Notes on sublattice-lattices
For a lattice \(L\), let \(\text{Sub} (L)\) denote the lattice of all sublattices of \(L\), including the empty set. The lattice \(\text{Sub}(L)\) is called the sublattice-lattice of \(L\). The lattice \(\text{Sub} (L)\) determines \(L\) if and only if, for an arbitrary lattice \(K, \text{Sub} (L)\cong \text{Sub}(K)\) implies \(L\cong K\) or \(L\cong K^d\). The author proves the following three results: Theorem 1. If \(L\) is a directly reducible lattice, then \(\text{Sub} (L)\) determines \(L\). Theorem 2. Selfdual subdirectly irreducible lattices of locally finite length are determined by their sublattice-lattices. Corollary 3. Selfdual splitting varieties are closed under the isomorphism of sublattice-lattices. The author also gives a class of selfdual varieties that are not closed under the isomorphism of sublattice-lattices, and observes that the characterization of closed varieties is still open.
scientific article; zbMATH DE number 1336271