Left and right distributivity between semi-uninorms and semi-S-uninorms

Abstract

This paper investigates the left and right distributivity between semi-uninorms and semi-S-uninorms without the commutativity and associativity. Firstly, we study the left and right distributivity for special disjunctive (resp. conjunctive) semi-uninorms in ${\mathcal{N}}_e^{\max }$ (resp. ${\mathcal{N}}_e^{\min }$) over S-uninorms with the underlying uninorms in ${\mathcal{N}}_{e^{\prime }}^{\min }$. Secondly, semi-S-uninorms are proposed by omitting the commutativity and associativity of S-uninorms. Thirdly, the left and right distributivity for semi-S-uninorms over semi-uninorms in ${\mathcal{N}}_e^{\min }$ (resp. ${\mathcal{N}}_e^{\max }$) are studied, where the underlying semi-uninorms of semi-S-uninorms are semi-uninorms in ${\mathcal{N}}_{e^{\prime }}^{\min }$. Especially, the distributivity between semi-uninorms and semi-S-uninorms has the distributivity between uninorms and S-uninorms as a special case.

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 $U_{\lambda }$ with the underlying semi-uninorm in ${\mathcal{N}}_{e^{\prime }}^{\min }$ coincides with the neutral element of a semi-uninorm $U\in {\mathcal{N}}_e^{\min }$, then the left distributivity, the right distributivity and the distributivity for the semi-S-uninorm $U_{\lambda }$ over the semi-uninorm U are equivalent to each other. Moreover, the IFC element e of a semi-S-uninorm $U_{\lambda }$ means that e is an idempotent element $U_{\lambda }(e,e)=e$ and fulfilling the fixed values $U_{\lambda }(e,1)=U_{\lambda }(1,e)=1$ and the continuity that both $U_{\lambda }(e,x)$ and $U_{\lambda }(x,e)$ 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.

Finally, the results in this paper show that the left and right distributivity between semi-uninorms and semi-S-uninorms, which may be applied in many fields, such as fuzzy logic, image processing, approximate reasoning, modeling some specific problems in the utility theory.

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.