A decomposition theory for complete modular meet-continuous lattices (Q535097)

Let \(L\) be a complete modular meet-continuous lattice, and let \([a,b]\) be an interval in \(L\). The author introduces the notions of large element in \([a,b]\), independent subset over \(a\), and small interval \([a,x_1]\) in \([a,b]\). A decomposition of \([a,b]\) is defined to be an indexed family \(X=\{x_i\mid i\in I\}\) of elements of \([a,b]\) satisfying the conditions (1) \(X\) is independent and \(a\); (2) \(\bigvee X\) is large in \([a,b]\); (3) for each \(i\in I\), the interval \([a,x_i]\) is small. The author finds necessary and sufficient conditions for \(L\) under which every interval of \(L\) has a decomposition; if these conditions are satisfied, a constructive description of the decomposition is presented. In the second part of the paper, the case when \(L\) is complemented is dealt with in detail. The relations to an analogous situation in ring theory are emphasized, but no result of ring theory is applied.