prove that if xy, T(x, z) ≤ T(y, z) for all zL. Subsequently, we demonstrate that T(x, z) ≤ T(y, z) for all zL. 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)V(y,z)=T(y,z).

      • 1.1.2. z ∈ [a, 1).

        T(x,z)=xzyz=T(y,z).

      • 1.1.3. zIa.

        T(x,z)=0=T(y,z).

    • 1.2. y ∈ [a, 1).

      • 1.2.1. z ∈ (0, a).

        T(x,z)=V(x,z)z=T(y,z).

      • 1.2.2. z ∈ [a, 1).

        T(x,z)=xzyz=T(y,z).

      • 1.2.3. zIa.

        T(x,z)=0=T(y,z).

    • 1.3. yIa

      Because x ∈ (0, a) and yIa, according to our constraint, x || y must hold. Therefore, we cannot consider the case of yIa.

    • 1.4. y = 1.

      • 1.4.1. z ∈ (0, a).

        T(x,z)=V(x,z)z=T(1,z).

      • 1.4.2. z ∈ [a, 1).

        T(x,z)=x<az=T(1,z).

      • 1.4.3. zIa.

        T(x,z)=0z=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)=xzyz=T(y,z).

      • 2.1.3. zIa.

        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)=xzz=T(1,z).

      • 2.1.3. zIa.

        T(x,z)=0z=T(1,z).

  • xIa.

    Therefore, it must be the case that yIa or y ∈ (a, 1].

    • 3.1 yIa.

      • 3.1.1. z ∈ (0, a) or z ∈ [a, 1).

        T(x,z)=0=T(y,z).

      • 3.1.2. zIa.

        T(x,z)=xzyz=T(y,z).

    • 3.2. y ∈ (a, 1).

      As xIa 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)=0z=T(1,z).

      • 3.3.2. zIa.

        T(x,z)=xzz=T(1,z).

  • x = 1. Subsequently, because y = 1, it is clear that T(x, z) = T(y, z) for all zL.

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).

        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).

      • 1.1.2. z ∈ [a, 1).

        T(x,T(y,z))=T(x,y)=V(x,y)=T(V(x,y),z)=T(T(x,y),z).

      • 1.1.3. zIa.

        T(x,T(y,z))=T(x,0)=0=T(V(x,y),z)=T(T(x,y),z).

    • 1.2. y ∈ [a, 1).

      • 1.2.1. z ∈ [0, a).

        T(x,T(y,z))=T(x,z)=T(T(x,y),z).

      • 1.2.2. z ∈ [a, 1).

        T(x,T(y,z))=T(x,yz)=x=T(x,z)=T(T(x,y),z).

      • 1.2.3. zIa.

        T(x,T(y,z))=T(x,0)=0=T(x,z)=T(T(x,y),z).

    • 1.3. yIa.

      • 1.3.1. z ∈ [0, a) or z ∈ [a, 1).

        T(x,T(y,z))=T(x,0)=0=T(0,z)=T(T(x,y),z).

      • 1.3.2. zIa.

        It holds that yzIa or yz = 0. Otherwise, if yz ∈ (0, a], because y || a, we obtain y || yz. Thus, there is a contradiction.

        T(x,T(y,z))=T(x,yz)=0=T(0,z)=T(T(x,y),z).

  • x ∈ [a, 1).

    • 2.1 y ∈ [0, a).

      • 2.1.1. z ∈ [0, a).

        T(x,T(y,z))=T(x,V(y,z))=V(y,z)=T(y,z)=T(T(x,y),z).

      • 2.1.2. z ∈ [a, 1).

        T(x,T(y,z))=T(x,y)=y=T(y,z)=T(T(x,y),z).

      • 2.1.3. zIa.

        T(x,T(y,z))=T(x,0)=0=T(y,z)=T(T(x,y),z).

    • 2.2. y ∈ [a, 1).

      • 2.2.1. z ∈ [0, a).

        T(x,T(y,z))=T(x,z)=z=T(xy,z)=T(T(x,y),z).

      • 2.2.2. z ∈ [a, 1).

        T(x,T(y,z))=T(x,yz)=xyz=T(xy,z)=T(T(x,y),z).

      • 2.2.3. zIa.

        T(x,T(y,z))=T(x,0)=0=T(xy,z)=T(T(x,y),z).

    • 2.3. yIa.

      • 2.3.1. z ∈ [0, a) or z ∈ [a, 1).

        T(x,T(y,z))=T(x,0)=0=T(0,z)=T(T(x,y),z).

      • 2.3.2. zIa.

        Then, yzIa or yz = 0 must be true.

        T(x,T(y,z))=T(x,yz)=0=T(0,z)=T(T(x,y),z).

  • xIa.

    • 3.1 y ∈ [0, a).

      • 3.1.1. z ∈ [0, a).

        T(x,T(y,z))=T(x,V(y,z))=0=T(0,z)=T(T(x,y),z).

      • 3.1.2. z ∈ [a, 1).

        T(x,T(y,z))=T(x,y)=0=0T(0,z)=T(T(x,y),z).

      • 3.1.3. zIa.

        T(x,T(y,z))=T(x,0)=0=T(0,z)=T(T(x,y),z).

    • 3.2. y ∈ [a, 1).

      • 3.2.1. z ∈ [0, a).

        T(x,T(y,z))=T(x,z)=0=T(0,z)=T(T(x,y),z).

      • 3.2.2. z ∈ [a, 1).

        T(x,T(y,z))=T(x,yz)=0=T(0,z)=T(T(x,y),z).

      • 3.2.3. zIa.

        T(x,T(y,z))=T(x,0)=0=T(0,z)=T(T(x,y),z).

    • 3.3. yIa.

      • 3.3.1. z ∈ [0, a) or z ∈ [a, 1).

        Thus, by assumption, xyIa or xy = 0 must hold.

        T(x,T(y,z))=T(x,0)=0=T(xy,z)=T(T(x,y),z).

      • 3.3.2. zIa.

        T(x,T(y,z))=T(x,yz)=xyz=T(xy,z)=T(T(x,y),z).

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)={0,if (x,y)[0,a)×IaIa×[0,a)[a,1)×Ia,Ia×[a,1),xy,otherwise.

Example 2

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

V(x,y)={xy,if a{x,y},0L2,otherwise.

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 aL \ {0, 1}. In Theorem 2, it can be observed that the condition for all xIa and y ∈ (0, a] holds; thus, x || y cannot be ignored. The following example illustrates that the function T : L2L defined by Theorem 2 is not a t-norm.

Example 3

Consider the bounded lattice (L3 = {0L3, b, c, e, d, a, p, 1L3}, ≤, 0L3, 1L3) described in Figure 3, which does not satisfy one of the constraints of Theorem 2; that is, there exists an element bL3, where b < e for eIa and b ∈ (0L3, a). Consider the t-norm V on [0L3, a] such that V (x, y) = xy.

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 ? 0L3 = T(b, e).

Remark 2

Let (L,≤, 0, 1) be a bounded lattice and aL \ {0, 1}. In Theorem 2, it can be observed that the condition for all xIa and y ∈ [a, 1) holds, and that x || y cannot be omitted. The following example illustrates that the function T : L2L defined by Theorem 2 is not a t-norm.

Example 4

Consider the bounded lattice (L4 = {0L4, b, c, e, d, a, p, 1L4}, ≤, 0L4, 1L5) described in Figure 4, which does not satisfy one of the constraints of Theorem 2; that is, there is an element eL4, where e < p for eIa and p ∈ (a, 1L4). Consider the t-norm V on [0L4, a] such that V (x, y) = xy.

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 ? 0L4= 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, TT* or T* ≤ T on any bounded lattice. We demonstrate these arguments using the following examples.

Example 5

Consider the bounded lattice (L2 = {0L2, b, c, e, d, a, p, 1L2},≤, 0L2, 1L2) described in Figure 2 and the t-norm V on [0L2, a] such that V (x, y) = xy, and consider the t-norm VT on [a, 1L2], VT (x, y) = xy. 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 = {0L5, p, t, s, m, a, 1L5},≤, 0L5, 1L5) described in Figure 5, and consider the t-norm V on [0L5, a] as follows:

V(x,y)={xy,if a{x,y},0L5,otherwise.

Moreover, consider any t-norm VT on [a, 1L5]. 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, TT*.

Remark 4

In Example 6, if we take the t-norm V on [0L5, a] such that V (x, y) = xy, we obtain T = T*.

Example 7

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

V(x,y)={xy,if a{x,y},0L6,otherwise.

Furthermore, consider any t-norm VT on [a, 1L6]. 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 TT*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 aL \ {0, 1}. If, for all xIa and y ∈ [a, 1), and for all xIa 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)={W(x,y),if (x,y)[a,1]2,1,if (x,y)(a,1]×IaIa×(a,1](0,a]×IaIa×(0,a],xy,otherwise.

Corollary 2

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

S(x,y)={1,if (x,y)(a,1]×IaIa×(a,1](0,a]×IaIa×(0,a],xy,otherwise.

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, ···, an} be a finite chain in L such that 0 = a0 < a1 < a2 < ··· < an = 1. Furthermore, let x || y for all xIai and y ∈ (0, ai], and for all xIaiand y ∈ [ai, 1). We use the t-norm V on [0, a1]. Thus, the operation Tn on L, which is defined as follows, is a t-norm, such that T1 = V, and for i ∈ {2, ···, n}, the function Ti on [0, ai] is expressed by

Ti(x,y)={Ti-1(x,y)if(x,y)[0,ai-1]2,0,if (x,y)[0,ai-1)×Iai-1Iai-1×[0,ai-1)[ai-1,ai)×Iai-1Iai-1×[ai-1,ai),xy,otherwise.

Theorem 5

Let (L,≤, 0, 1) be a bounded lattice and {a0, a1, a2, …, an} be a finite chain in L such that 1 = a0 > a1 > a2 > … > an = 0. Let x || y for all xIai and y ∈ [ai, 1), and for all xIai and y ∈ (0, ai]. We place the t-conorm S0 on [a1, 1]. Therefore, the operation Sn : L2L, which is defined as follows, is a t-conorm, such that S1 = S0, and for i ∈ {2, …, n}, the function Si : [ai, 1]2 → [ai, 1] is expressed by

Si(x,y),={Si-1(x,y)if (x,y)[ai-1,1]2,1,if (x,y)(ai-1,1]×Iai-1Iai-1×(ai-1,1](ai,ai-1]×Iai-1Iai-1×(ai,ai-1],xy,otherwise.

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.

Table. 1.

Table 1. The function TV on L1.

TV0L1bcdea1L1
0L10L10L10L10L10L10L10L1
b0L10L10L10L1bbb
c0L10L10L10L1bcc
d0L10L10L1db0L1d
e0L1bbbebe
a0L1bcbbaa
1L10L1bcdea1L1

Table. 2.

Table 2. The t-norm T on L2.

T0L2bcedap1L2
0L20L20L20L20L20L20L20L20L2
b0L20L20L20L20L2bbb
c0L20L2c0L2c0L20L2c
e0L20L20L2ee0L20L2e
d0L20L2ced0L20L2d
a0L2b0L20L20L2aaa
p0L2b0L20L20L2app
1L20L2bcedap1L2

Table. 3.

Table 3. The function T on L3.

T0L3bcedap1L3
0L30L30L30L30L30L30L30L30L3
b0L3b0L30L30L3bbb
c0L30L3cbc0L30L3c
e0L30L3beb0L30L3e
d0L30L3cbd0L30L3d
a0L3b0L30L30L3aaa
p0L3b0L30L30L3app
1L30L3bcedap1L3

Table. 4.

Table 4. The function T on L4.

T0L4bcedap1L4
0L40L40L40L40L40L40L40L40L4
b0L4b0L40L40L4bbb
c0L40L4c0L4c0L40L4c
e0L40L40L4ee0L40L4e
d0L40L4ced0L40L4d
a0L4b0L40L40L4aaa
p0L4b0L40L40L4app
1L40L4bcedap1L4

Table. 5.

Table 5. The t-norm T* on L2.

T*0L2bcedap1L2
0L20L20L20L20L20L20L20L20L2
b0L2b0L20L20L2bbb
c0L20L20L20L20L20L20L2c
e0L20L20L20L20L20L20L2e
d0L20L20L20L20L20L20L2d
a0L2b0L20L20L2aaa
p0L2b0L20L20L2app
1L20L2bcedap1L2

Table. 6.

Table 6. The t-norm T on L2.

T0L2bcedap1L2
0L20L20L20L20L20L20L20L20L2
b0L2b0L20L20L2bbb
c0L20L2c0L2c0L20L2c
e0L20L20L2ee0L20L2e
d0L20L2ced0L20L2d
a0L2b0L20L20L2aaa
p0L2b0L20L20L2app
1L20L2bcedap1L2

Table. 7.

Table 7. The t-norm T on L5.

T0L5ptsma1L5
0L50L50L50L50L50L50L50L5
p0L50L50L50L50L5pp
t0L50L50L50L50L5tt
s0L50L50L50L50L5ss
m0L50L50L50L50L5mm
a0L5ptsmaa
1L50L5ptsma1L5

Table. 8.

Table 8. The t-norm T* on L5.

T*0L5ptsma1L5
0L50L50L50L50L50L50L50L5
p0L5pp0L50L5pp
t0L5pt0L50L5tt
s0L50L50L5s0L5ss
m0L50L50L50L5mmm
a0L5ptsmaa
1L50L5ptsma1L5

Table. 9.

Table 9. The t-norm T on L6.

T0L6mnka1L6
0L60L60L60L60L60L60L6
m0L6m0L60L60L6m
n0L60L60L60L6nn
k0L60L60L6k0L6k
a0L60L6n0L6aa
1L60L6mnka1L6

Table. 10.

Table 10. The t-norm T* on L6.

T*0L6mnka1L6
0L60L60L60L60L60L60L6
m0L60L60L60L60L6m
n0L60L6n0L6nn
k0L60L60L60L60L6k
a0L60L6n0L6aa
1L60L6mnka1L6

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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

Copyright © The Korean Institute of Intelligent Systems.

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

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

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 aL \ {0, 1}. Further modifications were proposed in [1014].

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 aL \ {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 aL \ {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 aL \ {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 aL\{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

Definition 1 [15]

A lattice is a partially ordered set (L,≤) in which each two-element subset {x, y} has an infimum, denoted as xy, and a supremum, denoted as xy. 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 [1517]

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

Definition 3 [15,1820]

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

[a,b]={xL?axb}.

Similarly, [a, b) = {xL | ax < b}, (a, b] = {xL | a < xb}, and (a, b) = {xL | 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 xL.

Theorem 1 [9]

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

T*(x,y)={xy,if x=1or y=1,VT(x,y),if x,y[a,1),xya,otherwise.S*(x,y)={xy,if x=0or y=0,WS(x,y),if x,y(0,a],xya,otherwise.

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

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 aL \ {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, TV : L2L, which is defined by

TV(x,y)={V(x,y),if (x,y)[a,b]2,xy,otherwise.

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 SW be a t-conorm on [a, b]. Then, SW : L2L, which is defined by

SW(x,y)={W(x,y),if (x,y)[a,b]2,xy,otherwise.

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

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

V(x,y)={xy,if a{x,y},0L1,otherwise.

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

Theorem 2

Let (L,≤, 0, 1) be a bounded lattice and aL \ {0, 1}. If, for all xIa and y ∈ (0, a], and for all xIa 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)={V(x,y)if (x,y)[0,a]2,0,if (x,y)[0,a]×IaIa×[0,a)[a,1)×Ia,xy,otherwise.
Proof

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

i) Monotonicity

We prove that if xy, T(x, z) ≤ T(y, z) for all zL. Subsequently, we demonstrate that T(x, z) ≤ T(y, z) for all zL. 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)V(y,z)=T(y,z).

      • 1.1.2. z ∈ [a, 1).

        T(x,z)=xzyz=T(y,z).

      • 1.1.3. zIa.

        T(x,z)=0=T(y,z).

    • 1.2. y ∈ [a, 1).

      • 1.2.1. z ∈ (0, a).

        T(x,z)=V(x,z)z=T(y,z).

      • 1.2.2. z ∈ [a, 1).

        T(x,z)=xzyz=T(y,z).

      • 1.2.3. zIa.

        T(x,z)=0=T(y,z).

    • 1.3. yIa

      Because x ∈ (0, a) and yIa, according to our constraint, x || y must hold. Therefore, we cannot consider the case of yIa.

    • 1.4. y = 1.

      • 1.4.1. z ∈ (0, a).

        T(x,z)=V(x,z)z=T(1,z).

      • 1.4.2. z ∈ [a, 1).

        T(x,z)=x<az=T(1,z).

      • 1.4.3. zIa.

        T(x,z)=0z=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)=xzyz=T(y,z).

      • 2.1.3. zIa.

        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)=xzz=T(1,z).

      • 2.1.3. zIa.

        T(x,z)=0z=T(1,z).

  • xIa.

    Therefore, it must be the case that yIa or y ∈ (a, 1].

    • 3.1 yIa.

      • 3.1.1. z ∈ (0, a) or z ∈ [a, 1).

        T(x,z)=0=T(y,z).

      • 3.1.2. zIa.

        T(x,z)=xzyz=T(y,z).

    • 3.2. y ∈ (a, 1).

      As xIa 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)=0z=T(1,z).

      • 3.3.2. zIa.

        T(x,z)=xzz=T(1,z).

  • x = 1. Subsequently, because y = 1, it is clear that T(x, z) = T(y, z) for all zL.

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).

        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).

      • 1.1.2. z ∈ [a, 1).

        T(x,T(y,z))=T(x,y)=V(x,y)=T(V(x,y),z)=T(T(x,y),z).

      • 1.1.3. zIa.

        T(x,T(y,z))=T(x,0)=0=T(V(x,y),z)=T(T(x,y),z).

    • 1.2. y ∈ [a, 1).

      • 1.2.1. z ∈ [0, a).

        T(x,T(y,z))=T(x,z)=T(T(x,y),z).

      • 1.2.2. z ∈ [a, 1).

        T(x,T(y,z))=T(x,yz)=x=T(x,z)=T(T(x,y),z).

      • 1.2.3. zIa.

        T(x,T(y,z))=T(x,0)=0=T(x,z)=T(T(x,y),z).

    • 1.3. yIa.

      • 1.3.1. z ∈ [0, a) or z ∈ [a, 1).

        T(x,T(y,z))=T(x,0)=0=T(0,z)=T(T(x,y),z).

      • 1.3.2. zIa.

        It holds that yzIa or yz = 0. Otherwise, if yz ∈ (0, a], because y || a, we obtain y || yz. Thus, there is a contradiction.

        T(x,T(y,z))=T(x,yz)=0=T(0,z)=T(T(x,y),z).

  • x ∈ [a, 1).

    • 2.1 y ∈ [0, a).

      • 2.1.1. z ∈ [0, a).

        T(x,T(y,z))=T(x,V(y,z))=V(y,z)=T(y,z)=T(T(x,y),z).

      • 2.1.2. z ∈ [a, 1).

        T(x,T(y,z))=T(x,y)=y=T(y,z)=T(T(x,y),z).

      • 2.1.3. zIa.

        T(x,T(y,z))=T(x,0)=0=T(y,z)=T(T(x,y),z).

    • 2.2. y ∈ [a, 1).

      • 2.2.1. z ∈ [0, a).

        T(x,T(y,z))=T(x,z)=z=T(xy,z)=T(T(x,y),z).

      • 2.2.2. z ∈ [a, 1).

        T(x,T(y,z))=T(x,yz)=xyz=T(xy,z)=T(T(x,y),z).

      • 2.2.3. zIa.

        T(x,T(y,z))=T(x,0)=0=T(xy,z)=T(T(x,y),z).

    • 2.3. yIa.

      • 2.3.1. z ∈ [0, a) or z ∈ [a, 1).

        T(x,T(y,z))=T(x,0)=0=T(0,z)=T(T(x,y),z).

      • 2.3.2. zIa.

        Then, yzIa or yz = 0 must be true.

        T(x,T(y,z))=T(x,yz)=0=T(0,z)=T(T(x,y),z).

  • xIa.

    • 3.1 y ∈ [0, a).

      • 3.1.1. z ∈ [0, a).

        T(x,T(y,z))=T(x,V(y,z))=0=T(0,z)=T(T(x,y),z).

      • 3.1.2. z ∈ [a, 1).

        T(x,T(y,z))=T(x,y)=0=0T(0,z)=T(T(x,y),z).

      • 3.1.3. zIa.

        T(x,T(y,z))=T(x,0)=0=T(0,z)=T(T(x,y),z).

    • 3.2. y ∈ [a, 1).

      • 3.2.1. z ∈ [0, a).

        T(x,T(y,z))=T(x,z)=0=T(0,z)=T(T(x,y),z).

      • 3.2.2. z ∈ [a, 1).

        T(x,T(y,z))=T(x,yz)=0=T(0,z)=T(T(x,y),z).

      • 3.2.3. zIa.

        T(x,T(y,z))=T(x,0)=0=T(0,z)=T(T(x,y),z).

    • 3.3. yIa.

      • 3.3.1. z ∈ [0, a) or z ∈ [a, 1).

        Thus, by assumption, xyIa or xy = 0 must hold.

        T(x,T(y,z))=T(x,0)=0=T(xy,z)=T(T(x,y),z).

      • 3.3.2. zIa.

        T(x,T(y,z))=T(x,yz)=xyz=T(xy,z)=T(T(x,y),z).

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)={0,if (x,y)[0,a)×IaIa×[0,a)[a,1)×Ia,Ia×[a,1),xy,otherwise.

Example 2

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

V(x,y)={xy,if a{x,y},0L2,otherwise.

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 aL \ {0, 1}. In Theorem 2, it can be observed that the condition for all xIa and y ∈ (0, a] holds; thus, x || y cannot be ignored. The following example illustrates that the function T : L2L defined by Theorem 2 is not a t-norm.

Example 3

Consider the bounded lattice (L3 = {0L3, b, c, e, d, a, p, 1L3}, ≤, 0L3, 1L3) described in Figure 3, which does not satisfy one of the constraints of Theorem 2; that is, there exists an element bL3, where b < e for eIa and b ∈ (0L3, a). Consider the t-norm V on [0L3, a] such that V (x, y) = xy.

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 ? 0L3 = T(b, e).

Remark 2

Let (L,≤, 0, 1) be a bounded lattice and aL \ {0, 1}. In Theorem 2, it can be observed that the condition for all xIa and y ∈ [a, 1) holds, and that x || y cannot be omitted. The following example illustrates that the function T : L2L defined by Theorem 2 is not a t-norm.

Example 4

Consider the bounded lattice (L4 = {0L4, b, c, e, d, a, p, 1L4}, ≤, 0L4, 1L5) described in Figure 4, which does not satisfy one of the constraints of Theorem 2; that is, there is an element eL4, where e < p for eIa and p ∈ (a, 1L4). Consider the t-norm V on [0L4, a] such that V (x, y) = xy.

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 ? 0L4= 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, TT* or T* ≤ T on any bounded lattice. We demonstrate these arguments using the following examples.

Example 5

Consider the bounded lattice (L2 = {0L2, b, c, e, d, a, p, 1L2},≤, 0L2, 1L2) described in Figure 2 and the t-norm V on [0L2, a] such that V (x, y) = xy, and consider the t-norm VT on [a, 1L2], VT (x, y) = xy. 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 = {0L5, p, t, s, m, a, 1L5},≤, 0L5, 1L5) described in Figure 5, and consider the t-norm V on [0L5, a] as follows:

V(x,y)={xy,if a{x,y},0L5,otherwise.

Moreover, consider any t-norm VT on [a, 1L5]. 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, TT*.

Remark 4

In Example 6, if we take the t-norm V on [0L5, a] such that V (x, y) = xy, we obtain T = T*.

Example 7

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

V(x,y)={xy,if a{x,y},0L6,otherwise.

Furthermore, consider any t-norm VT on [a, 1L6]. 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 TT*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 aL \ {0, 1}. If, for all xIa and y ∈ [a, 1), and for all xIa 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)={W(x,y),if (x,y)[a,1]2,1,if (x,y)(a,1]×IaIa×(a,1](0,a]×IaIa×(0,a],xy,otherwise.

Corollary 2

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

S(x,y)={1,if (x,y)(a,1]×IaIa×(a,1](0,a]×IaIa×(0,a],xy,otherwise.

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

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, ···, an} be a finite chain in L such that 0 = a0 < a1 < a2 < ··· < an = 1. Furthermore, let x || y for all xIai and y ∈ (0, ai], and for all xIaiand y ∈ [ai, 1). We use the t-norm V on [0, a1]. Thus, the operation Tn on L, which is defined as follows, is a t-norm, such that T1 = V, and for i ∈ {2, ···, n}, the function Ti on [0, ai] is expressed by

Ti(x,y)={Ti-1(x,y)if(x,y)[0,ai-1]2,0,if (x,y)[0,ai-1)×Iai-1Iai-1×[0,ai-1)[ai-1,ai)×Iai-1Iai-1×[ai-1,ai),xy,otherwise.

Theorem 5

Let (L,≤, 0, 1) be a bounded lattice and {a0, a1, a2, …, an} be a finite chain in L such that 1 = a0 > a1 > a2 > … > an = 0. Let x || y for all xIai and y ∈ [ai, 1), and for all xIai and y ∈ (0, ai]. We place the t-conorm S0 on [a1, 1]. Therefore, the operation Sn : L2L, which is defined as follows, is a t-conorm, such that S1 = S0, and for i ∈ {2, …, n}, the function Si : [ai, 1]2 → [ai, 1] is expressed by

Si(x,y),={Si-1(x,y)if (x,y)[ai-1,1]2,1,if (x,y)(ai-1,1]×Iai-1Iai-1×(ai-1,1](ai,ai-1]×Iai-1Iai-1×(ai,ai-1],xy,otherwise.

5. Conclusion

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.

Fig 1.

Figure 1.

The lattice L1.

The International Journal of Fuzzy Logic and Intelligent Systems 2022; 22: 48-58https://doi.org/10.5391/IJFIS.2022.22.1.48

Fig 2.

Figure 2.

The lattice L2.

The International Journal of Fuzzy Logic and Intelligent Systems 2022; 22: 48-58https://doi.org/10.5391/IJFIS.2022.22.1.48

Fig 3.

Figure 3.

The lattice L3.

The International Journal of Fuzzy Logic and Intelligent Systems 2022; 22: 48-58https://doi.org/10.5391/IJFIS.2022.22.1.48

Fig 4.

Figure 4.

The lattice L4.

The International Journal of Fuzzy Logic and Intelligent Systems 2022; 22: 48-58https://doi.org/10.5391/IJFIS.2022.22.1.48

Fig 5.

Figure 5.

The lattice L5.

The International Journal of Fuzzy Logic and Intelligent Systems 2022; 22: 48-58https://doi.org/10.5391/IJFIS.2022.22.1.48

Fig 6.

Figure 6.

The lattice L6.

The International Journal of Fuzzy Logic and Intelligent Systems 2022; 22: 48-58https://doi.org/10.5391/IJFIS.2022.22.1.48

Table 1 . The function TV on L1.

TV0L1bcdea1L1
0L10L10L10L10L10L10L10L1
b0L10L10L10L1bbb
c0L10L10L10L1bcc
d0L10L10L1db0L1d
e0L1bbbebe
a0L1bcbbaa
1L10L1bcdea1L1

Table 2 . The t-norm T on L2.

T0L2bcedap1L2
0L20L20L20L20L20L20L20L20L2
b0L20L20L20L20L2bbb
c0L20L2c0L2c0L20L2c
e0L20L20L2ee0L20L2e
d0L20L2ced0L20L2d
a0L2b0L20L20L2aaa
p0L2b0L20L20L2app
1L20L2bcedap1L2

Table 3 . The function T on L3.

T0L3bcedap1L3
0L30L30L30L30L30L30L30L30L3
b0L3b0L30L30L3bbb
c0L30L3cbc0L30L3c
e0L30L3beb0L30L3e
d0L30L3cbd0L30L3d
a0L3b0L30L30L3aaa
p0L3b0L30L30L3app
1L30L3bcedap1L3

Table 4 . The function T on L4.

T0L4bcedap1L4
0L40L40L40L40L40L40L40L40L4
b0L4b0L40L40L4bbb
c0L40L4c0L4c0L40L4c
e0L40L40L4ee0L40L4e
d0L40L4ced0L40L4d
a0L4b0L40L40L4aaa
p0L4b0L40L40L4app
1L40L4bcedap1L4

Table 5 . The t-norm T* on L2.

T*0L2bcedap1L2
0L20L20L20L20L20L20L20L20L2
b0L2b0L20L20L2bbb
c0L20L20L20L20L20L20L2c
e0L20L20L20L20L20L20L2e
d0L20L20L20L20L20L20L2d
a0L2b0L20L20L2aaa
p0L2b0L20L20L2app
1L20L2bcedap1L2

Table 6 . The t-norm T on L2.

T0L2bcedap1L2
0L20L20L20L20L20L20L20L20L2
b0L2b0L20L20L2bbb
c0L20L2c0L2c0L20L2c
e0L20L20L2ee0L20L2e
d0L20L2ced0L20L2d
a0L2b0L20L20L2aaa
p0L2b0L20L20L2app
1L20L2bcedap1L2

Table 7 . The t-norm T on L5.

T0L5ptsma1L5
0L50L50L50L50L50L50L50L5
p0L50L50L50L50L5pp
t0L50L50L50L50L5tt
s0L50L50L50L50L5ss
m0L50L50L50L50L5mm
a0L5ptsmaa
1L50L5ptsma1L5

Table 8 . The t-norm T* on L5.

T*0L5ptsma1L5
0L50L50L50L50L50L50L50L5
p0L5pp0L50L5pp
t0L5pt0L50L5tt
s0L50L50L5s0L5ss
m0L50L50L50L5mmm
a0L5ptsmaa
1L50L5ptsma1L5

Table 9 . The t-norm T on L6.

T0L6mnka1L6
0L60L60L60L60L60L60L6
m0L6m0L60L60L6m
n0L60L60L60L6nn
k0L60L60L6k0L6k
a0L60L6n0L6aa
1L60L6mnka1L6

Table 10 . The t-norm T* on L6.

T*0L6mnka1L6
0L60L60L60L60L60L60L6
m0L60L60L60L60L6m
n0L60L6n0L6nn
k0L60L60L60L60L6k
a0L60L6n0L6aa
1L60L6mnka1L6

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