Completely hereditarily atomic OMLS (Q6634061)

An orthomodular lattice (OML) is an ortholattice \(\mathbf L=(L,\vee,\wedge,{}',0,1)\) satisfying the orthomodular law \(x\vee y\approx x\vee\big((x\vee y)\wedge x'\big)\). The height of \(\mathbf L\) is one less than the maximum cardinality of its chains. \(\mathbf L\) is said to have the covering property if \([b,b\vee a]\) has height at most \(1\) for every \(b\in L\) and every atom \(a\) of \(\mathbf L\). \(\mathbf L\) is said to have the \(2\)-covering property if \([b,b\vee a]\) has height at most \(2\) for every \(b\in L\) and every atom \(a\) of \(\mathbf L\). \(\mathbf L\) is said to be completely hereditarily atomic if it is complete and each of its complete subalgebras is atomic. Among others, results on algebraic OMLs, directly irreducible algebraic OMLs of finite height and algebraic OMLs having the covering property are proved. Further, it is shown that these results are sharp in that the covering property cannot be weakened to the \(2\)-covering property and algebraicity cannot be weakened to being completely hereditarily atomic. Constructions of Kalmbach and of Keller are used to produce OMLs having special properties.