Some methods to construct t-norms and t-conorms on bounded lattices

Abstract

In this paper, we give construction methods for triangular norms (t-norms) and triangular conorms (t-conorms) on appropriate bounded lattices. Then, we compare our methods and well-known methods proposed in [2 , 19]. Finally, we give different construction methods for t-norms and t-conorms on an appropriate bounded lattice by using recursion. Also, we provide some examples to discuss introduced methods.

Keywords

Triangular norms triangular conorms ordinal sum bounded lattice

1 Introduction and motivation

1.1 A brief review on the development of triangular norms and triangular conorms

In 1960, Schweizer and Sklar [21] presented t-norms and t-conorms on the unit interval [0, 1]. T-norms and t-conorms have numerous applications in different field, such as fuzzy sets, fuzzy logic, expert systems, decision making, image processing, etc. In several real problems we cannot deal with real scales, but with some more general scales, such as bounded chains, lattices or posets, in particular with different kinds of linguistic scales. Then the aggregation theory should be developed for theses more general scales, as it was stressed in a recent overview paper [17] and supported by numerous publications and conference contributions. In particular, Klement et al. in 2002, [13] researched t-norms as ordinal sums of semigroups in the way of Clifford [7]. First Saminger [19] introduced the ordinal sum of t-norms on bounded lattices in 2006. Also, she concentrated on ordinal sums of t-norms (t-conorms) acting on bounded chains or ordinal sums of posets. Saminger-Platz et al. [20] quarreled the expansion of t-norms (t-conorms) on bounded lattices in 2008. They used bounded sublattices as carriers for arbitrary summand t-norms. Then, in 2012, Medina [16] ensured a characterization of ordinal sums being t-norms (t-conorms) on bounded lattices. Moreover, Medina [16] ensured a characterization when an ordinal sum of t-norms is a t-norm on bounded lattice. In 2015, Ertuğrul et al. [8] proposed a modification of ordinal sums of t-norms (t-conorms) resulting to a t-norm (t-conorm) on an arbitrary bounded lattice.

1.2 The motivation

An interesting and natural study topic of t-norms and t-conorms is their research based on bounded lattices. It is worth noting that unlike the various classes of t-norms and t-conorms on the unit interval [0, 1], nowadays, the classes of t-norms and t-conorms on bounded lattices are not very clear yet, even though Ertuğrul et al. [8] provided some further investigations for the construction of ordinal sums of t-norms and t-conorms on bounded lattices, respectively. Therefore, from the theoretical point of view, the construction of ordinal sums of t-norms and t-conorms on bounded lattices becomes one of the most important and meaningful studies. Also, t-norms and t-conorms investigated on bounded lattices can be used to enrich their classes and to analyze their construction methods.

In 2020, Aşıcı and Mesiar [2] proposed some new methods to construct t-norm on a sublattice [0, a] and t-conorm on a sublattice [a, 1]. Also, when considering the existence of t-norms on a sublattice [0, a] and t-conorms on a sublattice [a, 1], respectively, there are no different corresponding investigations for the ordinal sums construction of t-norms and t-conorms on bounded lattices. So, in this paper, we introduce new ordinal sum constructions of t-norms and t-conorms on an appropriate bounded lattice, respectively, by using the existence of t-norms on a sublattice [0, a] and t-conorms on a sublattice [a, 1]. Thus, in this paper, we have two main motivations to investigate as follows.

• One of the motivations of this paper is that we suggest new ordinal sum constructions of t-norms and t-conorms on an appropriate bounded lattice, respectively, by using the existence of t-norms on a sublattice [0, a] and t-conorms on a sublattice [a, 1]. In particular, we show that the new construction approaches presented in this paper can be generalized by induction to a modified the ordinal sums construction of t-norms and t-conorms on appropriate bounded lattices.

• Another motivation of this paper is that, similarly to the investigation of ordinal sums of t-norms and t-conorms on the unit interval [0, 1], we provide constructions of ordinal sums of t-norms and t-conorms on bounded lattices and investigate their related properties.

In this study, we propose construction methods for t-norms and t-conorms on an appropriate bounded lattice which has some restrictions for a fixed element a ∈ L \ {0, 1}. We organized this paper as follows. In Section 2, we provide basic definitions to help our main study. In Section 3, we propose construction methods for t-norms and t-conorms on an appropriate bounded lattice, where a ∈ L \ {0, 1}, t-norms act on [0, a], and t-conorms act on [a, 1], respectively in Theorems 3 and 4. Also, we give some illustrative examples for clarity. Then, we ensure some examples to show that our construction approaches presented in this paper are different from the approaches proposed by Ertuğrul, Karaçal, Mesiar [8] and Aşıcı, Mesiar [2]. We submit our constructions in its full generality in Section 4. And we ensure some examples for clarity. Finally, some concluding remarks are added.

2 Preliminaries

Definition 1. [6] A bounded lattice (L, ≤) is a lattice which has the bottom and top elements, which are written as 0 and 1, respectively, that is, there exist two elements 0, 1 in L such that 0 ≤ x ≤ 1, for all x ∈ L.

Definition 2. [1 , 4–6] Let (L, ≤ , 0, 1) be a bounded lattice and a, b in L. If a and b are incomparable with each other, then we denote the notation a ∥ b. I_a = {x ∈ L ∣ x ∥ a}.

Definition 3. [3, 19] Let (L, ≤ , 0, 1) be a bounded lattice. A t-norm T (a t-conorm S) is a binary operation on L which is commutative, associative, monotonicity and it satisfies neutral element 1 (0), i.e., T (x, 1) = x (S (x, 0) = x) for all x in L.

Theorem 1. [8] 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], then the functions T^★ : L² → L and S^★ : L² → L are a t-norm and a t-conorm on L, respectively, where $T^{⋆} (x, y) = {\begin{matrix} i n f {x, y} & i f x =1 or y =1, \\ V_{T} (x, y) & i f x, y \in [a, 1), \\ i n f {x, y, a} & otherwise . \end{matrix}$ $S^{⋆} (x, y) = {\begin{matrix} s u p {x, y} & i f x o r y, \\ W_{S} (x, y) & i f x, y \in (0, a], \\ s u p {x, y, a} & otherwise . \end{matrix}$

Theorem 2. [2] Let (L, ≤ , 0, 1) be a bounded lattice and a ∈ L \ {0, 1}. Given t-norm V on [0, a] and t-conorm W on [a, 1].

i) If x ∥ y for all x ∈ I_a and y ∈ (0, a], then the function T^∼ : L² → L defined as follows is a t-norm on L $T^{~} (x, y) = {\begin{matrix} V (x, y) & i f (x, y) \in {[0, a]}^{2}, \\ 0 & i f (x, y) \in [0, a) \times I_{a} \\ \cup I_{a} \times [0, a) \cup I_{a} \times I_{a} \cup \\ [a, 1) \times I_{a} \cup I_{a} \times [a, 1), \\ i n f {x, y} & otherwise . \end{matrix}$ ii) If x ∥ y for all x ∈ I_a and y ∈ a r 1) rthen the function S^∼ : L² → L defined as follows is a t-conorm on L $S^{~} (x, y) = {\begin{matrix} W (x, y) & i f (x, y) \in {[a, 1]}^{2}, \\ 1 & i f (x, y) \in (a, 1] \times I_{a} \\ \cup I_{a} \times (a, 1] \cup I_{a} \times I_{a} \cup \\ (0, a] \times I_{a} \cup I_{a} \times (0, a], \\ s u p {x, y} & otherwise . \end{matrix}$

3 Some methods to obtain t-norms and t-conorms on appropriate bounded lattices

The following definition of an ordinal sum of t-norms defined on subintervals of a bounded lattice (L, ≤ , 0, 1) has been extracted from [19], which generalizes the methods are given in [14] on subintervals of [0, 1].

Definition 4. [19] Let (L, ≤ , 0, 1) be a bounded lattice and fix some subinterval [a, b] of L. Let V be a t-norm on [a, b]. Then T : L² → L defined by $T (x, y) = {\begin{matrix} V (x, y) & i f (x, y) \in {[a, b]}^{2}, \\ i n f {x, y} & otherwise . \end{matrix}$ (1) is an ordinal sum (< a, b, V >) of V on L.

Definition 4 can also be defined for t-conorm using duality. In Formula (1), the function T does not need to be a t-norm. Now, we will give an example only for t-norms because the example can be obtained for t-conorms from duality.

Example 1. Given a bounded lattice (L₁ = {0_{L
₁}, p, a, k, t, s, 1_{L
₁}} , ≤ , 0_{L
₁}, 1_{L
₁}) described by Fig. 1. Consider the t-norm V on [0_{L
₁}, a] as follows: $V (x, y) = {\begin{matrix} i n f {x, y} & i f a \in {x, y}, \\ 0_{L_{1}} & otherwise . \end{matrix}$ We see that the function T defined by Table 1 is not a t-norm on L₁. Clearly, T (a, T (t, p)) = T (a, p) = p ≠ 0_{L
₁} = T (p, p) = T (T (a, t) , p). So, it does not satisfy associativity.

Fig. 1

The lattice L₁.

Table 1

The function T on L₁

T	0_{L ₁}	p	a	k	t	s	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}
p	0_{L ₁}	0_{L ₁}	p	p	p	p	p
a	0_{L ₁}	p	a	a	p	p	a
k	0_{L ₁}	p	a	k	p	p	k
t	0_{L ₁}	p	p	p	t	t	t
s	0_{L ₁}	p	p	p	t	s	s
1_{L ₁}	0_{L ₁}	p	a	k	t	s	1_{L ₁}

In this section, we give a new way to construct t-norms and t-conorms on an arbitrary bounded lattice in Theorems 3 and 4, respectively, where a ∈ L \ {0, 1} and V is t-norm on [0, a] and W is t-conorm on [a, 1], respectively. Also, we give some examples to discuss introduced methods in Theorems 3, 4. Then, we investigate the relation between introduced methods in Theorems 3 and 4 and other methods proposed in [2, 8] and [19]. Also, we give some illustrative examples to well understand our approaches.

Theorem 3. 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) it holds x ∥ y, then the function T_V : L² → L defined as follows is a t-norm on L, where V is a t-norm on [0, a]. $T_{V} (x, y) = {\begin{matrix} V (x, y) & i f (x, y) \in {[0, a]}^{2}, \\ 0 & i f (x, y) \in [0, a) \times I_{a} \\ \cup I_{a} \times [0, a) \cup [a, 1) \times I_{a} \\ \cup I_{a} \times [a, 1), \\ a & i f (x, y) \in {[a, 1)}^{2}, \\ i n f {x, y} & otherwise . \end{matrix}$

Proof. We have T_V (x, 1) = inf {x, 1} = x. So, the fact that 1 ∈ L is a neutral element of T_V. It is easy to see commutativity of T_V.

i) Monotonicity: We prove that if x ≤ y, then T_V (x, z) ≤ T_V (y, z) for all z ∈ L. The proof can be split into all possible cases. If z = 1, then we have that T_V (x, z) = T_V (x, 1) = x ≤ y = T_V (y, 1) = T_V (y, z) for all x, y ∈ L. If x = 0, then we have that T_V (0, z) =0 ≤ T_V (y, z) for all y, z ∈ L. If z = 0, then we have that T_V (x, 0) =0 = T_V (y, 0) for all x, y ∈ L. If y = 0, then it is clear that monotonicity is held.

x ∈ (0, a)

y ∈ (0, a)

z ∈ (0, a) T_V (x, z) = V (x, z) ≤ V (y, z) = T_V (y, z)

z ∈ a, 1) T_V (x, z) = inf {x, z} ≤ inf {y, z} = T_V (y, z)

z ∈ I_a T_V (x, z) =0 = T_V (y, z)

y ∈ a, 1)

z ∈ (0, a) T_V (x, z) = V (x, z) ≤ z = T_V (y, z)

z ∈ a, 1) T_V (x, z) = x < a = T_V (y, z)

z ∈ I_a T_V (x, z) =0 = T_V (y, z)

y ∈ I_a.

Since x ∈ (0, a) and y ∈ I_a, then according to our constraint it must be x ∥ y. So, it can not be the case y ∈ I_a.

y = 1

z ∈ (0, a) T_V (x, z) = V (x, z) ≤ z = T_V (1, z)

z ∈ a, 1) T_V (x, z) = x < a ≤ z = T_V (1, z)

z ∈ I_a T_V (x, z) =0 ≤ T_V (1, z)

x ∈ a, 1).

Then, it must be the case that y ∈ a, 1].

y ∈ [a, 1)

z ∈ (0, a) T_V (x, z) = z = T_V (y, z)

z ∈ [a, 1) T_V (x, z) = a = T_V (y, z)

z ∈ I_a T_V (x, z) =0 = T_V (y, z)

y = 1

z ∈ (0, a) T_V (x, z) = z = T_V (1, z)

z ∈ [a, 1) T_V (x, z) = a ≤ z = T_V (1, z)

z ∈ I_a T_V (x, z) =0 ≤ z = T_V (1, z)

x ∈ I_a.

Then, it must be the case that y ∈ I_a or y ∈ (a, 1].

y ∈ I_a

z ∈ (0, a) or z ∈ [a, 1) T_V (x, z) =0 = T_V (y, z)

z ∈ I_a T_V (x, z) = inf {x, z} < inf {y, z} = T_V (y, z)

y ∈ (a, 1)

Since x ∈ I_a and y ∈ (a, 1), then according to our constraint it must be x ∥ y. So, it can not be the case y ∈ (a, 1).

y = 1

z ∈ (0, a) or z ∈ [a, 1) T_V (x, z) =0 ≤ z = T_V (1, z)

z ∈ I_a T_V (x, z) = inf {x, z} ≤ z = T_V (1, z)

x = 1. Then, since y = 1, it is clear that T_V (x, z) = T_V (y, z) for all z ∈ L.

ii) Associativity:

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

x ∈ [0, a)

y ∈ [0, a)

z ∈ [0, a) T_V (x, T_V (y, z)) = T_V (x, V (y, z)) = V (x, V (y, z))

= V (V (x, y) , z) = T_V (T_V (x, y) , z)

z ∈ [a, 1) T_V (x, T_V (y, z)) = T_V (x, y) = V (x, y) =

T_V (V (x, y) , z) = T_V (T_V (x, y) , z)

z ∈ I_a T_V (x, T_V (y, z)) = T_V (x, 0) =0 = T_V (V (x, y) , z)

= T_V (T_V (x, y) , z)

y ∈ [a, 1)

z ∈ [0, a) T_V (x, T_V (y, z)) = T_V (x, z) = T_V (T_V (x, y) , z)

z ∈ [a, 1) T_V (x, T_V (y, z)) = T_V (x, a) = x = T_V (x, z) =

T_V (T_V (x, y) , z)

z ∈ I_a T_V (x, T_V (y, z)) = T_V (x, 0) =0 = T_V (x, z) =

T_V (T_V (x, y) , z)

y ∈ I_a

z ∈ [0, a) or z ∈ [a, 1) T_V (x, T_V (y, z)) = T_V (x, 0) =0 = T_V (0, z) =

T_V (T_V (x, y) , z)

z ∈ I_a T_V (x, T_V (y, z)) = T_V (x, inf {y, z}) =0 = T_V (0, z)

= T_V (T_V (x, y) , z)

x ∈ [a, 1)

y ∈ [0, a)

z ∈ [0, a) T_V (x, T_V (y, z)) = T_V (x, V (y, z)) = V (y, z) =

T_V (y, z) = T_V (T_V (x, y) , z)

z ∈ [a, 1) T_V (x, T_V (y, z)) = T_V (x, y) = y =

T_V (y, z) = T_V (T_V (x, y) , z)

z ∈ I_a T_V (x, T_V (y, z)) = T_V (x, 0) =0 = T_V (y, z) =

T_V (T_V (x, y) , z)

y ∈ [a, 1)

z ∈ [0, a) T_V (x, T_V (y, z)) = T_V (x, z) = z = T_V (a, z) =

T_V (T_V (x, y) , z)

z ∈ [a, 1)

T_V (x, T_V (y, z)) = T_V (x, a) = a = T_V (a, z) =

T_V (T_V (x, y) , z)

z ∈ I_a T_V (x, T_V (y, z)) = T_V (x, 0) =0 = T_V (a, z) =

T_V (T_V (x, y) , z)

y ∈ I_a

z ∈ [0, a) or z ∈ [a, 1) T_V (x, T_V (y, z)) = T_V (x, 0) =0 = T_V (0, z) =

T_V (T_V (x, y) , z)

z ∈ I_a T_V (x, T_V (y, z)) = T_V (x, inf {y, z}) =0 =

T_V (0, z) = T_V (T_V (x, y) , z)

x ∈ I_a

y ∈ [0, a)

z ∈ [0, a) T_V (x, T_V (y, z)) = T_V (x, V (y, z)) =0 = T_V (0, z)

= T_V (T_V (x, y) , z)

z ∈ [a, 1) T_V (x, T_V (y, z)) = T_V (x, y) =0 = T_V (0, z) =

T_V (T_V (x, y) , z)

z ∈ I_a T_V (x, T_V (y, z)) = T_V (x, 0) =0 = T_V (0, z) =

T_V (T_V (x, y) , z)

y ∈ [a, 1)

z ∈ [0, a) T_V (x, T_V (y, z)) = T_V (x, z) =0 = T_V (0, z) =

T_V (T_V (x, y) , z)

z ∈ [a, 1)

T_V (x, T_V (y, z)) = T_V (x, a) =0 = T_V (0, z) = T_V (T_V (x, y) , z)

z ∈ I_a T_V (x, T_V (y, z)) = T_V (x, 0) =0 = T_V (0, z) = T_V (T_V (x, y) , z)

y ∈ I_a

z ∈ [0, a) or z ∈ [a, 1) T_V (x, T_V (y, z)) = T_V (x, 0) =0 = T_V (inf {x, y} , z) = T_V (T_V (x, y) , z)

z ∈ I_a T_V (x, T_V (y, z)) = T_V (x, inf {y, z}) =

inf {x, y, z} = T_V (inf {x, y} , z) =

T_V (T_V (x, y) , z)□

Corollary 1. If we take V = T_inf on [0, a] given in Theorem 3, then we obtain the following t-norm on L. $T_{V} (x, y) = {\begin{matrix} 0 & i f (x, y) \in [0, a) \times I_{a} \cup I_{a} \times [0, a) \cup [a, 1) \times I_{a} \cup I_{a} \times [a, 1), \\ a & i f (x, y) \in {[a, 1)}^{2}, \\ i n f {x, y} & otherwise . \end{matrix}$

Example 2. The lattice (L₂ = {0_{L
₂}, p, s, u, a, t, r, 1_{L
₂}} , ≤ , 0_{L
₂}, 1_{L
₂}) in Fig. 2. satisfies the constraints of Theorem 3 (for element a ∈ L₂). Consider the t-norm V on 0_{L
₂}, a] as follows: $V (x, y) = {\begin{matrix} i n f {x, y} & i f a \in {x, y}, \\ 0_{L_{2}} & otherwise . \end{matrix}$

Fig. 2

The lattice L₂.

Then, by using Theorem 3, the function T_V on L₂ defined by Table 2 is a t-norm.

Table 2

The t-norm T_V on L₂

T _V	0_{L ₂}	p	s	u	a	t	r	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}
p	0_{L ₂}	0_{L ₂}	0_{L ₂}	0_{L ₂}	p	0_{L ₂}	p	p
s	0_{L ₂}	0_{L ₂}	s	0_{L ₂}	0_{L ₂}	s	0_{L ₂}	s
u	0_{L ₂}	0_{L ₂}	0_{L ₂}	u	0_{L ₂}	0_{L ₂}	0_{L ₂}	u
a	0_{L ₂}	p	0_{L ₂}	0_{L ₂}	a	0_{L ₂}	a	a
t	0_{L ₂}	0_{L ₂}	s	0_{L ₂}	0_{L ₂}	t	0_{L ₂}	t
r	0_{L ₂}	p	0_{L ₂}	0_{L ₂}	a	0_{L ₂}	a	r
1_{L ₂}	0_{L ₂}	p	s	u	a	t	r	1_{L ₂}

Remark 1. Let (L, ≤ , 0, 1) be a bounded lattice and a ∈ L \ {0, 1}. In Theorem 3, observe that the condition for all x ∈ I_a and y ∈ (0, a] it holds x ∥ y can not be omitted, in general. The following example illustrates the fact that the function T_V : L² → L defined by Theorem 3 is not a t-norm.

Example 3. The lattice (L₃ = {0_{L
₃}, p, s, u, a, t, r, 1_{L
₃}} , ≤ , 0_{L
₃}, 1_{L
₃}) in Fig. 3. does not satisfy one of the constraints of Theorem 3. That is, there is the element p ∈ L₃ such that p < s for s ∈ I_a and p ∈ (0_{L
₃}, a). Consider the t-norm V on [0_{L
₃}, a] such that V (x, y) = inf {x, y}.

Fig. 3

The lattice L₃.

Then, the function T_V on L₃ defined by Table 3 is not a t-norm. Indeed, it does not satisfy monotonicity. Clearly, p < s and T_V (p, p) = p≰0_{L
₃} = T_V (s, p).

Table 3

The function T_V on L₃

T _V	0_{L ₃}	p	s	u	a	t	r	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}
p	0_{L ₃}	p	0_{L ₃}	0_{L ₃}	p	0_{L ₃}	p	p
s	0_{L ₃}	0_{L ₃}	s	p	0_{L ₃}	s	0_{L ₃}	s
u	0_{L ₃}	0_{L ₃}	p	u	0_{L ₃}	p	0_{L ₃}	u
a	0_{L ₃}	p	0_{L ₃}	0_{L ₃}	a	0_{L ₃}	a	a
t	0_{L ₃}	0_{L ₃}	s	p	0_{L ₃}	t	0_{L ₃}	t
r	0_{L ₃}	p	0_{L ₃}	0_{L ₃}	a	0_{L ₃}	a	r
1_{L ₃}	0_{L ₃}	p	s	u	a	t	r	1_{L ₃}

Remark 2. Let (L, ≤ , 0, 1) be a bounded lattice and a ∈ L \ {0, 1}. In Theorem 3, observe that the condition for all x ∈ I_a and y ∈ [a, 1) it holds x ∥ y can not be omitted, in general. The following example illustrates the fact that the function T_V : L² → L defined by Theorem 3 is not a t-norm.

Example 4. The lattice (L₄ = {0_{L
₄}, p, s, u, a, t, r, 1_{L
₄}} , ≤ , 0_{L
₄}, 1_{L
₄}) in Fig. 4. does not satisfy one of the constraints of Theorem 3. Namely, there is the element r ∈ L₄ such that t < r for t ∈ I_a and r ∈ (a, 1_{L
₄}). Consider the t-norm V on [0_{L
₃}, a] such that V (x, y) = inf {x, y}.

Fig. 4

The lattice L₄.

Then, the function T_V on L₄ defined by Table 4 is not a t-norm. Because, it does not satisfy monotonicity. Clearly, t < r and T_V (t, s) = s≰0_{L
₄} = T_V (r, s).

Table 4

The function T_V on L₄

T _V	0_{L ₄}	p	s	u	a	t	r	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}
p	0_{L ₄}	p	0_{L ₄}	0_{L ₄}	p	0_{L ₄}	p	p
s	0_{L ₄}	0_{L ₄}	s	0_{L ₄}	0_{L ₄}	s	0_{L ₄}	s
u	0_{L ₄}	0_{L ₄}	0_{L ₄}	u	0_{L ₄}	0_{L ₄}	0_{L ₄}	u
a	0_{L ₄}	p	0_{L ₄}	0_{L ₄}	a	0_{L ₄}	a	a
t	0_{L ₄}	0_{L ₄}	s	0_{L ₄}	0_{L ₄}	t	0_{L ₄}	t
r	0_{L ₄}	p	0_{L ₄}	0_{L ₄}	a	0_{L ₄}	a	r
1_{L ₄}	0_{L ₄}	p	s	u	a	t	r	1_{L ₄}

Proposition 1. 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) it holds x ∥ y. Suppose T_V and T are the t-norms on L defined as in Theorem 3 and Definition 4 with underlying t-norms V on [0, a], respectively. Then, T_V ≤ T.

Proof. Since T_V and T have the same t-norm V on [0, a], we need only to consider the cases such as on [0, a].

Let (x, y) ∈ [0, a) × I_a ∪ I_a × [0, a) ∪ [a, 1) × I_a ∪ I_a × [a, 1). Then, we have T_V (x, y) =0 ≤ inf {x, y} = T (x, y).

Let (x, y) ∈ [a, 1) ². Then, we have T_V (x, y) = a ≤ inf {x, y} = T (x, y).

For all other possible conditions, we have T_V (x, y) = inf {x, y} = T (x, y). □

Remark 3. Let’s examine the examples below to explain the differences between the two t-norms T_V and T defined in Theorem 3 and Definition 4, respectively.

Example 5. Consider the bounded lattice (L₄ = {0_{L
₄}, p, s, u, a, t, r, 1_{L
₄}} , ≤ , 0_{L
₄}, 1_{L
₄}) in Fig. 4. and consider the t-norm V : [0_{L
₄}, a] ² → [0_{L
₄}, a], V (x, y) = inf {x, y}. According to the Definition 4, the function T defined by Table 5 is a t-norm on L₄.

Table 5

The t-norm T on L₄

T	0_{L ₄}	p	s	u	a	t	r	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}
p	0_{L ₄}	p	0_{L ₄}	0_{L ₄}	p	0_{L ₄}	p	p
s	0_{L ₄}	0_{L ₄}	s	0_{L ₄}	0_{L ₄}	s	s	s
u	0_{L ₄}	0_{L ₄}	0_{L ₄}	u	0_{L ₄}	0_{L ₄}	u	u
a	0_{L ₄}	p	0_{L ₄}	0_{L ₄}	a	0_{L ₄}	a	a
t	0_{L ₄}	0_{L ₄}	s	0_{L ₄}	0_{L ₄}	t	t	t
r	0_{L ₄}	p	s	u	a	t	r	r
1_{L ₄}	0_{L ₄}	p	s	u	a	t	r	1_{L ₄}

On the other side, we proved that the function T_V defined by Table 4 in Example 4 is not a t-norm on L₄.

Remark 4. 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) it holds x ∥ y. Suppose T_V, T^∗ and T^∼ are the t-norms on L defined as in Theorem 3, Theorems 1 and 2 with underlying t-norms V and V_T on [0, a] and [a, 1], respectively. Then, we can not compare these t-norms for every bounded lattice. We can only compare these t-norms on any bounded lattice. Namely, it can be T_V < T^∗ or T^∗ < T_V or T_V = T^∗ on any special bounded lattice. Similarly, it can be T_V < T^∼ or T^∼ < T_V or T_V = T^∼ on any bounded lattice. To illustrate these claims we shall give the following examples.

Example 6. Consider the lattice (L₅ = {0_{L
₅}, m, n, k, a, q, 1_{L
₅}} , ≤ , 0_{L
₅}, 1_{L
₅}) in Fig. 5. Also, consider the t-norm V on [0_{L
₅}, a], V (x, y) = inf {x, y} and consider any t-norm on [a, 1_{L
₅}].

By using Theorem 3, Theorems 1 and 2 define the corresponding t-norms T_V, T^★ and T^∼ as given in Tables 6 and 7, respectively.

Fig. 5

The lattice L₅.

Table 6

The t-norm T_V on L₅

T _V	0_{L ₅}	m	n	k	a	q	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}
m	0_{L ₅}	m	0_{L ₅}	0_{L ₅}	0_{L ₅}	m	m
n	0_{L ₅}	0_{L ₅}	n	0_{L ₅}	n	0_{L ₅}	n
k	0_{L ₅}	0_{L ₅}	0_{L ₅}	k	0_{L ₅}	0_{L ₅}	k
a	0_{L ₅}	0_{L ₅}	n	0_{L ₅}	a	0_{L ₅}	a
q	0_{L ₅}	m	0_{L ₅}	0_{L ₅}	0_{L ₅}	q	q
1_{L ₅}	0_{L ₅}	m	n	k	a	q	1_{L ₅}

Table 7

The t-norm T^★ = T^∼ on L₅

T^★ = T^∼	0_{L ₅}	m	n	k	a	q	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}
m	0_{L ₅}	0_{L ₅}	0_{L ₅}	0_{L ₅}	0_{L ₅}	0_{L ₅}	m
n	0_{L ₅}	0_{L ₅}	n	0_{L ₅}	n	0_{L ₅}	n
k	0_{L ₅}	0_{L ₅}	0_{L ₅}	0_{L ₅}	0_{L ₅}	0_{L ₅}	k
a	0_{L ₅}	0_{L ₅}	n	0_{L ₅}	a	0_{L ₅}	a
q	0_{L ₅}	0_{L ₅}	0_{L ₅}	0_{L ₅}	0_{L ₅}	0_{L ₅}	q
1_{L ₅}	0_{L ₅}	m	n	k	a	q	1_{L ₅}

According to the Tables 6 and 7, it is clear that the T^★ = T^∼ < T_V on the bounded lattice L₅.

Example 7. Consider the lattice (L₆ = {0_{L
₆}, b, c, d, a, e, 1_{L
₆}} , ≤ , 0_{L
₆}, 1_{L
₆}) described in Fig. 6. Consider the t-norm V on [0_{L
₆}, a] defined as follows: $V (x, y) = {\begin{matrix} i n f {x, y} & i f a \in {x, y}, \\ 0_{L_{6}} & otherwise . \end{matrix}$ and the t-norm V_T on [a, 1_{L
₆}], V_T (x, y) = inf {x, y}.

Fig. 6

The lattice L₆.

By using Theorem 3, Theorems 1 and 2 define the corresponding t-norms T_V, T^★ and T^∼ as given in Table 8, Table 9 and Table 10, respectively.

Table 8

The t-norm T_V on L₆

T _V	0_{L ₆}	b	c	d	a	e	1_{L ₆}
0_{L ₆}	0_{L ₆}	0_{L ₆}	0_{L ₆}	0_{L ₆}	0_{L ₆}	0_{L ₆}	0_{L ₆}
b	0_{L ₆}	0_{L ₆}	0_{L ₆}	0_{L ₆}	b	b	b
c	0_{L ₆}	0_{L ₆}	0_{L ₆}	0_{L ₆}	c	c	c
d	0_{L ₆}	0_{L ₆}	0_{L ₆}	0_{L ₆}	d	d	d
a	0_{L ₆}	b	c	d	a	a	a
e	0_{L ₆}	b	c	d	a	a	e
1_{L ₆}	0_{L ₆}	b	c	d	a	e	1_{L ₆}

Table 9

The t-norm T^★ on L₆

T ^★	0_{L ₆}	b	c	d	a	e	1_{L ₆}
0_{L ₆}	0_{L ₆}	0_{L ₆}	0_{L ₆}	0_{L ₆}	0_{L ₆}	0_{L ₆}	0_{L ₆}
b	0_{L ₆}	b	0_{L ₆}	0_{L ₆}	b	b	b
c	0_{L ₆}	0_{L ₆}	c	0_{L ₆}	c	c	c
d	0_{L ₆}	0_{L ₆}	0_{L ₆}	d	d	d	d
a	0_{L ₆}	b	c	d	a	a	a
e	0_{L ₆}	b	c	d	a	e	e
1_{L ₆}	0_{L ₇}	b	c	d	a	e	1_{L ₆}

Table 10

The t-norm T^∼ on L₆

T ^∼	0_{L ₆}	b	c	d	a	e	1_{L ₆}
0_{L ₆}	0_{L ₆}	0_{L ₆}	0_{L ₆}	0_{L ₆}	0_{L ₆}	0_{L ₆}	0_{L ₆}
b	0_{L ₆}	0_{L ₆}	0_{L ₆}	0_{L ₆}	b	b	b
c	0_{L ₆}	0_{L ₆}	0_{L ₆}	0_{L ₆}	c	c	c
d	0_{L ₆}	0_{L ₆}	0_{L ₆}	0_{L ₆}	d	d	d
a	0_{L ₆}	b	c	d	a	a	a
e	0_{L ₆}	b	c	d	a	e	e
1_{L ₆}	0_{L ₆}	b	c	d	a	e	1_{L ₆}

According to the Tables 8, 9 and 10, it is clear that the T_V < T^★ and T_V < T^∼ on the bounded lattice L₆.

Remark 5. In Example 7, if we take the t-norm V on [0_{L
₆}, a] such that V (x, y) = inf {x, y} and the t-norm V_T on [a, 1_{L
₆}] such that $V_{T} (x, y) = {\begin{matrix} i n f {x, y} & i f 1_{L_{6}} \in {x, y}, \\ a & otherwise . \end{matrix}$ then we obtain T_V = T^★.

Example 8. Given a bounded lattice (L₂ = {0_{L
₂}, p, s, u, a, t, r, 1_{L
₂}} , ≤ , 0_{L
₂}, 1_{L
₂}) described by Fig. 2. Also we take the t-norm V on [0_{L
₂}, a] defined as follows: $V (x, y) = {\begin{matrix} i n f {x, y} & i f a \in {x, y}, \\ 0_{L_{2}} & otherwise . \end{matrix}$ and the t-norm V_T on [a, 1_{L
₂}], V_T (x, y) = inf {x, y}. By using Theorems 1 and 2 define the corresponding t-norms T^★ and T^∼ as given in Table 11 and Table 12, respectively. Then, neither T_V < T^★ nor T^★ ≤ T_V. Similarly, neither T_V < T^∼ nor T^∼ ≤ T_V.

Table 11

The t-norm T^★ on L₂

T ^★	0_{L ₂}	p	s	u	a	t	r	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}
p	0_{L ₂}	p	0_{L ₂}	0_{L ₂}	p	0_{L ₂}	p	p
s	0_{L ₂}	0_{L ₂}	0_{L ₂}	0_{L ₂}	0_{L ₂}	0_{L ₂}	0_{L ₂}	s
u	0_{L ₂}	0_{L ₂}	0_{L ₂}	0_{L ₂}	0_{L ₂}	0_{L ₂}	0_{L ₂}	u
a	0_{L ₂}	p	0_{L ₂}	0_{L ₂}	a	0_{L ₂}	a	a
t	0_{L ₂}	0_{L ₂}	0_{L ₂}	0_{L ₂}	0_{L ₂}	0_{L ₂}	0_{L ₂}	t
r	0_{L ₂}	p	0_{L ₂}	0_{L ₂}	a	0_{L ₂}	r	r
1_{L ₂}	0_{L ₂}	p	s	u	a	t	r	1_{L ₂}

Table 12

The t-norm T^∼ on L₂

T ^∼	0_{L ₂}	p	s	u	a	t	r	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}
p	0_{L ₂}	0_{L ₂}	0_{L ₂}	0_{L ₂}	p	0_{L ₂}	p	p
s	0_{L ₂}	0_{L ₂}	0_{L ₂}	0_{L ₂}	0_{L ₂}	0_{L ₂}	0_{L ₂}	s
u	0_{L ₂}	0_{L ₂}	0_{L ₂}	0_{L ₂}	0_{L ₂}	0_{L ₂}	0_{L ₂}	u
a	0_{L ₂}	p	0_{L ₂}	0_{L ₂}	a	0_{L ₂}	a	a
t	0_{L ₂}	0_{L ₂}	0_{L ₂}	0_{L ₂}	0_{L ₂}	0_{L ₂}	0_{L ₂}	t
r	0_{L ₂}	p	0_{L ₂}	0_{L ₂}	a	0_{L ₂}	r	r
1_{L ₂}	0_{L ₂}	p	s	u	a	t	r	1_{L ₂}

Theorem 4. 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] it holds x ∥ y, then the function S_W : L² → L defined as follows is a t-conorm on L, where W is a t-conorm on [a, 1]. $S_{W} (x, y) = {\begin{matrix} W (x, y) & i f (x, y) \in {[a, 1]}^{2}, \\ 1 & i f (x, y) \in (a, 1] \times I_{a} \cup I_{a} \times (a, 1] \cup (0, a] \times I_{a} \cup I_{a} \times (0, a], \\ a & i f (x, y) \in {(0, a]}^{2}, \\ s u p {x, y} & otherwise . \end{matrix}$

Corollary 2. If we take W = S_sup on [a, 1] given in Theorem 4, then we obtain the following t-conorm on L. $S_{W} (x, y) = {\begin{matrix} 1 & i f (x, y) \in (a, 1] \times I_{a} \cup I_{a} \times (a, 1] \cup (0, a] \times I_{a} \cup I_{a} \times (0, a], \\ a & i f (x, y) \in {(0, a]}^{2}, \\ s u p {x, y} & otherwise . \end{matrix}$

In the following, we provide three lattices L₇, L₈ and L₉ which satisfies and does not satisfy the constraints of Theorem 4, respectively.

Example 9. Consider the lattice (L₇ = {0_{L
₇}, b, c, t, a, d, s, 1_{L
₇}} , ≤ , 0_{L
₇}, 1_{L
₇}) in Fig. 7. satisfies the constraints of Theorem 4 (for element a ∈ L₇). Consider the t-conorm W : [a, 1_{L
₇}] ² → [a, 1_{L
₇}] as follows: $W (x, y) = {\begin{matrix} \sup {x, y} & i f a \in {x, y}, \\ 1_{L_{7}} & otherwise . \end{matrix}$

Fig. 7

The lattice L₇.

Then, the function S_W on L₇ defined by Table 13 is a t-conorm.

Table 13

The t-conorm S_W on L₇

S _W	0_{L ₇}	b	c	t	a	d	s	1_{L ₇}
0_{L ₇}	0_{L ₇}	b	c	t	a	d	s	1_{L ₇}
b	b	b	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}
c	c	1_{L ₇}	c	1_{L ₇}	1_{L ₇}	d	1_{L ₇}	1_{L ₇}
t	t	1_{L ₇}	1_{L ₇}	a	a	1_{L ₇}	s	1_{L ₇}
a	a	1_{L ₇}	1_{L ₇}	a	a	1_{L ₇}	s	1_{L ₇}
d	d	1_{L ₇}	d	1_{L ₇}	1_{L ₇}	d	1_{L ₇}	1_{L ₇}
s	s	1_{L ₇}	1_{L ₇}	s	s	1_{L ₇}	1_{L ₇}	1_{L ₇}
1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}

Example 10. Consider the lattice (L₈ = {0_{L
₈}, b, c, t, a, d, s, 1_{L
₈}} , ≤ , 0_{L
₈}, 1_{L
₈}) in Fig. 8. does not satisfy one of the constraints of Theorem 4. Namely, there is the element s ∈ L₈ such that d < s for d ∈ I_a and s ∈ (a, 1_{L
₈}). Consider the t-conorm W : [a, 1_{L
₈}] ² → [a, 1_{L
₈}], W (x, y) = sup {x, y}.

Fig. 8

The lattice L₈.

The function S_W on L₈ defined by Table 14 is not a t-conorm. Indeed, it does not hold monotonicity. Clearly, d < s and S_W (s, d) =1_{L
₈}≰s = S_W (s, s).

Table 14

The function S_W on L₈

S _W	0_{L ₈}	b	c	t	a	d	s	1_{L ₈}
0_{L ₈}	0_{L ₈}	b	c	t	a	d	s	1_{L ₈}
b	b	b	s	1_{L ₈}	1_{L ₈}	s	1_{L ₈}	1_{L ₈}
c	c	s	c	1_{L ₈}	1_{L ₈}	d	1_{L ₈}	1_{L ₈}
t	t	1_{L ₈}	1_{L ₈}	a	a	1_{L ₈}	s	1_{L ₈}
a	a	1_{L ₈}	1_{L ₈}	a	a	1_{L ₈}	s	1_{L ₈}
d	d	s	d	1_{L ₈}	1_{L ₈}	d	1_{L ₈}	1_{L ₈}
s	s	1_{L ₈}	1_{L ₈}	s	s	1_{L ₈}	s	1_{L ₈}
1_{L ₈}	1_{L ₈}	1_{L ₈}	1_{L ₈}	1_{L ₈}	1_{L ₈}	1_{L ₈}	1_{L ₈}	1_{L ₈}

Example 11. The lattice (L₉ = {0_{L
₉}, b, c, t, a, d, s, 1_{L
₉}} , ≤0_{L
₉}, 1_{L
₉}) in Fig. 9 does not satisfy one of the constraints of Theorem 4. Namely, there is the element t ∈ L₉ such that t < c for c ∈ I_a and t ∈ (0_{L
₉}, a). Consider the t-conorm W : [a, 1_{L
₉}] ² → [a, 1_{L
₉}], W (x, y) = sup {x, y}.

Fig. 9

The lattice L₉.

The function S_W on L₉ defined by Table 15 is not a t-conorm. Indeed, it does not satisfy monotonicity. Clearly, t < c and S_W (t, d) =1_{L
₉}≰d = S_W (c, d).

Table 15

The function S_W on L₉

S _W	0_{L ₉}	b	c	t	a	d	s	1_{L ₉}
0_{L ₉}	0_{L ₉}	b	c	t	a	d	s	1_{L ₉}
b	b	b	1_{L ₉}	1_{L ₉}	1_{L ₉}	1_{L ₉}	1_{L ₉}	1_{L ₉}
c	c	1_{L ₉}	c	1_{L ₉}	1_{L ₉}	d	1_{L ₉}	1_{L ₉}
t	t	1_{L ₉}	1_{L ₉}	a	a	1_{L ₉}	s	1_{L ₉}
a	a	1_{L ₉}	1_{L ₉}	a	a	1_{L ₉}	s	1_{L ₉}
d	d	1_{L ₉}	d	1_{L ₉}	1_{L ₉}	d	1_{L ₉}	1_{L ₉}
s	s	1_{L ₉}	1_{L ₉}	s	s	1_{L ₉}	s	1_{L ₉}
1_{L ₉}	1_{L ₉}	1_{L ₉}	1_{L ₉}	1_{L ₉}	1_{L ₉}	1_{L ₉}	1_{L ₉}	1_{L ₉}

Proposition 2. Consider the bounded lattice (L, ≤ , 0, 1) with a ∈ L \ {0, 1} such that for all x ∈ I_a and y ∈ [a, 1), and for all x ∈ I_a and y ∈ (0, a] it holds x ∥ y. Suppose S_W and S are the t-conorms on L defined as in Theorem 4 and Definition 4 with underlying t-conorms W on [a, 1], respectively. Then, S ≤ S_W.

Remark 6. Let’s examine the examples below to explain the differences between the two t-conorms S_W and S defined in Theorem 4 and Definition 4, respectively.

Example 12. Consider the bounded lattice (L₉ = {0_{L
₉}, b, c, t, a, d, s, 1_{L
₉}} , ≤ , 0_{L
₉}, 1_{L
₉}) in Fig. 9 and consider the t-conorm W : [a, 1_{L
₄}] ² → [a, 1_{L
₄}], W (x, y) = x ∨ y. According to the Definition 4, the function S defined by Table 16 is a t-conorm on L₉.

Table 16

The t-norm S on L₉

S	0_{L ₉}	b	c	t	a	d	s	1_{L ₉}
0_{L ₉}	0_{L ₉}	b	c	t	a	d	s	1_{L ₉}
b	b	b	1_{L ₉}	b	1_{L ₉}	1_{L ₉}	1_{L ₉}	1_{L ₉}
c	c	1_{L ₉}	c	c	1_{L ₉}	d	1_{L ₉}	1_{L ₉}
t	t	b	c	t	a	d	s	1_{L ₉}
a	a	1_{L ₉}	1_{L ₉}	a	a	1_{L ₉}	s	1_{L ₉}
d	d	1_{L ₉}	d	d	1_{L ₉}	d	1_{L ₉}	1_{L ₉}
s	s	1_{L ₉}	1_{L ₉}	s	s	1_{L ₉}	s	1_{L ₉}
1_{L ₉}	1_{L ₉}	1_{L ₉}	1_{L ₉}	1_{L ₉}	1_{L ₉}	1_{L ₉}	1_{L ₉}	1_{L ₉}

On the other side, we proved that the function S_W defined by Table 15 in Example 11 is not a t-conorm on L₉.

Remark 7. Consider the bounded lattice (L, ≤ , 0, 1) with a ∈ L \ {0, 1} such that for all x ∈ I_a and y ∈ [a, 1), and for all x ∈ I_a and y ∈ (0, a] it holds x ∥ y. Suppose S_W, S^∗ and S^∼ are the t-conorms on L defined as in Theorem 4, Theorems 1 and 2 with underlying t-conorms W and W_T on [a, 1] and [0, a], respectively. Then, we can not compare these t-conorms for every bounded lattices. We can only compare these t-conorms on any bounded lattice. Namely, it can be S_W < S^∗ or S^∗ < S_W or S_W = S^∗ on any special bounded lattice. Similarly, it can be S_W < S^∼ or S^∼ < S_W or S_W = S^∼ on any bounded lattice. To illustrate these claims we shall give the following examples.

Example 13. Consider the lattice (L₁₀ = {0_{L
₁₀}, k, a, m, n, u, 1_{L
₁₀}} , ≤ , 0_{L
₁₀}, 1_{L
₁₀}) in Fig. 10. Consider the t-conorm W on [a, 1_{L
₁₀}], W (x, y) = sup {x, y} and consider any t-conorm on [0_{L
₁₀}, a].

Fig. 10

The lattice L₁₀.

By using Theorem 4, Theorems 1 and 2 define the corresponding t-conorms S_W, S^★ and S^∼ as given in Tables 17 and 18, respectively.

Table 17

The t-conorm S_W on L₁₀

S _W	0_{L ₁₀}	k	a	m	n	u	1_{L ₁₀}
0_{L ₁₀}	0_{L ₁₀}	k	a	m	n	u	1_{L ₁₀}
k	k	k	1_{L ₁₀}	1_{L ₁₀}	1_{L ₁₀}	1_{L ₁₀}	1_{L ₁₀}
a	a	1_{L ₁₀}	a	1_{L ₁₀}	1_{L ₁₀}	u	1_{L ₁₀}
m	m	1_{L ₁₀}	1_{L ₁₀}	m	n	1_{L ₁₀}	1_{L ₁₀}
n	n	1_{L ₁₀}	1_{L ₁₀}	n	n	1_{L ₁₀}	1_{L ₁₀}
u	u	1_{L ₁₀}	u	1_{L ₁₀}	1_{L ₁₀}	u	1_{L ₁₀}
1_{L ₁₀}	1_{L ₁₀}	1_{L ₁₀}	1_{L ₁₀}	1_{L ₁₀}	1_{L ₁₀}	1_{L ₁₀}	1_{L ₁₀}

Table 18

The t-conorm S^★ = S^∼ on L₁₀

S^★ = S^∼	0_{L ₁₀}	k	a	m	n	u	1_{L ₁₀}
0_{L ₁₀}	0_{L ₁₀}	k	a	m	n	u	1_{L ₁₀}
k	k	1_{L ₁₀}	1_{L ₁₀}	1_{L ₁₀}	1_{L ₁₀}	1_{L ₁₀}	1_{L ₁₀}
a	a	1_{L ₁₀}	a	1_{L ₁₀}	1_{L ₁₀}	u	1_{L ₁₀}
m	m	1_{L ₁₀}	1_{L ₁₀}	1_{L ₁₀}	1_{L ₁₀}	1_{L ₁₀}	1_{L ₁₀}
n	n	1_{L ₁₀}	1_{L ₁₀}	1_{L ₁₀}	1_{L ₁₀}	1_{L ₁₀}	1_{L ₁₀}
u	u	1_{L ₁₀}	u	1_{L ₁₀}	1_{L ₁₀}	u	1_{L ₁₀}
1_{L ₁₀}	1_{L ₁₀}	1_{L ₁₀}	1_{L ₁₀}	1_{L ₁₀}	1_{L ₁₀}	1_{L ₁₀}	1_{L ₁₀}

According to the Table 17 and Table 18, it is clear that the S_W < S^★ = S^∼ on the bounded lattice L₁₀.

Example 14. Given a bounded lattice (L₁₁ = {0_{L
₁₁}, p, a, s, r, t, 1_{L
₁₁}} , ≤ , 0_{L
₁₁}, 1_{L
₁₁}) in Fig. 11. Consider the t-conorm W on [a, 1_{L
₁₁}] defined as follows: $W (x, y) = {\begin{matrix} \sup {x, y} & i f a \in {x, y}, \\ 1_{L_{11}} & otherwise . \end{matrix}$ and the t-conorm W_S on [0_{L
₁₁}, a], W_S (x, y) = sup {x, y}.

Fig. 11

The lattice L₁₁.

By using Theorem 4, Theorems 1 and 2 define the corresponding t-conorms S_W, S^★ and S^∼ as given in Tables 19, 20 and 21, respectively.

Table 19

The t-conorm S_W on L₁₁

S _W	0_{L ₁₁}	p	a	s	r	t	1_{L ₁₁}
0_{L ₁₁}	0_{L ₁₁}	p	a	s	r	t	1_{L ₁₁}
p	p	a	a	s	r	t	1_{L ₁₁}
a	a	a	a	s	r	t	1_{L ₁₁}
s	s	s	s	1_{L ₁₁}	1_{L ₁₁}	1_{L ₁₁}	1_{L ₁₁}
r	r	r	r	1_{L ₁₁}	1_{L ₁₁}	1_{L ₁₁}	1_{L ₁₁}
t	t	t	t	1_{L ₁₁}	1_{L ₁₁}	1_{L ₁₁}	1_{L ₁₁}
1_{L ₁₁}	1_{L ₁₁}	1_{L ₁₁}	1_{L ₁₁}	1_{L ₁₁}	1_{L ₁₁}	1_{L ₁₁}	1_{L ₁₁}

Table 20

The t-conorm S^★ on L₁₁

S ^★	0_{L ₁₁}	p	a	s	r	t	1_{L ₁₁}
0_{L ₁₁}	0_{L ₁₁}	p	a	s	r	t	1_{L ₁₁}
p	p	p	a	s	r	t	1_{L ₁₁}
a	a	a	a	s	r	t	1_{L ₁₁}
s	s	s	s	s	1_{L ₁₁}	1_{L ₁₁}	1_{L ₁₁}
r	r	r	r	1_{L ₁₁}	r	1_{L ₁₁}	1_{L ₁₁}
t	t	t	t	1_{L ₁₁}	1_{L ₁₁}	t	1_{L ₁₁}
1_{L ₁₁}	1_{L ₁₁}	1_{L ₁₁}	1_{L ₁₁}	1_{L ₁₁}	1_{L ₁₁}	1_{L ₁₁}	1_{L ₁₁}

Table 21

The t-norm S^∼ on L₁₁

S ^∼	0_{L ₁₁}	p	a	s	r	t	1_{L ₁₁}
0_{L ₁₁}	0_{L ₁₁}	p	a	s	r	t	1_{L ₁₁}
p	p	p	a	s	r	t	1_{L ₁₁}
a	a	a	a	s	r	t	1_{L ₁₁}
s	s	s	s	1_{L ₁₁}	1_{L ₁₁}	1_{L ₁₁}	1_{L ₁₁}
r	r	r	r	1_{L ₁₁}	1_{L ₁₁}	1_{L ₁₁}	1_{L ₁₁}
t	t	t	t	1_{L ₁₁}	1_{L ₁₁}	1_{L ₁₁}	1_{L ₁₁}
1_{L ₁₁}	1_{L ₁₁}	1_{L ₁₁}	1_{L ₁₁}	1_{L ₁₁}	1_{L ₁₁}	1_{L ₁₁}	1_{L ₁₁}

According to the Tables 19, 20 and 21, it is clear that the S^★ < S_W and S^∼ < S_W on the bounded lattice L₁₁.

Remark 8. In Example 14, if we take the t-conorm W on [a, 1_{L
₁₁}] such that W (x, y) = sup {x, y} and the t-conorm W_T on [0_{L
₁₁}, a] such that $W_{T} (x, y) = {\begin{matrix} \sup {x, y} & i f a \in {x, y}, \\ 1_{L_{11}} & otherwise . \end{matrix}$ then we obtain S_W = S^★.

Example 15. Consider the lattice (L₇ = {0_{L
₇}, b, c, t, a, d, s, 1_{L
₇}} , ≤ , 0_{L
₇}, 1_{L
₇}) in Fig. 7. Consider the t-conorm W on [a, 1_{L
₇}] defined as follows: $W (x, y) = {\begin{matrix} \sup {x, y} & i f a \in {x, y}, \\ 1_{L_{7}} & otherwise . \end{matrix}$ and the t-conorm W_T on [0_{L
₇}, a], W_T (x, y) = sup {x, y}. By using Theorems 1 and 2 define the corresponding t-conorms S^★ and S^∼ as given in Table 22 and Table 23, respectively. Then, neither S_W < S^★ nor S^★ ≤ S_W. Similarly, neither S_W < S^∼ nor S^∼ ≤ S_W.

Table 22

The t-norm S^★ on L₇

S ^★	0_{L ₇}	b	c	t	a	d	s	1_{L ₇}
0_{L ₇}	0_{L ₇}	b	c	t	a	d	s	1_{L ₇}
b	b	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}
c	c	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}
t	t	1_{L ₇}	1_{L ₇}	t	a	1_{L ₇}	s	1_{L ₇}
a	a	1_{L ₇}	1_{L ₇}	a	a	1_{L ₇}	s	1_{L ₇}
d	d	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}
s	s	1_{L ₇}	1_{L ₇}	s	s	1_{L ₇}	s	1_{L ₇}
1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}

Table 23

The t-norm S^∼ on L₇

S ^∼	0_{L ₇}	b	c	t	a	d	s	1_{L ₇}
0_{L ₇}	0_{L ₇}	b	c	t	a	d	s	1_{L ₇}
b	b	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}
c	c	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}
t	t	1_{L ₇}	1_{L ₇}	t	a	1_{L ₇}	s	1_{L ₇}
a	a	1_{L ₇}	1_{L ₇}	a	a	1_{L ₇}	s	1_{L ₇}
d	d	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}
s	s	1_{L ₇}	1_{L ₇}	s	s	1_{L ₇}	1_{L ₇}	1_{L ₇}
1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}	1_{L ₇}

4 Modified ordinal sum constructions of t-norms and t-conorms on appropriate bounded lattices

We know that new t-norms and t-conorms on bounded lattices can be obtained using recursion. In this section, based on the approaches of constructing t-norms and t-conorms proposed in Section 3, we introduce new ordinal sum constructions of t-norms and t-conorms on an arbitrary bounded lattice L using recursion.

Theorem 5. Consider the bounded lattice (L, ≤ , 0, 1) and suppose that there exist a₁, a₂, . . . , a_n-1 ∈ L \ {0, 1} such that {a₁, a₂, ⋯ , a_n-1} be a chain. If we take a₀ = 0 and a_n = 1, then we obtain 0 = a₀ < a₁ < a₂ < ⋯ a_n-1 < a_n = 1. Also, 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) and we take the t-norm V on [0, a₁]. Then, the function T_n on L defined as the following is a t-norm, such that T₁ = V and for i ∈ {2, ⋯ , n}, the function T_i : [0, a_i] ² → [0, a_i] is given by $T_{i} (x, y) = {\begin{matrix} T_{i - 1} (x, y) & i f (x, y) \in {[0, a_{i - 1}]}^{2}, \\ 0 & i f (x, y) \in [0, a_{i - 1}) \times I_{a_{i - 1}} \cup I_{a_{i - 1}} \times [0, a_{i - 1}) \cup [a_{i - 1}, a_{i}) \times I_{a_{i - 1}} \cup I_{a_{i - 1}} \times [a_{i - 1}, a_{i}) \\ a_{i - 1} & i f (x, y) \in {[a_{i - 1}, a_{i})}^{2} \\ i n f {x, y} & otherwise . \end{matrix}$ (2)

We omit this proof because of the proof of Theorem 3.

If we put L is a chain rthen we can rewrite the Formula (2) as the following: $T_{i} (x, y) = {\begin{matrix} T_{i - 1} (x, y) & i f (x, y) \in {[0, a_{i - 1}]}^{2}, \\ a_{i - 1} & i f (x, y) \in {[a_{i - 1}, 1)}^{2}, \\ i n f {x, y} & otherwise . \end{matrix}$

Example 16. Consider the lattice (L₁₂ = {0_{L
₁₂}, a₁, a₂, d, a₃, c, 1_{L
₁₂}} , ≤ , 0_{L
₁₂}, 1_{L
₁₂}) described in Fig. 12. with the finite chain 0_{L
₁₂} < a₁ < a₂ < a₃ < 1_{L
₁₂} in L₁₂ and define the t-norm V : [0_{L
₁₂}, a₁] ² → [0_{L
₁₂}, a₁] by V = T_inf. By using Theorem 5, where V = T₁, t-norms T₂ : [0_{L
₁₂}, a₂] ² → [0_{L
₁₂}, a₂], T₃ : [0_{L
₁₂}, a₃] ² → [0_{L
₁₂}, a₃], T₄ : L₁₂² → L₁₂ are defined in Tables 24–26.

Fig. 12

The lattice L₁₂.

Table 24

The t-norm T₂ on L₁₂

T ₂	0_{L ₁₂}	a ₁	a ₂
0_{L ₁₂}	0_{L ₁₂}	0_{L ₁₂}	0_{L ₁₂}
a ₁	0_{L ₁₂}	a ₁	a ₁
a ₂	0_{L ₁₂}	a ₁	a ₂

Table 25

The t-norm T₃ on L₁₂

T ₃	0_{L ₁₂}	a ₁	a ₂	d	a ₃
0_{L ₁₂}	0_{L ₁₂}	0_{L ₁₂}	0_{L ₁₂}	0_{L ₁₂}	0_{L ₁₂}
a ₁	0_{L ₁₂}	a ₁	a ₁	0_{L ₁₂}	a ₁
a ₂	0_{L ₁₂}	a ₁	a ₂	0_{L ₁₂}	a ₂
d	0_{L ₁₂}	0_{L ₁₂}	0_{L ₁₂}	d	d
a ₃	0_{L ₁₂}	a ₁	a ₂	d	a ₃

Table 26

The t-norm T₄ on L₁₂

T ₄	0_{L ₁₂}	a ₁	a ₂	d	a ₃	c	1_{L ₁₂}
0_{L ₁₂}	0_{L ₁₂}	0_{L ₁₂}	0_{L ₁₂}	0_{L ₁₂}	0_{L ₁₂}	0_{L ₁₂}	0_{L ₁₂}
a ₁	0_{L ₁₂}	a ₁	a ₁	0_{L ₁₂}	a ₁	0_{L ₁₂}	a ₁
a ₂	0_{L ₁₂}	a ₁	a ₂	0_{L ₁₂}	a ₂	0_{L ₁₂}	a ₂
d	0_{L ₁₂}	0_{L ₁₂}	0_{L ₁₂}	d	d	0_{L ₁₂}	d
a ₃	0_{L ₁₂}	a ₁	a ₂	d	a ₃	0_{L ₁₂}	a ₃
c	0_{L ₁₂}	0_{L ₁₂}	0_{L ₁₂}	0_{L ₁₂}	0_{L ₁₂}	c	c
1_{L ₁₂}	0_{L ₁₂}	a ₁	a ₂	d	a ₃	c	1_{L ₁₂}

Theorem 6. Consider the bounded lattice (L, ≤ , 0, 1) and suppose that there exist a₁, a₂, . . . , a_n-1 ∈ L \ {0, 1} such that {a₁, a₂, ⋯ , a_n-1} be a chain. If we take a₀ = 0 and a_n = 1, then we obtain 1 = a₀ > a₁ > a₂ > . . . a_n-1 > 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], and we put the t-conorm W on [a₁, 1]. Then, the function S_n : L² → L defined as the following is a t-conorm, such that S₁ = W and for i ∈ {2, . . . , n}, the function S_i : [a_i, 1] ² → [a_i, 1] is given by $S_{i} (x, y) = {\begin{matrix} S_{i - 1} (x, y) & i f (x, y) \in {[a_{i - 1}, 1]}^{2}, \\ 1 & i f (x, y) \in (a_{i - 1}, 1] \times I_{a_{i - 1}} \cup I_{a_{i - 1}} \times (a_{i - 1}, 1] \cup (a_{i}, a_{i - 1}] \times I_{a_{i - 1}} \cup I_{a_{i - 1}} \times (a_{i}, a_{i - 1}] \\ a_{i - 1} & i f (x, y) \in {(a_{i - 1}, a_{i}]}^{2} \\ s u p {x, y} & otherwise . \end{matrix}$ (3)

If we put L is a chain then, we can rewrite the Formula (3) as the following: $S_{i} (x, y) = {\begin{matrix} S_{i - 1} (x, y) & i f (x, y) \in {[a_{i - 1}, 1]}^{2}, \\ a_{i - 1} & i f (x, y) \in {(0, a_{i - 1}]}^{2}, \\ s u p {x, y} & otherwise . \end{matrix}$

Example 17. Consider the lattice (L₁₃ = {0_{L
₁₃}, t, a₃, a₂, k, a₁, 1_{L
₁₃}} , ≤ , 0_{L
₁₃}, 1_{L
₁₃}) described in Fig. 13. with the finite chain 1_{L
₁₃} > a₁ > a₂ > a₃ > 0_{L
₁₃} in L₁₃ and define the t-conorm W : [a₁, 1_{L
₁₃}] ² → [a₁, 1_{L
₁₃}] by W = S_sup. By using Theorem 6, where W = S₁, t-conorms S₂ : [a₂, 1_{L
₁₃}] ² → [a₂, 1_{L
₁₃}], S₃ : [a₃, 1_{L
₁₃}] ² → [a₃, 1_{L
₁₃}], $S_{4} : L_{13}^{2} \to L_{13}$ are defined in Tables 27-29.

Fig. 13

The lattice L₁₃.

Table 27

The t-conorm S₂ on L₁₃

S ₂	a ₂	a ₁	1_{L ₁₃}
a ₂	a ₂	a ₁	1_{L ₁₃}
a ₁	a ₁	a ₁	1_{L ₁₃}
1_{L ₁₃}	1_{L ₁₃}	1_{L ₁₃}	1_{L ₁₃}

Table 28

The t-conorm S₃ on L₁₃

S ₃	a ₃	a ₂	k	a ₁	1_{L ₁₃}
a ₃	a ₃	a ₂	k	a ₁	1_{L ₁₃}
a ₂	a ₂	a ₂	1_{L ₁₃}	a ₁	1_{L ₁₃}
k	k	1_{L ₁₃}	k	1_{L ₁₃}	1_{L ₁₃}
a ₁	a ₁	a ₁	1_{L ₁₃}	a ₁	1_{L ₁₃}
1_{L ₁₃}	1_{L ₁₃}	1_{L ₁₃}	1_{L ₁₃}	1_{L ₁₃}	1_{L ₁₃}

Table 29

The t-conorm S₄ on L₁₃

S ₄	0_{L ₁₃}	t	a ₃	a ₂	k	a ₁	1_{L ₁₃}
0_{L ₁₃}	0_{L ₁₃}	t	a ₃	a ₂	k	a ₁	1_{L ₁₃}
t	t	t	1_{L ₁₃}	1_{L ₁₃}	1_{L ₁₃}	1_{L ₁₃}	1_{L ₁₃}
a ₃	a ₃	1_{L ₁₃}	a ₃	a ₂	k	a ₁	1_{L ₁₃}
a ₂	a ₂	1_{L ₁₃}	a ₂	a ₂	1_{L ₁₃}	a ₁	1_{L ₁₃}
k	k	1_{L ₁₃}	k	1_{L ₁₃}	k	1_{L ₁₃}	1_{L ₁₃}
a ₁	a ₁	1_{L ₁₃}	a ₁	a ₁	1_{L ₁₃}	a ₁	1_{L ₁₃}
1_{L ₁₃}	1_{L ₁₃}	1_{L ₁₃}	1_{L ₁₃}	1_{L ₁₃}	1_{L ₁₃}	1_{L ₁₃}	1_{L ₁₃}

5 Concluding remarks

A t-norm is a kind of binary operation used in the framework of probabilistic metric spaces and in multi-valued logic, specifically in fuzzy logic. A t-norm generalizes intersection in a lattice and conjunction in logic. The name t-norm refers to the fact that in the framework of probabilistic metric spaces t-norms are used to generalize triangle inequality of ordinary metric spaces. T-norms are a generalization of the usual two-valued logical conjunction, studied by classical logic, for fuzzy logics. Indeed, the classical Boolean conjunction is both commutative and associative. The monotonicity property ensures that the degree of truth of conjunction does not decrease if the truth values of conjuncts increase. The requirement that 1 be an identity element corresponds to the interpretation of 1 as true (and consequently 0 as false). Continuity, which is often required from fuzzy conjunction as well, expresses the idea that very small changes in truth values of conjuncts should not macroscopically affect the truth value of their conjunction. T-norms are also used to construct the intersection of fuzzy sets or as a basis for aggregation operators (see fuzzy set operations). In probabilistic metric spaces, t-norms are used to generalize triangle inequality of ordinary metric spaces. Individual t-norms may of course frequently occur in further disciplines of mathematics, since the class contains many familiar functions. T-norms have numerous applications in different field, such as fuzzy logics, expert systems, decision making, image processing, etc. One of the earliest applications of idempotent t-norms appeared in the framework of decision making in a fuzzy environment, see [11], see also [10]. In bipolar decision making, Grabisch [12] has introduced and applied a particular t-norm. Luo and Wang [15] have presented the interval-valued fuzzy reasoning triple I algorithms based on left-continuous interval-valued t-representable t-norms.

Our results contribute to the theory of aggregation on bounded lattices. T-norms on bounded lattices have been studied widely. In particular, the construction of t-norms related to algebraic structures on bounded lattices is still an active research area. In this paper, we have continued to study the topic of t-norms on bounded lattices from a mathematical point of view.

In this aim, we have investigated and introduced new construction methods for building t-norms and t-conorms on an appropriate bounded lattice with a constraint concerning the specific splitting point a. Based on this ordinal sum method, we have introduced a new class of t-norms T and t-conorms S on an arbitrary bounded lattice, respectively, 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 3 and 4. In order to well understand the constructed t-norms T and t-conorms S, we have given some illustrative examples. Also, in Proposition 1, Example 5, Remark 7, Example 6, Example 7, Example 8 Proposition 2, Example 12, Remark 7, Example 13, Example 14, Example 15 we have investigated the relation between introduced methods Theorems 3, 4 and other approaches proposed in [2 , 19]. Finally, we have shown that the new construction methods can be generalized by induction to a modified ordinal sum for t-norms and t-conorms on arbitrary bounded lattice, respectively.

Though our results are purely theoretical, we expect possible application in several domains dealing with lattice-valued scales. Up to the logics dealing with truth values forming a bounded lattice, where the introduced t-norms and t-conorms can play the role of conjunction or disjunction, we should stress the domain of qualitative decision theory. For example, our results can help to advance the resent paper [9, 15] dealing with some basic t-norms to evaluate decisions under incomplete information. Similarly, lattice-valued integrals where t-norms play the role of multiplication could be a promising field for application of our results.

Footnotes

Acknowledgment

We are grateful to the anonymous reviewers and editors for their valuable comments, which helped to improve the original version of our manuscript greatly.

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