Nullnorms on bounded lattices derived from t-norms and t-conorms
Introduction
T-operators and nullnorms with an annihilator a in the interior of the unit interval that are generalizations and unifications of t-norms and t-conorms were introduced by Mas et al. [29] and Calvo et al. [7], respectively. Due to the fact that both of them have the identical block structures on [0, 1]2, it was observed that nullnorms are equivalent to t-operators on the unit interval [0,1] in [30]. These operators play a crucial role in lots of areas like fuzzy quantifiers, fuzzy logic, fuzzy system modeling, decision making and expert systems. Compared with the concept of t-norms and t-conorms, nullnorms allow an annihilator a to be anywhere in the unit interval. In particular, a nullnorm is a t-norm (resp. t-conorm) when the case (resp. ). Nullnorms that can be taken as aggregation operators or in fuzzy logic maintain many of their logical characteristics [1], [8], [19], [20], [21], [22], [26], [27], [33], [34]. Nullnorms on the unit interval were investigated in many discussions [15], [17], [18], [23], [28], [30], [31], [32], [35], [36], [37], [38], [40].
In recent times, nullnorms related to algebraic structures on bounded lattices become a substantial study from the theoretical point of view. It is known that the concept of nullnorms on a bounded lattice was introduced by Karaçal et al. [25]. They also showed the presence of nullnorms and obtained the greatest and the smallest nullnorms on an arbitrary bounded lattice. It should be stated that the classes of nullnorms on a bounded lattice obtained in [25] are not idempotent, in general. For this reason, Çaylı and Karaçal [9] investigated the presence of idempotent nullnorms on bounded lattices having an annihilator. They demonstrated that an idempotent nullnorm on a bounded lattice may not always exist. Moreover, it was proposed a method to yield an idempotent nullnorm on a bounded lattice having an annihilator a when there is just one element incomparable with a. Afterwards, Wang, Zhan and Liu [39] introduced two methods for building idempotent nullnorms on bounded lattices under some additional assumptions on the annihilator. Furthermore, Çaylı [14] presented two methods to get nullnorms on a bounded lattice M having an annihilator a with some constraints. Although the constructions in [14] base on the presence of a t-norm T on [a, 1M]2 such that T(x, y) ∈ ]a, 1M] for all x, y ∈ ]a, 1M] and a t-conorm S on [0M, a]2 such that S(x, y) ∈ [0M, a[ for all x, y ∈ [0M, a[, those in [39] exploit only idempotent t-norm on [a, 1M]2 and idempotent t-conorm on [0M, a]2. By considering that there always exists an idempotent t-norm and t-conorm on a bounded lattice M, two idempotent nullnorms on M were obtained in [14]. Hence, the constructions described in [14] are generalizations of the ones proposed in [39].
It is an interesting point that contrary to the classes of nullnorms on the unit interval, nowadays, the classes of nullnorms on bounded lattices are quite unknown yet, although some researches for nullnorms on bounded lattices are presented in [2], [3], [5], [9], [10], [11], [14], [24], [25], [39]. Hence, the research of nullnorms on bounded lattices is significant from the theoretical point of view. In this paper, we demonstrate some new methods to construct nullnorms on bounded lattices having an annihilator with the underlying t-norms and t-conorms.
This study comprises five main parts. We first briefly discuss some notions and results dealing with nullnorms on bounded lattices in Section 2. In Section 3, considering a bounded lattice M and an element a ∈ M\{0M, 1M}, we introduce some constructions to have two new classes of nullnorms FT on M based on a t-norm T on [a, 1M]2 and FS on M based on a t-conorm S on [a, 1M]2, where some necessary and sufficient conditions on the element a playing the role of an annihilator are required. Further, we also present some corresponding examples showing the role of these conditions for the constructions. Considering the fact that the one idempotent t-norm (inf) and the one idempotent t-conorm (sup) S∨: M2 → M, respectively, exist, two classes of idempotent nullnorms F∧ and F∨ are obtained. In Section 4, we propose two other classes of nullnorms and on a bounded lattice M having an annihilator a ∈ M\{0M, 1M} derived from both t-norms on [a, 1M]2 and t-conorms on [0M, a]2. It should be pointed out that although the nullnorms FT and FS are obtained by using the one of t-norms and t-conorms, the nullnorms and are obtained by using both t-norms and t-conorms. It is worthy of note that there is no restriction on the underlying t-norms and t-conorms in our constructions in contrast to those in [14]. As a result, we describe the nullnorms FT,S and FT,S that show the presence of idempotent nullnorms on bounded lattices different from the nullnorms and . Further, we present some illustrative examples in order to clarify the structure of these nullnorms on bounded lattices. Finally, Section 5 includes some concluding remarks.
Section snippets
Preliminaries
We will just present some main elements dealing with t-norms, t-conorms and nullnorms in this part.
A bounded lattice (M, ≤ , 0M, 1M) is a lattice that has the top element and bottom element denoted as 1M and 0M, respectively. In the following, we denote M as a bounded lattice unless otherwise stated.
Let u, v ∈ M. We use the notation u∥v to denote that u and v are incomparable. denotes the family of all incomparable elements with u, that is, We use the notation u∦v to denote
Nullnorms on bounded lattices
In this section, some basic characteristics of idempotent nullnorms on bounded lattices are researched. Two new constructions of nullnorms on bounded lattices are introduced with some additional constraints on theirs annihilator by using the existence of t-norms and t-conorms on a bounded lattice. In the same time, some corresponding examples are provided in order to demonstrate that these additional constraints play a crucial role in our methods. In particular, we obtain two idempotent
Some other constructions of nullnorms on bounded lattices
In this section, we introduce two other new methods that yield nullnorms on a bounded lattice M in Theorems 4 and 5, where some sufficient and necessary conditions on the element a ∈ M\{0M, 1M} playing the role of an annihilator are required. These constructions consider the presence of a t-norm T: [a, 1M]2 → [a, 1M] and t-conorm S: [0M, a]2 → [0M, a]. As a result of our constructions, we have two idempotent nullnorms on bounded lattices. Further, we also present some corresponding examples to
Concluding remarks
Similarly to the concept of nullnorms on unit interval [0, 1], the concept of nullnorms has been extended to bounded lattices by Karaç al et al. [25]. Following the demonstration of the existence of nullnorms on bounded lattices in [25], recently, the existence of idempotent nullnorms on bounded lattices has been studied. Since the investigations for nullnorms in [25] do not clarify the existence of idempotent nullnorms on a bounded lattice, it was proposed a method to construct idempotent
Declaration of Competing Interest
I declare that this manuscript is original, has not been published before and is not currently being considered for publication elsewhere. I wish to confirm that there are no known conflicts of interest associated with this publication and there has been no financial support for this work that could have influenced its outcome. I confirm that there are no other persons who satisfied the criteria for authorship but are not listed and that I have provided a current, correct email address which is
Acknowledgements
The author expresses her sincere thanks to the editors and reviewers for their most valuable comments and suggestions in improving this paper greatly.
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