In order to provide a unified framework for studying non-commutative algebraic logic, Rump and Yang used three axioms to define quantum B-algebras, which can be seen as implicational subreducts of quantales. Based on the work of Rump and Yang, in this paper we shall continue to investigate the properties of three axioms in quantum B-algebras. First, using two axioms we introduce the concept of generalized quantum B-algebras and prove that the opposite of the category GqBAlg of generalized quantum B-algebras is equivalent to the category LogPQ of logical pre-quantales, but we can not prove that pre-quantales can be used as the injective objects in GqBAlg. Next, we use one axiom to propose the concept of C-algebras and show that a C-algebra is a group if and only if each of its elements is dualizing. Further, by dualizing elements of a C-algebra X, we can define different binary operations on X such that X is a moniod. Finally, we by the Zig–Zag relation discuss some properties of quantum B-algebras.

为了为研究非交换代数逻辑提供一个统一的框架，Rump 和 Yang 使用三个公理来定义量子B-代数，可以看作是量子数的蕴涵子约。在 Rump 和 Yang 工作的基础上，本文将继续研究量子B代数中三个公理的性质。首先，使用两个公理我们引入广义量子的概念乙-代数和证明类别的相对GqBAlg广义量子的乙-代数相当于类别LogPQ逻辑预quantales的，但是我们也不能证明预quantales 可以用作GqBAlg 中的单射对象. 接下来，我们使用一个公理来提出C -代数的概念，并证明C -代数是一个群当且仅当它的每个元素都是对偶的。此外，通过对C代数X 的元素进行二元化，我们可以在X上定义不同的二元运算，使得X是一个单体。最后，我们通过 Zig-Zag 关系讨论量子B代数的一些性质。