Left and right distributivity between semi-uninorms and semi-S-uninorms
Introduction
Aggregation operations combine and merge a given number of values into a representative value, which are applied in many fields, such as decision theory, pattern recognition, image processing, approximate reasoning, fuzzy control and so on [3], [7], [12]. The distributivity for two binary operations was proposed in [1] by Aczél. The distributivity plays a fundamental and essential role in the study of fuzzy connectives, such as the distributivity between t-norms and t-conorms [14], aggregation operations [5] and fuzzy implications [2], [13], [20], [21], [33].
Uninorms firstly proposed by Yager and Rybalov [31] are a special kind of aggregation operations. The structure of uninorms is a special combination of t-norms and t-conorms [11]. Due to the great quantity of applications of aggregation operations, uninorms are also applied in many fields, such as expert systems [8], fuzzy system modeling [28], [29], [30], image processing [6], neural networks [4], data mining [32] and so on. That great quantity of applications have inspired many authors to study uninorms from a theoretical point of view. The distributivity between uninorms and other aggregation operations were investigated in [9], [17], [22], [25], [26], [27], such as uninorms [26], [27], idempotent uninorms [25], nullnorms [9], [22], t-operators [17] and so on. Semi-uninorms were discussed in [9], [15], [16] by omitting the commutativity and associativity of uninorms. Especially, Liu [16] studied the distributivity for semi-uninorms over continuous t-conorms and t-norms. Inspired by semi-uninorms, Qin [19] further investigated the distributivity between semi-uninorms and semi-t-operators. S-uninorms were firstly proposed in Mas et al. [18] as the aggregation operations with annihilators. Fang and Hu [10] studied the distributivity between uninorms and S-uninorms. However, the characterizations of distributivity for semi-uninorms over S-uninorms are not obtained.
In this paper, we mainly study the left and right distributivity between semi-uninorms and semi-S-uninorms. Our research is motivated by three directions of consideration.
- •
When semi-uninorms are left or right distributive over S-uninorms, S-uninorms must be idempotent. Meanwhile, considering the left (resp. right) distributivity for semi-uninorms over semi-S-uninorms, semi-S-uninorms are verified to be idempotent. In fact, those idempotent S-uninorms coincide with idempotent semi-S-uninorms. Hence, we characterize the left and right distributivity for semi-uninorms over S-uninorms instead of semi-S-uninorms.
- •
We propose semi-S-uninorms without the commutativity and associativity, and solve the characterizations of the left and right distributivity for semi-S-uninorms over semi-uninorms. Especially, if the IFC element of a semi-S-uninorm with the underlying semi-uninorm in coincides with the neutral element of a semi-uninorm , then the left distributivity, the right distributivity and the distributivity for the semi-S-uninorm over the semi-uninorm U are equivalent to each other. Moreover, the IFC element e of a semi-S-uninorm means that e is an idempotent element and fulfilling the fixed values and the continuity that both and are continuous.
- •
Because the distributivity between S-uninorms and uninorms is a special case of the distributivity between semi-S-uninorms and semi-uninorms, this paper also shows that Theorems 3.3, 4.3, 5.1 and 5.2 in [10] are faulty.
The content of this paper is organized as follows. In Section 2, we recall some fundamental concepts and related properties of basic fuzzy logic connectives and distributive equations applied in this research. Section 3 characterizes the left and right distributivity for semi-uninorms over S-uninorms. In Section 4, we propose semi-S-uninorms and study their properties by omitting the commutativity and associativity of S-uninorms. Moreover, we also discuss the left and right distributivity for semi-S-uninorms over semi-uninorms. We present some conclusions of our research and further work in the final section.
Section snippets
Preliminaries
In this section, we recall some basic notations applied in this research. Definition 2.1 A binary operation is a semi-uninorm, if it is increasing with respect to each variable and has a neutral element , that is, for all .See [9], [15], [16]
Left and right distributivity for semi-uninorms over S-uninorms
In this section, we study the left and right distributivity for semi-uninorms over S-uninorms. As uninorms can be viewed as special semi-uninorms, the distributivity for semi-uninorms over S-uninorms has the distributivity for uninorms over S-uninorms in [10] as a special case.
Left and right distributivity for semi-S-uninorms over semi-uninorms
In this section, we firstly propose semi-S-uninorms without the commutativity and associativity of S-uninorms. In fact, it is important to avoid verifying the associativity of S-uninorms in distributive equations. Secondly, the properties of semi-S-uninorms are investigated. Thirdly, we study the left and right distributivity for semi-S-uninorms over semi-uninorms. Definition 4.1 A binary operation is called a semi-S-uninorm, if it is increasing with respect to each variable and satisfying
Conclusions
This paper studies the left and right distributivity between semi-uninorms in (resp. ) and semi-S-uninorms with the underlying semi-uninorms in by omitting the commutativity and associativity. Considering the left or right distributivity for semi-uninorms over S-uninorms, S-uninorms are proven to be idempotent. Moreover, semi-S-uninorms must be idempotent, while semi-uninorms are left or right distributive over semi-S-uninorms. Those idempotent S-uninorms are equal to
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.
Acknowledgements
The authors would like to thank the Editors and the anonymous reviewers for their valuable comments and suggestions in improving this paper. This research was supported by the National Natural Science Foundation of China (Grant No. 11501334) and the Natural Science Foundation of Shandong Province (Grant No. ZR2015FQ014).
References (33)
- et al.
Some remarks on the distributive equation of fuzzy implication and the contrapositive symmetry for continuous, Archimedean t-norms
Int. J. Approx. Reason.
(2013) - et al.
Automorphisms, negations and implication operators
Fuzzy Sets Syst.
(2003) - et al.
Construction of fuzzy indices from fuzzy DI-subsethood measures: application to the global comparison of images
Inf. Sci.
(2007) - et al.
Melle's combining function in MYCIN is a representable uninorm: an alternative proof
Fuzzy Sets Syst.
(1999) - et al.
Distributivity between uninorms and nullnorms
Fuzzy Sets Syst.
(2008) - et al.
Distributivity and conditional distributivity for S-uninorms
Fuzzy Sets Syst.
(2019) Semi-uninorms and implications on a complete lattice
Fuzzy Sets Syst.
(2012)Distributivity and conditional distributivity of semi-uninorms over continuous t-conorms and t-norms
Fuzzy Sets Syst.
(2015)- et al.
The distributivity condition for uninorms and t-operators
Fuzzy Sets Syst.
(2002) Distributivity between semi-uninorms and semi-t-operators
Fuzzy Sets Syst.
(2016)
Distributive equations of implications based on nilpotent triangular norms
Int. J. Approx. Reason.
On the distributivity property for uninorms
Fuzzy Sets Syst.
The distributivity equation for uninorms revisited
Fuzzy Sets Syst.
Uninorms in fuzzy systems modeling
Fuzzy Sets Syst.
Universal approximation theorem for uninorm-based fuzzy systems modeling
Fuzzy Sets Syst.
Uninorm aggregation operators
Fuzzy Sets Syst.
Cited by (1)
A unified way to studies of t-seminorms, t-semiconorms and semi-uninorms on a complete lattice in terms of behaviour operations
2023, International Journal of Approximate Reasoning