Cross-connections of linear transformation semigroups (Q1756693)

Let \(V\) be a vector space over a field, \(Sing(V)\) the multiplicative semigroup of all singular linear transformations on \(V\), and \(\mathcal{S}(V)\) the category of proper subspaces of \(V\) with linear transformations as morphisms. A \textit{cross-connection} from a normal category \(\mathcal{D}\) to a normal category \(\mathcal{C}\) is a triplet (\(\mathcal{D}, \mathcal{C}; \Gamma\)) where \(\Gamma\colon \mathcal{D}\to N^* \mathcal{C}\) is a certain local isomorphism (\(N^* \mathcal{C}\) denotes the normal dual category of the category \( \mathcal{C}\)). The following theorems are proved: 1) \(\mathcal{S}(V)\) is a normal category and the semigroup of normal cones in \(\mathcal{S}(V)\) is isomorphic to \(Sing(V)\); 2) If \(V\) is a finite dimensional vector space, then \(N^*\mathcal{S}(V)\) is isomorphic as a normal category to \(\mathcal{S}(V^*)\), where \(V^*\) is the algebraic dual of \(V\); 3) Every cross-connection semigroup arising from the cross-connections between a certain \textit{annihilator category} \(\mathcal{A}(V)\) and \(\mathcal{S}(V)\) is isomorphic to \(Sing(V)\); 4) The cross-connection semigroup \((\mathcal{B}(V), \mathcal{R}(V);\Gamma)\) is isomorphic to the semigroup of all regular elements of \(\mathcal{T}_V^{\theta}\) where \(\mathcal{R}(V)\) and \(\mathcal{B}(V)\) are certain normal full-subcategories of \(\mathcal{S}(V)\) and \(\mathcal{A}(V)\) respectively, and \(\Gamma\) a suitably defined local isomorphism. Here \(\mathcal{T}_V^{\theta}\) is the variant of the full linear transformation semigroup \(\mathcal{T}_V\) on \(V\) with the binary composition \(*\) defined by \(\alpha*\beta=\alpha\cdot\theta\cdot\beta\) for \(\alpha,\beta\in \mathcal{T}_V\) (\(\theta\) - an arbitrary linear transformation).