On characterizations of atomistic lattices (Q1866796)

A lattice \(L\) is called atomistic if every of its elements is a join of atoms of \(L\). An element \(a\) of a complete lattice \(L\) is pure in \(L\) if for each \(c\in K([a,1])\) (the set of all compact elements of the interval \([a,1]\)) there exists an element \(b\in L\) such that \(c=a\vee b\) and \(a\wedge b=0\). An element \(a\in L\) (\(L\) is a lattice with \(0\)) is called neat, if \(b\succ a\) (\(b\in L\)) implies the existence of an element \(c\in L\) with \(b=a\vee c\) and \(a\wedge c=0\). In the paper several characterizations of atomistic lattices are given in terms of concepts related to pure and neat elements.