Construction of Quantum B-algebras over Posets

In order to provide a unified semantics for non-commutative algebraic logic, based on posets, Rump and Yang introduced the concept of quantum B-algebras. In this paper, we mainly consider the construction of quantum B-algebras over posets. We prove that a finite poset can support a quantum B-algebra if and only if its every connected component has a greatest element. However, such a result for infinite posets is not necessarily true. Under certain conditions, some interesting results for a poset to support quantum B-algebra are provided.