In this note we contribute to the recently developing study of “almost Boolean” quantum logics (i.e. to the study of orthomodular partially ordered sets that are naturally endowed with a symmetric difference). We call them enriched quantum logics (EQLs). We first consider set-representable EQLs. We disprove a natural conjecture on compatibility in EQLs. Then we discuss the possibility of extending states and prove an extension result for \(\mathbb {Z}_{2}\)-states on EQLs. In the second part we pass to general orthoposets with a symmetric difference (GEQLs). We show that a simplex can be a state space of a GEQL that has an arbitrarily high degree of noncompatibility. Finally, we find an appropriate definition of a “parametrization” as a mapping between GEQLs that preserves the set-representation.

在这篇笔记中，我们对最近发展的“几乎布尔”量子逻辑的研究做出了贡献（即对自然赋予对称差异的正交模偏序集的研究）。我们称它们为富集量子逻辑 (EQL)。我们首先考虑可表示集合的 EQL。我们反驳了关于 EQL 兼容性的自然猜想。然后我们讨论扩展状态的可能性并证明对 EQL 上的\(\mathbb {Z}_{2}\) -状态的扩展结果。在第二部分中，我们将介绍具有对称差异 (GEQL) 的一般矫形器。我们证明了单纯形可以是具有任意高度不兼容性的 GEQL 状态空间。最后，我们找到了“参数化”的适当定义，即保留集合表示的 GEQL 之间的映射。