论文标题
Sasaki地图对矫形空间的表征
A characterisation of orthomodular spaces by Sasaki maps
论文作者
论文摘要
鉴于Hilbert Space $ h $，一维子空间的集合$ p（h）$ $ h $的$ h $成为矫形器，当配备了$ h $上的内部产品引起的正交关系$ \ perp $。在这里，\ emph {Orthoset}是$ x $的一对$（x，\ perp）$和$ x $上的对称，irreflexive二进制关系$ \ perp $。在这项贡献中，我们研究了矫正器$（x，\ perp）$的哪些条件足以结论，即矫正器对$（p（h），\ perp）$同构成某些正数空间$ h $，其中\ emph emph {orthomodular space}是线性的空间。为了实现这一目标，我们在矫正器上介绍了\ emph {sasaki maps}，这与萨萨基（Sasaki）在正数晶格上的预测密切相关。我们表明，对于某些正数空间的任何校正$（x，\ perp）$具有足够多的sasaki地图是同构至$（p（h），\ perp）$，我们在$（x，\ perp）上提供更多条件（x，\ perp）$，以确保$ h $实际上是$ \ nath $ \ nath $ \ nmath $ \ $ \ $ \ $ \ c.
Given a Hilbert space $H$, the set $P(H)$ of one-dimensional subspaces of $H$ becomes an orthoset when equipped with the orthogonality relation $\perp$ induced by the inner product on $H$. Here, an \emph{orthoset} is a pair $(X,\perp)$ of a set $X$ and a symmetric, irreflexive binary relation $\perp$ on $X$. In this contribution, we investigate what conditions on an orthoset $(X,\perp)$ are sufficient to conclude that the orthoset is isomorphic to $(P(H),\perp)$ for some orthomodular space $H$, where \emph{orthomodular spaces} are linear spaces that generalize Hilbert spaces. In order to achieve this goal, we introduce \emph{Sasaki maps} on orthosets, which are strongly related to Sasaki projections on orthomodular lattices. We show that any orthoset $(X,\perp)$ with sufficiently many Sasaki maps is isomorphic to $(P(H),\perp)$ for some orthomodular space, and we give more conditions on $(X,\perp)$ to assure that $H$ is actually a Hilbert space over $\mathbb R$, $\mathbb C$ or $\mathbb H$.
