Abstract Bipolar fuzzy relation equations arise when unknown variables together with their logical negations appear simultaneously in fuzzy relation equations. This paper gives a characterization of the solvability of bipolar max-product fuzzy (relation) equations with the standard negation. In addition, some properties associated with the existence of the greatest/least solution or maximal/minimal solutions are shown, when these (relation) equations are solvable. Different examples are included in order to clarify the developed theory.

中文翻译：

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带标准否定的双极最大积模糊关系方程的可解性

摘要 当未知变量及其逻辑否定同时出现在模糊关系方程中时，就产生了双极模糊关系方程。本文给出了具有标准否定的双极最大乘积模糊（关系）方程的可解性表征。此外，当这些（关系）方程可解时，还显示了与最大/最小解或最大/最小解的存在相关的一些属性。包括不同的例子以阐明发展的理论。