Simple quantales were introduced by Paseka and are closely related to \(C^{*}\)-algebras. The sublattice \({\mathcal {S}}(L)\) of \(L^{L}\) is a prototypical example of a simple quantale under the multiplication defined by composition \(\circ \), where \({\mathcal {S}}(L)\) denotes the set of sup-preserving endomaps of a nontrivial complete lattice L and \(L^{L}\) denotes the set of order-preserving endomaps of L. However, \((L^{L},\circ )\) is not a quantale. We define a new composition operation \(\cdot \) on the set of order-preserving maps so that \((L^{L},\cdot )\) forms a quantale, where L is a completely distributive lattice. Moreover, \(({\mathcal {S}}(L),\circ )\) is quantale isomorphic to a quotient of \((L^{L},\cdot )\). In addition, if there exists a nontrivial surjective quantale homomorphism from \(L^{L}\) to a unital quantale M, then M is a simple quantale.

简单量子由 Paseka 引入，与\(C^{*}\)代数密切相关。\(L^{L}\)的子格\({\mathcal {S}}(L) \)是由组合\(\circ \)定义的乘法下的简单量子的典型示例，其中\( {\mathcal {S}}(L)\)表示非平凡完全格L的保序内图集，而\(L^{L}\)表示L的保序内图集。然而，\((L^{L},\circ )\)不是量子。我们在保序映射集上定义一个新的组合操作\(\cdot \) ，以便\((L^{L},\cdot )\)形成量子，其中L是完全分布格。此外，\(({\mathcal {S}}(L),\circ )\)与\((L^{L},\cdot )\)的商量子同构。此外，如果存在从\(L^{L}\)到单位量子M 的非平凡满射量子同态，则M是简单量子。