The reflexivity index of a lattice of sets. (Q2925690)

In the background of finite lattice \(\mathcal L\) of subsets of a set \(X\), the authors derive a formula for the reflexivity index \(k_X(\mathcal L)\), by first obtaining an upper bound for \(k_X(\mathcal L)\), then the finite subset lattices are characterized with necessary conditions in order that \(k_X(\mathcal L)=1\) and \(k_X(\mathcal L)\leq 2\). A sufficient condition on \(X\) is obtained so that \(k_X(\mathcal L)=2\). The reflexivity indices of some known infinite lattices are determined and sufficient condition obtained to ensure that \(k_X(\mathcal L)\) is finite. Also certain properties of nests are studied so that their reflexivity index could be analyzed in the finite case. The results promise to have applications in known lattices in the case of topological spaces.