## The Jordan-Hölder chain condition and annihilators in finite lattices.(English)Zbl 0721.06008

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.
Reviewer: E.Fuchs (Brno)

### MSC:

 06B10 Lattice ideals, congruence relations 06B05 Structure theory of lattices
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