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The circle star-composition of fuzzy implications: Closures with respect to properties, powers and families

机译：模糊含义的圆星组成：关于性质，权力和家庭的封闭

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Recently, Vemuri and Jayaram proposed a novel method of generating fuzzy implications from a given pair of fuzzy implications. Viewing this as a binary operation circle star on the set I of fuzzy implications they obtained, for the first time, a monoid structure (I, circle star) on the set I. Some algebraic aspects of (I, circle star) had already been explored and hitherto unknown representation results for the Yager's families of fuzzy implications were obtained in [53] (N.R. Vemuri and B. Jayaram, Representations through a monoid on the set of fuzzy implications, fuzzy sets and systems, 247 (2014) 51-67). However, the properties of fuzzy implications generated or obtained using the circle star-composition have not been explored. In this work, the preservation of the basic properties like neutrality, ordering and exchange principles, the functional equations that the obtained fuzzy implications satisfy, the powers w.r.t. circle star and their convergence, and the closures of some families of fuzzy implications w.r.t. the operation circle star, specifically the families of (S, N)-, R-, f-and g-implications, are studied. This study shows that the circle star-composition carries over many of the desirable properties of the original fuzzy implications to the generated fuzzy implications and further, due to the associativity of the circle star-composition one can obtain, often, infinitely many new fuzzy implications from a single fuzzy implication through self-composition w.r.t. the circle star-composition. (C) 2014 Elsevier B.V. All rights reserved.
机译：最近，Vemuri和Jayaram提出了一种从给定的一对模糊含义中生成模糊含义的新颖方法。他们首次将其视为具有模糊含义的集合I上的二元运算圆星，这是他们第一次在集合I上获得了一个单面体结构（I，圆星）。（I，圆星）的一些代数方面已经在[53]中获得了Yager的模糊含意族的探索和迄今未知的表示结果（NR Vemuri和B. Jayaram，关于模糊含意集，模糊集和系统的表示法，247（2014）51-67） ）。但是，尚未探索使用圆星组成生成或获得的模糊含义的属性。在这项工作中，保留了诸如中立性，有序性和交换性原理等基本属性，所获得的模糊含义满足的功能方程式，幂w.r.t。圆星及其收敛，以及一些模糊含义系列的关闭。研究了操作圆星，特别是（S，N）-，R-，f-和g蕴涵的族。这项研究表明，圆星组成将原始模糊含义的许多理想特性传递给生成的模糊含义，而且，由于圆星组成的关联性，人们常常可以无限地获得许多新的模糊含义。从单一的模糊蕴涵到自我构成圆星组成。 （C）2014 Elsevier B.V.保留所有权利。

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