The lattice of subsemilattices of a semilattice
The main result of this paper is the following characterization of the subsemilattice-lattice: A lattice \(\mathcal L\) is isomorphic to \(\text{Sub}(L)\) for some semilattice \(L\) iff it is algebraic, biatomic (that is, for any two non-zero \(x,y\in {\mathcal L}\) and any atom \(z\leq x\lor y\) there exist atoms \(x'\leq x\), \(y'\leq y\) such that \(z\leq x'\lor y'\)) and satisfies \((S_ 3)\) (we say that a biatomic lattice \(\mathcal L\) satisfies properties \((S_ n)\) if for any ideal \(V\) generated by \(n\) atoms \(a_ 1,\dots,a_ n\in {\mathcal L}\) there exists a finite semilattice \(L_ V\) such that \(V\simeq\text{Sub}(L_ V)\) and the natural embedding of ideals \(V\mapsto W\) induces the embedding of semilattices \(L_ V\mapsto L_ W\).
scientific article; zbMATH DE number 549249