The reflexivity index of a finite distributive lattice of subspaces (Q2015076)

Let \({\mathcal S}(X)\) be the set of all subspaces of a vector space \(X\) (no topology in \(X\) is assumed) and let \({\mathcal L}(X)\) be the set of all linear operators on \(X\). For any \({\mathcal L}\subset {\mathcal S}(X)\) and \({\mathcal F}\subset {\mathcal L}(X)\), define \[ \text{Alg}{\mathcal L}= \{ T\in {\mathcal L}(X): T(M)\subset M \text{ for all }M\in {\mathcal L} \}, \] \[ \text{Lat}{\mathcal F}= \{ M\in {\mathcal S}(X):T(M)\subset M \text{ for all }T\in {\mathcal F} \}. \] A subset \({\mathcal L}\subset {\mathcal S}(X)\) is \textit{reflexive} if \({\mathcal L}=\text{Lat}\text{Alg}{\mathcal L}\). It is proved that any finite distributive lattice of subspaces of \(X\) is reflexive. The \textit{reflexivity index} \(\kappa_{X}(\mathcal L)\) of a reflexive lattice \(\mathcal L\) is defined by \( \kappa_{X}(\mathcal L) =\min \{ |\mathcal F|: {\mathcal L}=\text{Lat}{\mathcal F} \}, \) where \(|{\mathcal F}|\) denotes the cardinality of \(\mathcal F\). As usual, the \textit{dimension} of the vector space~\(X\) is the common cardinality of all its Hamel bases. Let \(\dim X>\aleph_{0}\), and let \(\mathcal L\) be a finite distributive lattice of subspaces of \(X\). It is proved that in this case \(\kappa_{X}(\mathcal L)=\dim X\). There are also investigated the cases \(\dim X\leq\aleph_{0}\) (then \(\kappa_{X}(\mathcal L)\leq 2\)) and \(\dim X=\aleph_{0}\) (then \(\kappa_{X}(\mathcal L) = 2\)). In the case \(\dim X<\infty\), there are obtained necessary and sufficient conditions for the equality \(\kappa_{X}(\mathcal L) = 1\).