Lattices of subspaces of vector spaces with orthogonality

Author:

Chajda Ivan1,Länger Helmut12

Affiliation:

1. Palacký University Olomouc, Faculty of Science, Department of Algebra and Geometry, 17. listopadu 12, 771 46 Olomouc, Czech Republic

2. TU Wien, Faculty of Mathematics and Geoinformation, Institute of Discrete Mathematics and Geometry, Wiedner Hauptstraße 8-10, 1040 Vienna, Austria

Abstract

It is well known that the lattice of subspaces of a vector space over a field is modular. We investigate under which conditions this lattice is orthocomplemented with respect to the orthogonality operation. Using this operation, we define closed subspaces of a vector space and study the lattice of these subspaces. In particular, we investigate when this lattice is modular or orthocomplemented. Finally, we introduce splitting subspaces as special closed subspaces and we prove that the poset of splitting subspaces and the poset of projections are isomorphic orthomodular posets. The vector spaces under consideration are of arbitrary dimension and over arbitrary fields.

Funder

OAD

IGA

Publisher

World Scientific Pub Co Pte Lt

Subject

Applied Mathematics,Algebra and Number Theory

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