The law of importation is an important property of fuzzy implication functions with interesting applications in approximate reasoning and image processing. This property has been extensively studied and some open problems have been proposed in the literature. In particular, in this paper, we partially solve an open problem related to this property posed some years ago. Specifically, given a fixed uninorm, all fuzzy implication functions that satisfy the law of importation with respect to this uninorm, and having an α-section that is a continuous negation, are characterized. This characterization is specially detailed for the case of uninorms lying in each one of the most usual classes of uninorms. This is done in two different papers, this paper and the forthcoming paper (S. Massanet, D. Ruiz-Aguilera, and J. Torrens, “Characterization of a class of fuzzy implication functions satisfying the law of importation with a fixed uninorm-Part II,” IEEE Trans. Fuzzy Syst., to be published). In particular, in this paper the case of uninorms in Umin is solved, whereas the cases where the uninorm is in the other usual classes (that is, idempotent, representable, and continuous in the open unit square) are left for the above-mentioned paper.

中文翻译：

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一类满足固定非正则输入定律的模糊蕴涵函数的刻画__art I


输入定律是模糊蕴涵函数的一个重要性质，在近似推理和图像处理中有着有趣的应用。该属性已被广泛研究，并且文献中提出了一些开放性问题。特别是，在本文中，我们部分解决了几年前提出的与该属性相关的开放问题。具体来说，给定一个固定的单范数，所有满足关于该单范数的输入定律并且具有连续否定的α-部分的模糊蕴涵函数都被表征。这种表征是针对最常见的单范数类别中的每一个单范数的情况特别详细的。这是在两篇不同的论文中完成的，即这篇论文和即将发表的论文（S. Massanet、D. Ruiz-Aguilera 和 J. Torrens，“Characterization of a class of fuzzy implytion Functions Saturation the law of importing with afixed uninorm-Part”） II，”IEEE Trans。，待出版）。特别是，本文解决了 Umin 中的单范数的情况，而单范数属于其他常见类（即幂等、可表示、在开单位正方形中连续）的情况则留给上述问题纸。