\(n\)-transitivity and the complementation property (Q5935585)

Let \(\mathbb{M}_k(\mathbb{F})\) and \(\mathbb{M}_{k\times n}(\mathbb{F})\) denote the spaces of all \(k\) by \(k\) matrices and of all \(k\) by \(n\) matrices, respectively. A semigroup \(\mathcal S\) in \(\mathbb{M}_k(\mathbb{F})\) is a non-empty subset of \(\mathbb{M}_k(\mathbb{F})\) which is closed under matrix multiplication. The maximum of the ranks of the matrices in \(\mathcal S\) is called the rank of \(\mathcal S\). \(\mathcal S\) is said to be \(n\)\textit{-transitive} (\(n\leq k\)) if \(\mathcal SX=\mathbb{M}_{k\times n}(\mathbb{F})\) for every rank-\(n\) matrix \(X\) in \(\mathbb{M}_{k\times n}(\mathbb{F})\). An \(n\)-transitive semigroup with no proper \(n\)-transitive subsemigroup is called a minimal \(n\)-transitive semigroup. The paper under review studies properties of \(n\)-transitive rank-\(n\) semigroups and shows that the problem of determining all minimal \(n\)-transitive rank-\(n\) semigroups of \(\mathbb{M}_k(\mathbb{F})\) is equivalent to the problem of determining all collections of \((k-n)\)-dimensional subspaces of \(\mathbb{F}^k\) which are minimal with the complementation property, i.e. every \(n\)-dimensional subspace is complemented by some member of the collection.NEWLINENEWLINENEWLINEOne of the main theorems states that, for a collection \(\mathcal L\) of \((k-n)\)-dimensional subspaces of \(\mathbb{F}^k\), the set \(\mathcal S_{\mathcal L}\) of all those matrices \(T\in \mathbb{M}_k(\mathbb{F})\) which have kernels contain some \(M\in \mathcal L\) is a minimal \(n\)-transitive rank-\(n\) semigroup if and only if \(\mathcal L\) is a minimal collection possessing the complementation property. The paper also gives detailed discussions on \(2\)-transitive rank-\(2\) semigroups in \(\mathbb{M}_4(\mathbb{F})\) and collections of \(2\)-dimensional subspaces of \(\mathbb{F}^4\) with the complementation property.