Biatomic lattices
The author studies ``biatomic'' lattices which are atomic lattices L satisfying: \(p\in L\) atom, a, \(b\in L\), \(p\leq a\vee b\) imply \(p\leq c\vee d\) for atoms \(c\leq a\) and \(d\leq b\). The subspace lattice of a vector space is biatomic. For an ordered division ring D the lattice of convex subsets of D n is biatomic. Let C \(n\subseteq D\) n be the n-cube. Then its face lattice \({\mathcal F}(C\) n) is biatomic. Products and ideals of biatomic lattices are biatomic. The MacNeille completion of a biatomic lattice is biatomic. For a biatomic lattice L there exists in the variety generated by L a complete algebraic biatomic lattice which contains L as a sublattice. For an atomic lattice L are equivalent: (i) L is modular, (ii) L is biatomic and has the exchange property.
scientific article; zbMATH DE number 4047785