Recent zbMATH articles in MSC 06Chttps://zbmath.org/atom/cc/06C2024-05-13T19:39:47.825584ZWerkzeugA characterisation of orthomodular spaces by Sasaki mapshttps://zbmath.org/1532.460232024-05-13T19:39:47.825584Z"Lindenhovius, Bert"https://zbmath.org/authors/?q=ai:lindenhovius.bert"Vetterlein, Thomas"https://zbmath.org/authors/?q=ai:vetterlein.thomasSummary: Given a Hilbert space \(H\), the set \(P(H)\) of one-dimensional subspaces of \(H\) becomes an orthoset when equipped with the orthogonality relation \(\bot\) induced by the inner product on \(H\). Here, an \textit{orthoset} is a pair \((\boldsymbol{X},\bot)\) of a set \(\boldsymbol{X}\) and a symmetric, irreflexive binary relation \(\bot\) on \(\boldsymbol{X}\). In this contribution, we investigate what conditions on an orthoset \((\boldsymbol{X},\bot)\) are sufficient to conclude that the orthoset is isomorphic to \((P(H),\bot)\) for some orthomodular space \(H\), where \textit{orthomodular spaces} are linear spaces that generalize Hilbert spaces. In order to achieve this goal, we introduce \textit{Sasaki maps} on orthosets, which are strongly related to Sasaki projections on orthomodular lattices. We show that any orthoset \((\boldsymbol{X},\bot)\) with sufficiently many Sasaki maps is isomorphic to \((P(H),\bot)\) for some orthomodular space, and we give more conditions on \((\boldsymbol{X},\bot)\) to assure that \(H\) is actually a Hilbert space over \(\mathbb{R}, \mathbb{C}\) or \(\mathbb{H}\).