Minimum Cutsets for an Element of a Subspace Lattice over a Finite Vector Space

Let ℒn (q) denote the lattice of subspaces of an n-dimensional vector space over the finite field of q elements, ordered by inclusion. In this note, we prove that for all n and m the minimum cutset for an element A with dim(A) = m of ℒn(q) is just L(A) if m < n/2, is U (A) if m > n/2, and both L(A) and U (A) if m = n/2, where L(A) is the collection of all X ∈ ℒn(q) such that X ⊈ A and dim(X ∩ A) = dim(X) - 1, and U (A) the collection of all Y ∈ ℒn(q) such that A ⊈ Y and dim(Y + A) = dim(Y) + 1. Hence a finite vector space analog is given for the theorem of Griggs and Kleitman that determines all the minimum cutsets for an element of a Boolean algebra.