The aim of this paper is to define and study the involutive and weakly involutive quantum B-algebras. We prove that any weakly involutive quantum B-algebra is a residuated poset. As an application, we introduce and investigate the notions of existential and universal quantifiers on involutive quantum B-algebras. It is proved that there is a one-to-one correspondence between the quantifiers on weakly involutive quantum B-algebras. One of the main results consists of proving that any pair of quantifiers is a monadic operator on weakly involutive quantum B-algebras. We investigate the relationship between quantifiers on bounded sup-commutative pseudo BCK-algebras and quantifiers on other related algebraic structures, such as pseudo MV-algebras and bounded Wajsberg hoops.

中文翻译：

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对合的量子 B 代数

本文的目的是定义和研究对合和弱对合的量子B-代数。我们证明了任何弱对合的量子 B 代数都是一个剩余的偏序集。作为一个应用程序，我们介绍并研究了对合量子 B 代数的存在量词和全称量词的概念。证明了弱对合量子B-代数上的量词之间存在一一对应的关系。主要结果之一包括证明任何一对量词都是弱对合量子 B 代数上的一元算子。我们研究了有界上交换伪 BCK 代数上的量词与其他相关代数结构（例如伪 MV 代数和有界 Wajsberg 环）上的量词之间的关系。