The d-elements of precoherent preidempotent quantales and their applications

In this paper, we introduce the notion of d-elements on precoherent preidempotent quantale (PIQ), construct Zariski topology on $Max(Q_{d})$ and explore its various properties. Firstly, we give a sufficient condition of a topological space $Max(Q_{d})$ being Hausdorff. Secondly, we prove that if $ P=\mathfrak{B}(P) $ and $ Q=\mathfrak{B}(Q) $, then $P$ is isomorphic to $Q$ iff $ Max(P_{d}) $ is homeomorphic to $ Max(Q_{d}) $. Moreover, we prove that $ (P\otimes Q)_{d} $ is isomorphic to $ P_{d} \otimes Q_{d} $ iff $ P_{d} \otimes Q_{d}=(P_{d} \otimes Q_{d})_{d} $. Finally, we prove that the category $ \textbf{dPFrm} $ is a reflective subcategory of $\textbf{PIQuant}.$