Elsevier

Fuzzy Sets and Systems

Volume 395, 15 September 2020, Pages 40-70
Fuzzy Sets and Systems

New construction of t-norms and t-conorms on bounded lattices

https://doi.org/10.1016/j.fss.2019.05.017Get rights and content

Abstract

This article studies the construction of ordinal sums of triangular norms (t-norms) and triangular conorms (t-conorms) on bounded lattices. First, we present a new class of t-norms and t-conorms on an arbitrary bounded lattice. Some examples are provided to illustrate that our new construction method is different from some existing methods for the construction of ordinal sums of t-norms and t-conorms on an arbitrary bounded lattice. Then we illustrate that our new construction method can be generalized by induction to a modified ordinal sum construction for t-norms and t-conorms on an arbitrary bounded lattice. Finally, for any bounded sublattice S of a bounded lattice L, we provide a sufficient condition that does not need S and L to have the same bottom and top elements such that an ordinal sum function is a t-norm on L, whereas the function is determined by an arbitrary selection of bounded sublattices as carriers for arbitrary summand t-norms. In particular, we give a necessary and sufficient condition that needs S to have the same bottom element as L such that an ordinal sum function is a t-norm on L.

Section snippets

A brief review of the development of t-norms and t-conorms

Triangular norms (t-norms) with the neutral 1 and triangular conorms (t-conorms) with the neutral 0 on the unit interval were introduced by Menger [37] in 1942 and by Schweizer and Sklar [45] in 1961, respectively. As an extension of the logic connectives conjunction and disjunction in classical two-valued logic, respectively, t-norms and t-conorms play an important role in many fields, such as fuzzy set theory [51], fuzzy logic [2], fuzzy systems modeling [48], and probabilistic metric spaces

Preliminaries

In this section, we recall some basic concepts and notation for bounded lattices and some properties related to them, which will be used in the sequel.

A bounded lattice (L,,0L,1L) [21] is a lattice that has top element 1L and bottom element 0L. A bounded sublattice (S,,a,b) of L is a sublattice of L that has top element b and bottom element a. In the following, unless stated otherwise, we denote L as a bounded lattice.

Definition 2.1

(See Birkhoff [6] and Wang et al. [47].) Let a,bL. We use the notation ab

A new method to construct t-norms and t-conorms on bounded lattices

In this section, we introduce a new class of t-norms and t-conorms on an arbitrary bounded lattice L. The new construction method in this article to obtain t-norms and t-conorms is different from the proposals of Çaylı [8] and Ertuğrul et al. [19]. Apart from the different values for t-norms and t-conorms on the same domains, the greatest difference is that we give a new ordinal sum construction for t-norms and t-conorms on an arbitrary bounded lattice L, where a,bL{0L,1L}, the summand t-norm

Modified construction of ordinal sums of t-norms and t-conorms on bounded lattices

Ertuğrul et al. [19] provided a modification of ordinal sums of t-norms and t-conorms on an arbitrary bounded lattice. Moreover, the construction method they presented can be generalized by induction to a modified ordinal sum construction for t-norms and t-conorms on an arbitrary bounded lattice. Çaylı [8] introduced a new class of t-norms and t-conorms on an arbitrary bounded lattice and also provided a method to study the construction of ordinal sums of t-norms and t-norms on an arbitrary

Characterizing ordinal sums of t-norms with one summand only

As we stated in Section 1.2, on the basis of the constraint that the bounded sublattice S has the same bottom and top elements as L, Saminger-Platz et al. [44] provided a necessary and sufficient condition such that the function TTSL given in Eq. (3) is a t-norm on L. In this section, on the basis of [44], we consider whether there exists a necessary and sufficient condition without satisfying the constraint that needs the bounded sublattice S has the same bottom and top elements as L such that

Concluding remarks

Both t-norms and t-conorms on bounded lattices have been extensively investigated similarly to research into t-norms and t-conorms on the unit interval. In particular, the construction of ordinal sums of t-norms and t-conorms related to algebraic structures on bounded lattices is still an important research area. In this article, we have further studied the construction of ordinal sums of t-norms and t-conorms on bounded lattices from a mathematical point of view. The main contributions are as

Acknowledgements

The authors express their sincere thanks to the editors and anonymous reviewers for their most valuable comments and suggestions for greatly improving this article. Yexing Dan and Bao Qing Hu acknowledge support by the National Natural Science Foundation of China (grant nos. 11571010 and 61179038). Junsheng Qiao acknowledges support by the Scientific Research Fund for Young Teachers of Northwest Normal University (grant no. 5007/384) and by the Doctoral Research Fund of Northwest Normal

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