A new structure for uninorms on bounded lattices
Introduction
Uninorms [2], [11], [19], [21], [22], [23], [26], [28], [32], [33] with the neutral element e on the unit interval, as an important generalization of triangular norms (t-norms) and triangular conorms (t-conorms) [18], [22], play an important role in lots of fields, such as in fuzzy logic [1], fuzzy set theory, and fuzzy system modeling [34]. In contrast to t-norms and t-conorms, uninorms allow the neutral element e to lie anywhere in the unit interval. In particular, a uninorm U is a t-norm T when and a t-conorm S when . For uninorms with the neutral element e different from 1 (the top element) and 0 (the bottom element) on the unit interval, the construction of those functions is an important task not only from the theoretical point of view, such as and [11], but also for their applications.
On the other hand, t-norms and t-conorms on the unit interval, as an extension of the conjunction and disjunction in classical two-valued logic, respectively, have been extended to bounded lattices in the literature (see, e.g., [10], [20], [24]). In addition, uninorms on bounded lattices have been investigated in many studies (see, e.g., [7], [14], [16], [25], [27], [29], [30], [31]), especially in work related to the construction of uninorms on bounded lattices.
Uninorms on bounded lattices were introduced by Karaçal and Mesiar [15] in 2015. They introduced the existence of uninorms with the neutral element on an arbitrary bounded lattice L. They obtained the greatest uninorm and the smallest uninorm with the neutral element .
In 2016, Çaylı et al. [9] proposed a new class of uninorms on bounded lattices. They showed the existence of idempotent uninorms with the neutral element on bounded lattices. In addition, they discussed uninorms with neutral element e that is incomparable with and provided an example of an idempotent uninorm that is neither disjunctive nor conjunctive.
In 2017, Çaylı and Karaçal [8] provided some new methods for the construction of uninorms on bounded lattices. Compared with the previous work, they presented new methods for constructing uninorms on an arbitrary bounded lattice L with some additional constraints on the neutral element . Some illustrative examples for the construction of uninorms on bounded lattices were provided.
In 2018, based on [8], Çaylı [5] proposed new construction approaches for uninorms on bounded lattices without the need for some additional constraints on the neutral element and studied the main characteristics of uninorms on bounded lattices. Also, the relationship between the uninorms the author proposed and previous uninorms was discussed.
The construction of uninorms on bounded lattices by some other approaches can be found in the literature [4], [6].
In this article, we propose new construction approaches for defining uninorms on an arbitrary bounded lattice with the neutral element. In addition, we exemplify the differences between our new construction approaches and the previous approaches.
When considering the specific structures of uninorms on bounded lattices, we mainly illustrate the relationship between the uninorms on bounded lattices we present in this article and the uninorms proposed by Karaçal and Mesiar [15], Bodjanova and Kalina [4], Çaylı et al. [9], Çaylı and Karaçal [8], and Çaylı [5] as Theorem 3.1, Theorem 3.2, Theorem 3.3, Theorem 3.4, Theorem 3.5, Theorem 3.6, respectively. The relationship between the uninorms we present in Theorem 3.7, Theorem 3.14 and other uninorms such as those proposed by Çaylı [6] can be discussed in an analogous way; we leave this task to interested readers. Moreover, readers can refer to the literature [5], [6] for more comparabilities.
The article is organized as follows. In Section 2 we recall some basic notions of bounded lattices and some properties related to them. In Section 3 we propose a new construction of uninorms with underlying t-norms and t-conorms on bounded lattices. In addition, we provide some examples to illustrate the differences between the new construction of uninorms on bounded lattices proposed by us and some existing uninorms. In Section 4 we provide concluding remarks.
Section snippets
Preliminaries
In this section, we recall some basic concepts of bounded lattices and some properties related to them, which will be used in the sequel.
A bounded lattice [12] is a lattice that has top element and bottom element . In the following, unless stated otherwise, we denote L as a bounded lattice.
Definition 2.1 (See Birkhoff [3].) Let . We use the notation to denote that a and b are incomparable. In the following, denotes the family of all incomparable elements with a; that is,
Construction of uninorms with underlying t-norms and t-conorms on bounded lattices
Similarly to the investigation of uninorms on a unit interval, the notion of uninorms has been extended to bounded lattices and the existence of uninorms on bounded lattices was introduced by Karaçal and Mesiar [15]. Bodjanova and Kalina [4], Çaylı et al. [9], Çaylı and Karaçal [8], and Çaylı [5] introduced new methods for constructing uninorms on bounded lattices. Their results can be used to enrich the class of uninorms on bounded lattices and to analyze their structure.
In this section,
Concluding remarks
Uninorms on bounded lattices have been extensively studied similarly to research into uninorms on the unit interval. In particular, the construction of uninorms related to algebraic structures on bounded lattices is still an active research area. In this article, from the mathematical point of view, we have further investigated the topic of uninorms on bounded lattices with the neutral element . Precisely, we propose a new structure of uninorms on bounded lattices with the neutral
Acknowledgements
The authors express their sincere thanks to the editors and anonymous reviewers for their most valuable comments and suggestions for improving this article greatly. The work described in this article was supported by grants from the National Natural Science Foundation of China (grant nos. 11571010 and 61179038).
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