An orthogonality space is a set equipped with a symmetric and irreflexive binary relation. We consider orthogonality spaces with the additional property that any collection of mutually orthogonal elements gives rise to the structure of a Boolean algebra. Together with the maps that preserve the Boolean substructures, we are led to the category $\mathcal{N}\mathcal{O}\mathcal{S}$ of normal orthogonality spaces.

Moreover, an orthogonality space of finite rank is called linear if for any two distinct elements e and f there is a third one g such that exactly one of f and g is orthogonal to e and the pairs $e,f$ and $e,g$ have the same orthogonal complement. Linear orthogonality spaces arise from finite-dimensional Hermitian spaces. We are led to the full subcategory $\mathcal{L}\mathcal{O}\mathcal{S}$ of $\mathcal{N}\mathcal{O}\mathcal{S}$ and we show that the morphisms are the orthogonality-preserving lineations.

Finally, we consider the full subcategory $\mathcal{E}\mathcal{O}\mathcal{S}$ of $\mathcal{L}\mathcal{O}\mathcal{S}$ whose members arise from positive definite Hermitian spaces over Baer ordered ⋆-fields with a Euclidean fixed field. We establish that the morphisms of $\mathcal{E}\mathcal{O}\mathcal{S}$ are induced by generalised semiunitary mappings.

正交空间是具有对称和自反二元关系的集合。我们考虑具有附加属性的正交空间，即任何相互正交元素的集合都会产生布尔代数的结构。连同保留布尔子结构的映射，我们被引导到类别$\mathcal{N}\mathcal{哦}\mathcal{秒}$ 正规正交空间。

此外，有限秩的正交空间被称为线性的，如果对于任何两个不同的元件Ë和˚F有第三个克，使得恰好一个˚F和克正交Ë和对$\mathrm{电子},F$ 和 $\mathrm{电子},G$有相同的正交补。线性正交空间来自有限维厄米空间。我们被引导到完整的子类别$\mathcal{升}\mathcal{哦}\mathcal{秒}$ 的 $\mathcal{N}\mathcal{哦}\mathcal{秒}$ 并且我们证明了态射是保持正交性的线型。

最后，我们考虑完整的子类别 $\mathcal{乙}\mathcal{哦}\mathcal{秒}$ 的 $\mathcal{升}\mathcal{哦}\mathcal{秒}$其成员来自具有欧几里得固定域的 Baer 有序 ⋆ 域上的正定 Hermitian 空间。我们建立的态射$\mathcal{乙}\mathcal{哦}\mathcal{秒}$ 是由广义半酉映射引起的。