Orthogonality and complementation in the lattice of subspaces of a finite-dimensional vector space over a finite field

Varieties corresponding to classes of complemented posets

As algebraic semantics of the logic of quantum mechanics there are usually used orthomodular posets, i.e. bounded posets with a complementation which is an antitone involution and where the join of orthogonal elements exists and the orthomodular law is satisfied. When we omit the condition that the complementation is an antitone involution, then we obtain skew-orthomodular posets. To each such poset we can assign a bounded lambda-lattice in a non-unique way. Bounded lambda-lattices are lattice-like algebras whose operations are not necessarily associative. We prove that any of the following pr…

Gradual transitivity in orthogonality spaces of finite rank

An orthogonality space is a set together with a symmetric and irreflexive binary relation. Any linear space equipped with a reflexive and anisotropic inner product provides an example: the set of one-dimensional subspaces together with the usual orthogonality relation is an orthogonality space. We present simple conditions to characterise the orthogonality spaces that arise in this way from finite-dimensional Hermitian spaces. Moreover, we investigate the consequences of the hypothesis that an orthogonality space allows gradual transitions between any pair of its elements. More precisely, give…

借助正交关系刻画希尔伯特域上的二次空间

Characterisation of quadratic spaces over the Hilbert field by means of the orthogonality relation

本文通过对正交集施加传递性、极小性等条件，逐步刻画埃尔米特空间，最终证明希尔伯特域上秩n≥4的二次空间对应的正交集是同秩二次正集中的极小元。