In this work, we have proposed a new relationship among rough set, soft set and quantales with the help of soft compatible relation. This typical relationship is used to approximate the fuzzy substructures in quantales in association with soft compatible relations by using aftersets and foresets. This type of approximation is extended notation of rough quantales, rough fuzzy subquantales and soft subquantales. We have corroborated this work by considering some test examples containing soft compatible relations over quantales. Moreover, by using soft compatible relations, we will describe the relationship between upper (lower) generalized rough fuzzy soft substructures of quantale and the upper (lower) approximations of their homomorphic images with the help of weak quantale homomorphism. The comparison of this type approximations and their results affirms the superiority of our new approximation method over current methods on the topic.

中文翻译：

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关于软关系的量子模糊子结构的广义粗糙度

在这项工作中，我们借助软兼容关系提出了粗糙集、软集和量子之间的新关系。这种典型的关系用于通过使用后集和前集来近似与软兼容关系相关联的 quantale 中的模糊子结构。这种类型的近似是粗糙量子、粗糙模糊子量子和软子量子的扩展符号。我们通过考虑一些包含量子数的软兼容关系的测试示例来证实了这项工作。此外，通过使用软兼容关系，我们将借助弱量子同态来描述量子的上（下）广义粗糙模糊软子结构与其同态图像的上（下）近似之间的关系。