Isomorphisms of linear semigroups (Q1360284)

Let \(K\) be a division ring and \(V\) a right vector space over \(K\). Let \(\text{End}_K(V)\) denote the semigroup of all \(K\)-linear maps from \(V\) to \(V\). A linear semigroup is a subsemigroup of \(\text{End}_K(V)\). A rank-one-operator of \(V\) is a \(K\)-linear map of \(V\) such that the dimension of the image is one. A subsemigroup of \(\text{End}_K(V)\) is said to be wide if it contains all rank-one-operators of \(V\). The authors prove that every isomorphism of wide linear semigroups is induced by a semilinear bijection between the corresponding vector spaces, unless these vector spaces have dimension one.