Minimum cutsets for an element of a subspace lattice over a finite vector space (Q1267605)

The authors give a finite vector space analog for the theorem that determines all the minimum cutsets for an element of a Boolean algebra, given by \textit{J. R. Griggs} and \textit{D. J. Kleitman} [Order 6, No. 1, 31-37 (1989; Zbl 0689.06010)]. Let \({\mathcal L}_n(q)\) be the lattice of subspaces of an \(n\)-dimensional vector space over the finite field of \(q\) elements, ordered by inclusion. The main result of the present paper is the following one: For an element \(A\in{\mathcal L}_n(q)\), the minimum cutset is just \(L(A)\) if \(\dim A<n/2\), is \(U(A)\) if \(\dim A>n/2\) and both \(L(A)\) and \(U(A)\) if \(\dim A=n/2\), where \(L(A)\) is the collection of all \(X\in{\mathcal L}_n(q)\) such that \(X\not\subseteq A\) and \(\dim(X\cap A)=\dim(X)-1\), and \(U(A)\) the collection of all \(Y\in{\mathcal L}_n(q)\) such that \(A\not\subseteq Y\) and \(\dim(Y+A)=\dim(Y)+1\).