Boolean Vector Spaces

This article discusses the basic properties of finite-dimensional Boolean vector spaces and their linear transformations. We first introduce a natural Boolean-valued inner product and discuss orthonormal bases. We show that all bases are orthonormal and have the same cardinality. It is shown that the set of subspaces form an atomistic orthomodular poset. We then demonstrate that an operator that is diagonal relative to one basis is diagonal relative to all bases and that all projections are diagonal. It is proved that an operator is diagonal if and only if any basis consists of eigenvectors of the operator. We characterize extremal states and show that a state is extremal if and only if it is pure. Finally, we introduce tensor products and direct sums of Boolean vector spaces.