Elsevier

Information Sciences

Volume 490, July 2019, Pages 191-209
Information Sciences

Semi-t-operators on bounded lattices

https://doi.org/10.1016/j.ins.2019.03.077Get rights and content

Abstract

We study semi-t-operators on bounded lattices and show the existence of semi-t-operators using the fact that there exist some pseudo-t-norms and pseudo-t-conorms on the arbitrary fixed bounded lattice L. As a by-product some constructive approaches to semi-t-operators are discussed. Using these approaches, we obtain the least semi-t-operators and the greatest semi-t-operators on bounded lattices. A partial order relation is also defined based on the semi-t-operator and some properties of this relation are examined.

Introduction

Aggregation functions are widely used in many theoretical and practical areas. A high level of attention in this field is directed towards construction and characterization of various aggregation operators [5], [7], [13], [18], [21], [23], [24], [25], [34], [38] and order relations induced by relative aggregation operators [1], [2], [15], [20], [22], [27], [28]. In recent years, the focus of investigations is on t-norms and t-conorms [5], [10], [14], [16], [19], [28], uninorms and nullnorms [1], [6], [8], [15], [25], [27].

The characterization of t-norms and t-conorms on boundary lattice has been fully studied in different forms [5], [14], [29], [40]. Different methods of constructing uninorms on bounded lattices were proposed in [7], [23], [25]. Çaylı and Drygás [8] showed some properties of idempotent uninorms on a special class of bounded lattices. Nullnorms were generalized to any arbitrary bounded lattices by Karaçal, Ince and Mesiar [21], [34]. Meanwhile, Çaylı and Karaçal [6] gave a construction method of constructing idempotent nullnorms on bounded lattices. Two different methods to obtain nullnorms on bounded lattices were shown by Ertuǧrul in [13]. Moreover, an equivalence relation based on a class of nullnorms was defined. Wang, Zhan and Liu [41] illustrated the existence of uni-nullnorms on bounded lattices and proposed two different methods to obtain a uni-nullnorm.

Nullnorms [4] and t-operators [31] are both generalizations of t-norms and t-conorms. It has been pointed out that they are equivalent in [32] because they have the same block structures in [0, 1]2. Namely, if an operation F is a t-operator then it is also a nullnorm and vice versa. However, in some cases, commutativity is not desired property for the aggregation operators, such as when the inputs do not have the same weight. Recently, Drygaś introduced the concept of semi-t-operators [9] by eliminating commutative law from the axiom of t-operators. A semi-t-operator F has different block structures in [0, 1]2 with semi-nullnorms (eliminating commutative law from the axiom of nullnorms), when F(0,1)=aF(1,0)=b. Nowadays, a lot of attention is focused on semi-t-operators [9], [11], [12], [17], [35], [36], [37], [38], [39], [42] from a theoretical point of view.

In this paper, we study semi-t-operators on bounded lattices. By using the existence of some pseudo-t-norms and pseudo-t-conorms on an arbitrary bounded lattice L, we introduce semi-t-operators F on L with F(0,1)=a,F(1,0)=b. With the partial order relation on bounded lattice L, there exists a situation that elements are incomparable, which is different from L=[0,1]. So the existence of a semi-t-operator on bounded lattice is not known yet in the literature, at least according to the best of our knowledge. The main aim of this paper is to fill this gap. We will consider three cases for constructing semi-t-operators: a ≤ b, b ≤ a and a, b are incomparable. As a by-product, the least (resp. greatest) semi-t-operators in the first two situations are obtained. Moreover, we study the relation induced by a semi-t-operator and its properties. In the real world, there exist data with the partial order relation, for instance interval-valued data. However, there has been no comparative analysis of the properties that hold for semi-t-operators based on partial order relation. This suggests that semi-t-operators on bounded lattices should be studied more thoroughly in this context. Following on from this, we apply our results to interval mathematics theory.

The remainder of this paper is organized as follows: Section 2 reviews briefly related basic concepts and proposes our motivations. Section 3 introduces the semi-t-operator on bounded lattices and its properties. Section 4 constructs the semi-t-operators with underlying pseudo-t-norms and pseudo-t-conorms. Section 5 studies the relation induced by a semi-t-operator and its properties. Section 6 applies to the semi-t-operators on interval-valued lattice. Finally, Section 7 concludes the paper.

Section snippets

Associative, monotonic binary operators on bounded lattices

We start to give some basic definitions and facts on t-norms, t-conorms and their extensions, such as pseudo-t-norm, pseudo-t-conorm, nullnorm, semi-nullnorm, t-operator, semi-t-operator and so on.

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

Definition 2.1

([3]) Given a bounded lattice (L,  ≤ , 0L, 1L), and a, b ∈ L, if a and b are incomparable, in this

Semi-t-operators on bounded lattices

As we know, the structure of a t-operator is same as the structure of a nullnorm on [0,1], which can be generalized on bounded lattices. There exist semi-nullnorms on bounded lattices, which can be defined as follows.

Definition 3.1

Let (L,  ≤ , 0L, 1L) be a bounded lattice. The operation VL: L2 → L is called a semi-nullnorm if it is associative, increasing, and has a zero element k ∈ L that satisfiesVL(0L,x)=V(x,0L)=x,forallxk,VL(1L,x)=V(x,1L)=x,forallxk.

Next, the existence of a semi-t-operator on bounded

Semi-t-operators with underlying pseudo-t-norms and pseudo-t-conorms

In this section, we construct semi-t-operators on arbitrary bounded lattices in three cases for a, b ∈ L∖{0, 1}: a ≤ b, b ≤ a and ab.

We denote FL1×L2 as a restriction of semi-t-operator F on L1 × L2, where L1, L2L.

An order induced by semi-t-operators on bounded lattices

Recently, the generating problem of order from aggregation operators and the properties have been studied in [1], [2], [15], [20], [22], [27], [28], [33]. As an extension of ordering based on nullnorm, the ordering based on semi-t-operator on bounded lattice is defined as follows.

Definition 5.1

Let (L,  ≤ , 0L, 1L) be a bounded lattice and FFa,bL. Define the relation ⪯F, ∀x, y ∈ L, we say xFy if and only if one of the following statements hold.

  • (i)

    x, y ∈ [0L, ab] and there exists k[0L,ab]suchthatF(x,k)=y.

  • (ii)

    x, y

Application to semi-t-operators on L([0, 1])

Let L=L([0,1])={[a,b]0ab1}, [a, b] ≤ L[c, d]: ⇔a ≤ c and b ≤ d, and [a, b] < L[c, d]: ⇔[a, b] ≤ L[c, d] and [a, b] ≠ [c, d]. It is easy to verify that (L,  ≤ L) is a complete lattice. The following results can be derived from the method proposed in [23], [33], considering representable aggregation function on L([0, 1]).

We denote a¯=[a,a] for all a ∈ [0, 1].

Definition 6.1

A semi-t-operator F on L([0, 1]) is called representable if and only if there exist two semi-t-operators F1 and F2 on [0,1] with F1 ≤ F2

Conclusion

In this work, we studied the semi-t-operators on bounded lattices from the theoretical point of view. We showed the existence of the semi-t-operator FFa,bL on an arbitrary given bounded lattice L in three cases: (i) a ≤ b, (ii) a ≥ b and (iii) ab. Constructive approaches to semi-t-operators on bounded lattices were studied. As a by-product, the least (greatest) semi-t-operators in the first two cases were obtained. Moreover, we introduced a partial order relation based on a semi-t-operator.

Acknowledgments

The authors would like to express their sincere thanks to the Editors and anonymous reviewers for their most valuable comments and suggestions in improving this paper greatly. This research was supported by the National Natural Science Foundation of China (Grant nos. 11571010 and 61179038).

References (42)

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