Distributivity and conditional distributivity for S-uninorms
Introduction
The literature [1] laid the foundation of the topic about distributivity equations for two binary operators. Since the distributive law plays a critical role in both theoretical and practical fields, the focus of researchers is on characterizations of distributivity involving two binary operators. Firstly, the distributivity between t-norms and t-conorms was considered in [4], [15]. Then the characterizations of distributivity for some binary operations [5] and fuzzy implications [3] were studied. In 2002, Mas et al. [30] studied the solution of distributivity involving two t-operators [31] (equivalently nullnorms [6]), a t-operator and a uninorm, and two uninorms. Then Ruiz and Torrens [40] examined the distributivity equation involving two idempotent uninorms. The characterizations of distributivity equations involving two nullnorms were studied by Rak in [37]. Inspired by nullnorms and uninorms, Qin and Zhou [36] studied the characterization of distributivity between nullnorms and idempotent uninorms. Then the solution of distributivity between nullnorms and uninorms was proposed by J. Drewniak and P. Drygaś [8]. In 2016, Su et al. examined the distributivity equations for most studied classes of uninorms [43], [44].
By eliminating the assumption of commutativity and associativity from the axiom of uninorm, semi-uninorm is obtained. Meanwhile, semi-t-operator (resp. semi-nullnorm) is defined by omitting commutative law from the axiom of t-operator (resp. nullnorm). Moreover, semi-nullnorm is a subclass of semi-t-operator. The solution of distributivity between semi-nullnorms and semi-t-operators was examined by Drygaś in [9]. Then Qin [34] studied the distributivity equations between semi-t-operators and semi-uninorms. The solution of distributivity for uninorms over semi-t-operators was proposed in [11], [45], [47]. Then the complementary side, i.e., characterizations of distributivity for semi-t-operators over uninorms, was proposed by Su et al. [45]. The distributivity equations involving two semi-t-operators were studied by Drygaś and Rak [17]. On the other hand, a new class of aggregation operators called n-uninorms was obtained by Akella [2], which generalizes the concept of neutral element in uninorms. It is an extension of nullnorms and uninorms. Then characterizations of distributivity equations for some special classes of 2-uninorms are examined by Drygaś and Rak [10]. Otherwise, a class of aggregation operators called Mayor's aggregation operators was proposed by Mayor in [33]. Then distributivity equations between Mayor's aggregation operators and some special classes of aggregation operators were studied by Jočić and Štajner-Papuga [24]. Moreover, Qin and Wang [35] studied the characterizations of distributivity equations between Mayor's aggregation operators and semi-t-operators.
Conditional distributivity, i.e., distributivity on the restricted domain, is also called the restricted distributivity. A lot of literature is appeared to study this particular problem. Firstly, the research is focused on the utility function and the solution that a t-norm is conditional distributive over a t-conorm [13], [14], [25]. The problem of integration [26], [41] has been studied, e.g., -integral, -integral and general fuzzy integral. Then Ruiz and Torrens [39] solved the open problem [42] related to distributivity and conditional distributivity for uninorms over continuous t-conorms. In 2013, Rak [38] solved the same problem for several classes of semi-uninorms, i.e., a special class of conjunctive semi-uninorms, a special class of disjunctive semi-uninorms and the class of idempotent semi-uninorms. Moreover, Jočić and Štajner-Papuga [20], [21] studied the conditional distributivity for operators with absorbing element and applied for modeling a decision maker's behavior in utility theory. The solutions of distributivity and conditional distributivity for semi-uninorms over continuous t-conorms and t-norms were proposed by Liu [29]. Characterizations of the conditional distributivity for uninorms with continuous underlying t-norms and t-conorms over continuous t-conorms were examined by Li et al. [27]. Then Li and Liu [28] studied the characterizations of conditional distributivity for nullnorms over uninorms with continuous underlying t-conorms. Moreover, Jočić and Štajner-Papuga proposed the solutions of conditional distributivity for continuous semi-t-operators over uninorms [22] and T-uninorms over uninorms [23]. Recently, Wang et al. [48] studied the distributivity and conditional distributivity for uni-nullnorms.
In the paper, we mainly study the distributivity equations between S-uninorms and some aggregation operators and the conditional distributivity equations for some aggregation operators over S-uninorms. Our research is motivated by two directions of consideration.
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The first aim is to extend research on regular distributivity from [8], [30], [37], [43], [44], [46] towards S-uninorms which are a generalization of conjunctive uninorms and nullnorms. Though we study the regular distributivity equations from the mathematical point of view, new pairs of logical connectives are obtained which may be applied in fuzzy logic and image processing as well as in approximate reasoning.
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The second one is to study conditional distributivity equations for T-uninorms and semi-t-operators over S-uninorms. The existing literature about conditional distributivity mainly focuses on the first aggregation operator, where the second aggregation operator is fixed as t-conrom [19], [21], [23], [27], [39] or uninorm [20], [22], [23], [28], [48]. In this paper, we want to extend the second aggregation operator to S-uninorms. Though we study the conditional distributivity equations from the theoretical point of view, new pairs of aggregation operators are obtained which may be applied for modeling some specific problems in the utility theory.
Section snippets
Basic concepts
In this section, we recall some concepts relevant to some classes of aggregation operators, functional equations of distributivity and conditional distributivity.
Distributivity for S-uninorms over some binary aggregation operators
In this section, we mainly consider the distributivity for S-uninorms in over t-norms, t-conorms, uninorms from and T-uninorms in (see Fig. 2). Since S-uninorms are commutative, the left distributivity equation (8) and right distributivity equation (9) are equivalent.
Theorem 3.1 Let in and T be a t-norm. Then is distributive over T if and only if . Proof First, we show that T is an idempotent t-norm, i.e., for all . If , then
Distributivity for some binary aggregation operators over S-uninorms
In this section, we mainly consider the complementary side, i.e., distributivity for t-norms, t-conorms, uninorms from , T-uninorms in and semi-t-operators over S-uninorms in (see Fig. 6). The left distributivity equation (8) and right distributivity equation (9) are equivalent except when is a semi-t-operator.
Lemma 4.1 Let in and T be a t-norm (resp. S be a t-conorm). If T (resp. S) is distributive over , then is an idempotent S-uninorm of the following form.
Distributivity equation in a class of S-uninorms
In this section, main topic is distributivity equation involving two S-uninorms in . Since the case that annihilator of S-uninorms equal to 1 is studied above, we assume annihilator of S-uninorms are less than 1 in distributivity equation involving two S-uninorms. Sufficient conditions of distributivity are discussed in the following two lemmas. Lemma 5.1 Let and be two S-uninorms in with neutral elements and , respectively. If is distributivity over , then and . Proof
Conditional distributivity for S-uninorms
In this section, we mainly consider the conditional distributivity for T-uninorms in and semi-t-operators over S-uninorms in (see Fig. 14). It is obvious that the solution of distributivity is one of the solutions of conditional distributivity. With the restriction of the domain for distributivity law, some new solutions that are non-idempotent are obtained. According to Theorem 2.7, some kinds of continuity for and are required. That is, T-uninorms in , semi-t-operators and
Conclusion
In this paper, both regular and conditional distributivity equations for S-uninorms in with an absorbing element are considered through the context from the theoretical point of view. The main results of this paper are listed as follows.
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The paper examined distributively equations for S-uninorms in with an absorbing element over t-norms, t-conorms, uninorms from , T-uninorms in with an absorbing element and semi-t-operators.
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The paper studied
Acknowledgements
The authors would like to express their sincere thanks to the Editors and anonymous reviewers for their most valuable comments and suggestions in improving this paper greatly. This research was supported by the National Natural Science Foundation of China (Grant nos. 11571010 and 61179038).
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2020, International Journal of Approximate ReasoningCitation Excerpt :□ As the distributivity for uninorms over S-uninorms is a special case of the distributivity for semi-uninorms over S-uninorms, Propositions 3.12 and 3.13 show that there are some faults in Theorem 4.3 in [10]. We only verify item (2).