Representation of nullnorms on bounded lattices
Introduction
T-norms and t-conorms were firstly introduced by Schweizer and Sklar in [25]. They serve as the semantic interpretations of logical connectives [27] and are systematically studied in both practical applications and theoretic investigations. While t-norms and t-conorms always fix their absorbing elements on the left or right end of the real unit interval, nullnorms, introduced in Ref. [6], unify the two notions by letting the absorbing elements lie in anywhere of the interval. Structurally speaking, a nullnorm is a combination of a t-norm and a t-conorm. This fact brings nullnorms a lot of good properties, which enable them to play important roles in applied aspects of fuzzy logic, fuzzy sets theory, expert systems, neural networks and so on [1], [12], [13], [15], [20], [23]. Nullnorms have been deeply studied on the real unit interval and a lot of profound results have been proposed [10], [11], [16], [21], [22], [24].
Karaçal et al. first proposed the notion of nullnorms on bounded lattices in Ref. [18] with the intent of replacing the real unit interval by more general algebraic structures. This kind of generalizations, including the generalizations of t-norms [3], t-conorms [2], uninorms [19], semi-t-operators [14], uni-nullnorms [26] on bounded lattices are meaningful research activities because bounded lattices (e.g. interval-valued lattices and finite chains) are more applicable for practical applications. Besides, these studies on bounded lattices can provide us with new insights on the universal properties of these functions. The existence of nullnorms on arbitrary bounded lattices is proved in Ref. [18] by providing the smallest and greatest nullnorms. Since then the construction methods and characterization of nullnorms considering various classes of nullnorms have been frequently discussed. Çaylı and Karaçal investigated the class of idempotent nullnorms on bounded lattices in Ref. [7] and showed that idempotent nullnorms do not always exist on arbitrary bounded lattices. Abundant results on the constructions of idempotent nullnorms or more general classes of nullnorms have been already proposed in literature [7], [8], [9], [17], [26]. However, many results in the aforementioned contributions were obtained with quite strong restrictions on the structures of the given lattices, which motivates us to give a further research on the nullnorms on bounded lattices.
Since the values of a nullnorm are comparable with the absorbing element if only one of the variables is comparable with the absorbing element, it is reasonable to study the nullnorms whose values are all comparable with their absorbing elements first. This quite general class of nullnorms is called in this paper. We give the representation of the nullnorms in and show that every nullnorm in is totally determined by the underlying functions and the values on boundary. Furthermore, we find that for a general nullnorm V with an absorbing element a on a bounded lattice L, the value when is also totally determined in a same way. The full characterization of arbitrary nullnorms on bounded lattices can be achieved if only the structures of them on are determined. In the end, we give some necessary conditions for this part of the problem.
This paper comprises five sections. In Section 2, we give some basic definitions and properties for nullnorms and lattices. In Section 3, we give the representation for the nullnorms in on bounded lattices and give an example. In Section 4, we give some further investigations and show that the structure of arbitrary nullnorms on bounded lattices is also determined in a same way when only one of the two variables is comparable with the absorbing elements. For the rest cases, we give some necessary conditions. In Section 5, we list the results in this paper and give some further perspectives.
Section snippets
Preliminaries
In this section we present some basic definitions and properties about lattices and nullnorms. Definition 2.1 Let be a partially ordered set. For a subset S, an element is called an upper bound of S if and only if for all . An upper bound s of S is called the least upper bound if and only if for all upper bounds a of S. An element is called a bottom element of P if for all .[5]
The notions of lower bound, greatest lower bound and top element can be analogically defined. Definition 2.2 A partially [4]
Representation of nullnorms in on bounded lattices
Definition 3.1 Let V be a nullnorm on with an absorbing element . We say V belongs to if and only if is always comparable with a.
By Proposition 2.7, only when can happen. So is quite a general class of nullnorms. We first give a construction method of this class of nullnorms. Theorem 3.2 Let be a bounded lattice and . and are t-norm and t-conorm on and . For two order-preserving functions and , we define two sets A and B
Characterization of general nullnorms on bounded lattices
Now we consider a general nullnorm V with an absorbing element . From the proof of Theorem 3.4, the values of V can also be characterized when only one variable belongs to . Theorem 4.1 Consider a nullnorm V on with an absorbing element , an underlying t-norm and t-conorm . We have (i) for , (ii) for ,
where and .
Proof
For arbitrary , by the
Conclusion
In this paper, we have given the representation for the nullnorms in on bounded lattices. In this class, the structure of every nullnorm is totally determined by its values on boundary and the underlying functions. Considering the more general cases, for an arbitrary nullnorm is determined in a same way if . A full characterization of arbitrary nullnorms on bounded lattices can be achieved if only we figure out what happens on the area .
Surely, the case when is
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
CRediT authorship contribution statement
Xiang-Rong Sun: Conceptualization, Methodology, Writing - original draft, Writing - review & editing, Supervision. Hua-Wen Liu: Validation, Writing - review & editing, Supervision, Project administration, Funding acquisition.
Acknowledgment
This work was supported by the National Key R&D Program of China (No. 2018YFA0703900) and National Natural Science Foundation of China (Nos. 61573211 and 11531009).
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