Lattices characterized by their sublattice-lattices
For a lattice \(L\) let Sub\((L)\) denote the lattice of all sublattices of \(L\) and \(L^d\) the dual of \(L\). It is proved that if \(L\) and \(K\) are lattices with Sub\((L)\cong \text{Sub} (K)\) and \(L\) is weakly complemented or \(L\) is modular and ordinal sum indecomposable or \(L\) is semimodular, strongly atomic and ordinal sum indecomposable, then \(L\cong K\) or \(L \cong K^d\). Hence, a complemented modular lattice \(L\) is characterized by Sub\((L)\).
scientific article; zbMATH DE number 1226213