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Bijective transformations of fuzzy implications - An algebraic perspective

机译：模糊含义的双射变换-代数观点

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摘要

Bijective transformations play an important role in generating fuzzy implications from fuzzy implications. In [Representations through a Monoid on the set of Fuzzy Implications, Fuzzy Sets and Systems, 247, 51-67], Vemuri and Jayaram proposed a monoid structure on the set of fuzzy implications, which is denoted by I, and using the largest subgroup S of this monoid discussed some group actions on the set I. In this context, they obtained a bijective transformation which ultimately led to hitherto unknown representations of the Yager's families of fuzzy implications, viz., f-, g-implications. This motivates us to consider whether the bijective transformations proposed by Baczynski & Drewniak and Jayaram & Mesiar, in different but purely analytic contexts, also possess any algebraic connotations. In this work, we show that these two bijective transformations can also be seen as being obtained from some group actions of Son I. Further, we consider the most general bijective transformation that generates fuzzy implications from fuzzy implications and show that it can also be obtained as a composition of group actions of S on I. Thus this work tries to position such bijective transformations from an algebraic perspective. (C) 2015 Elsevier B.V. Allrightsreserved.
机译：双射变换在从模糊含意产生模糊含意中起重要作用。 Vemuri和Jayaram在[通过模糊蕴涵集，模糊集和系统的象素表示，247，51-67]中提出了关于模糊蕴涵集的类id象结构，用I表示，并使用最大子集该类半人像的S讨论了集合I上的一些群动作。在这种情况下，他们获得了双射变换，最终导致了Yager模糊含意族的迄今未知的表示，即f-，g-蕴涵。这促使我们考虑Baczynski＆Drewniak和Jayaram＆Mesiar在不同但纯粹是分析性背景下提出的双射变换是否也具有任何代数含义。在这项工作中，我们表明这两个双射变换也可以看作是从Son I的某些群体动作中获得的。此外，我们考虑了最一般的双射变换，它从模糊含义中产生了模糊含义，并表明也可以得到它。因此，本工作试图从代数的角度定位这种双射变换。 （C）2015 Elsevier B.V.保留所有权利。

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