Distributivity equation in the class of 2-uninorms
Introduction
The functional equations involving aggregation operators (e.g., [4], [5], [8], [26], [27]) play an important role in theories of fuzzy sets and fuzzy logic. A new direction of investigations is concerned with distributivity equation and inequalities for uninorms and nullnorms [3], [9], [10], [13], [14], [16], [17], [20], [27], [29], [30], [31], [32]. Uninorms, introduced by Yager and Rybalov [35], and studied by Fodor et al. [18], are special aggregation operators that have proven to be useful in many fields like fuzzy logic, expert systems, neural networks, utility theory and fuzzy system modeling (see [15], [19], [22], [24], [34]). Uninorms are interesting because their structure is a special combination of t-norms and t-conorms having a neutral element lying somewhere in the unit interval. The first notions of nullnorms and t-operators appeared respectively in [5], [25]. Both of these operators, which also generalize concepts of t-norms and t-conorms, where an absorbing (or zero) element is from the whole unit interval, have been studied further (e.g., [11], [28]). Moreover, in [5] and [27] it is proved that nullnorms and t-operators have the same structure so they are equivalent.
Our consideration was motivated by the logical connectives called 2-uninorms, which generalize both nullnorms and uninorms. Such generalization, further extended to the n-uninorms, was introduced by P. Akella in [2]. A 2-uninorm belongs to the class of increasing, associative and commutative binary operators on the unit interval with an absorbing element separating two subintervals having their own neutral elements.
In the case of nullnorms these neutral elements are respectively, 0 and 1 while for 2-uninorms they lie anywhere in the subintervals. Hence, in the structure of 2-uninorm we have two operators isomorphic with some uninorms, where in the case of nullnorms these operators are isomorphic with respectively, some t-conorm and t-norm. This makes the class of 2-uninorms significant generalization of both nullnorms and uninorms, which also includes certain other generalizations known from the literature e.g., S-uninorms, T-uninorms and Bi-uninorms [28].
This paper is organized as follows. In Section 2, we considered the algebraic structures of uninorms, nullnorms and 2-uninorms. We also reminded there the characterization of three subclasses of 2-uninorms. Then, the functional equation of distributivity is recalled (Section 3). In Sections 4 and 5, most of the possible solutions of distributivity equation for described subclasses of 2-uninorms are characterized. In particular, we investigate the distributivity between the couple of 2-uninorms as well as the distributivity between 2-uninorm and continuous triangular norm/conorm. It turns out that the full characterization depends on a number of cases what makes this problem interesting. In the last Section 6 we summarized the considerations and explained some of the encountered difficulties.
Section snippets
The class of 2-uninorms
We start with basic definitions and facts.
Definition 2.1 (See [35].) Let . An operation is called a uninorm if it is associative, commutative, increasing with respect to both variables and fulfilling By we denote the family of all uninorms with neutral element .
Now we recall the general structure of a uninorm (for more details see [7], [12], [18]). We use the following notation for .
Theorem 2.2 (Cf. [18].) Let
Functional equation of distributivity
Now, we consider the distributivity equation of two binary operations. Let us remind some of the most important facts relating to this topic.
Definition 3.1 (Cf. [1], p. 318.) Let . We say that operation F is distributive over G, if
Lemma 3.2 (See [29].) Let , , . If operation F with neutral element is distributive over operation G fulfilling , then G is idempotent in Y.
Lemma 3.3 (See [29].) Every increasing operation is
Distributivity between operators from the class of 2-uninorms and continuous t-norms and t-conorms
At first our consideration will concern the distributivity of 2-uninorm F from all defined subclasses over , where T, S denote the set of all triangular norms and triangular conorms, respectively.
Theorem 4.1 Let . A 2-uninorm is distributive over a continuous triangular norm G if and only if .
Proof Let be distributive over a continuous triangular norm G. To prove the necessary condition at first we show that and . Since G is t-norm we
Distributivity of over
Now we consider the distributivity between operators and distinguishing both the order of their zero elements as well as their specific structures. We investigate the four following cases in which essential is the form of operator F and the proper order for the respective neutral elements that distributivity could occur.
Theorem 5.1 Let and . A 2-uninorm is distributive over a 2-uninorm if and only if , G is idempotent,
Concluding remarks
Inspired by uninorms and nullnorms, we examined the functional equation of distributivity in the class of 2-uninorms. We gave the characterization of Eq. (7) including 3 subclasses of 2-uninorms, introduced in the paper of Akella from 2007, where he used . Surely, it is clear that it is possible to create much more other subclasses, in particular, using and/or as representable uninorms. Then the solutions of distributivity equation will be in another type. Nevertheless,
Acknowledgements
Authors wish to thank the editors and the anonymous referees whose valuable comments helped to improve final version of the paper.
This work was partially supported by the Centre for Innovation and Transfer of Natural Sciences and Engineering Knowledge No. RPPK.01.03.00-18-001/10.
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2022, Fuzzy Sets and SystemsCitation Excerpt :It is well known that null-uninorms (uni-nullnorms) are very special 2-uninorms and the structures of 2-uninorms. However, other papers addressed this theme, demonstrating different approaches concerning the structural characterization of 2-uninorms like in Zong et al. [22], or in Drygaś and Rak where it is evaluated the solving of the functional equation of distributivity between aggregation operators [23]. Other papers in the literature highlighted the theme based on some different architectures of the 2-uninorms (see more in [24–27]).