Characterizations of complete sublattices of a given complete lattice
Let \(L\) be a complete lattice. A mapping \(\varphi\) of \(L\) into itself is called a closure operator if \(\varphi (x)\geq x\) and \(\varphi (\varphi ( x))=\varphi (x)\) for all \(x\in X\) and \(\varphi (x)\leq\varphi (y)\) whenever \(x\leq y\) for all \(x,y\in L\). The dual notion is that one of a kernel operator. It is proved that \(U\) is a complete sublattice of \(L\) if and only if there exists a closure operator \(\varphi\) preserving all suprema with \(U=\{x\in L;\varphi (x)=x\}\) (or, dually, if and only if there exists a kernel operator \(\psi\) preserving all infima with \(U=\{x\in L;\psi (x)=x\}\)). By this fact the author derives and formulates a characterization of all complete sublattices of \(L\) in terms of Galois-closed relations.
scientific article; zbMATH DE number 1747989