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Micanorm aggregation operators: basic logico-algebraic properties

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Abstract

This paper investigates basic logico-algebraic properties of micanorms. For this, as preliminaries, we first introduce micanorms and consider their logic related properties such as conjunctiveness, left-continuity, and continuity. We then give a characterization of left-continuous micanorms and consider two kinds of micanorm analogues of the \(\L \)ukasiewicz, Gödel, and product t-norms and their residuated implications. We finally generalize basic algebraic concepts of t-norms and t-conorms to those of micanorms and investigate related properties, along with those of the micanorm analogues.

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Notes

  1. Recently, some literature are considering commutative semi-uninorms on [0, 1] (e.g., Li and Liu (2015); Su et al. (2019)), and thus, they are in fact micanorms.

  2. See Yang (2015) for an example of product micanorms.

  3. For the reason to investigate the weak forms of associativity, see Yang (2017). The basic motivation is to introduce weak associative operations as a generalization of t-norms and uninorms.

  4. (3) and (6) are often called probabilistic t-norm and t-conorm, respectively. In appearance, \(T^{\L }\) and \(T^{\Pi }\) are not defined using identity 1 but we may consider (2) and (3) as (2\('\)) \(T^{G}(x, y) = min\{x, y, 1\}\) and (3\('\)) \(T^{\Pi }(x, y) = \frac{xy}{1}\), respectively, and similarly for \(S^{\L }\) and \(S^{\Pi }\). Henceforth, we consider the Gödel and product t-norms and t-conorms to be given by (2\('\)) and (3\('\)), respectively.

  5. The conjunctive \(\L \)ukasiewicz micanorm \(*^{c\L }\) is the same as the \(\L \)ukasiewicz micanorm \(*^{\L }\) introduced in Yang (2015) and that the conjunctive and disjunctive Gödel micanorms are the uninorms \(\hbox {R}^{*}\) and R\(_{*}\), respectively, introduced in Yager and Rybalov (1996).

  6. In pointed rlu-groupoids used as semantics for substructural logics, negations are defined using residuated implications and a point f. For instance, in algebraic semantics for uninorm-based logics, n(x) is defined as \(x \Rightarrow f\) using the residuated implication \(\Rightarrow \) and f, and this negation does not necessarily satisfy \(n(1) = 0\), see, e.g., Galatos et al. (2007); Metcalfe and Montagna (2007).

  7. The term “subminimal negation” was first introduced by Hazen (1992) as an operation satisfying contraposition on a partially ordered set. Following his idea, Dunn (1996) introduced it as the weakest negation.

  8. More precisely, some of (i) to (iii) are easy consequences of the known facts.

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Funding

This work was supported by National University Development Project in 2020.

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  1. Department of Philosophy & Institute of Critical Thinking and Writing, Jeonbuk National University, Rm 417, Center for Humanities & Social Sciences, Jeonju, 54896, Korea

    Eunsuk Yang

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A draft of this paper was presented in 2019 International Conference on Data Intelligence & Neutrosophic Sets with Applications in Xi’an, China. I must thank Prof. X. Zhang for his giving me a chance to give a talk on this subject in the conference. I also would like to thank the reviewers for their valuable comments helped me to improve the paper.

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Yang, E. Micanorm aggregation operators: basic logico-algebraic properties. Soft Comput 25, 13167–13180 (2021). https://doi.org/10.1007/s00500-021-06097-2

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