The Jordan-Hölder chain condition and annihilators in finite lattices (Q2640628)

If L is a lattice, then for a,b\(\in L\) the set \(<a,b>=\{x\); \(x\wedge a\leq b\}\) is called an annihilator, and its dual \(<a,b>_ d=\{x\); \(x\vee a\geq b\}\) is called a dual annihilator. An annihilator \(<a,b>\neq L\) is called prime, if \(<a,b>\cup <b,a>_ d=L\) and \(<a,a\wedge b>\cap <a\wedge b,a>_ d=\emptyset\). The author characterizes the Jordan-Hölder chain condition by means of prime annihilators in finite lattices. The intersection property of prime annihilators is also considered.