Annihilators in modular lattices. (English) Zbl 0613.06004

Davey, Brian A.; Nieminen, Juhani

Algebra Univers. 22, 154-158 (1986).

For elements a, b of a lattice L, the (dual) annihilator is \[ <a,b>_ d:=<x\in L| \quad x\vee a\geq b\}\quad and\quad <a,b>:=\{x\in L| \quad x\wedge a\leq b\}. \] The annihilator \(<a,b>\) is called prime if \(<a,b>\cup <b,a>_ d=L\) and \(<a,a\wedge b>\cap <a\wedge b,a>_ d=\emptyset\). The Prime-Annihilator Condition for L requires that every annihilator is an intersection of prime annihilators. The authors prove an analogue of the result that a lattice is distributive iff every ideal is an intersection of prime ideals: A lattice is modular and weakly atomic iff it satisfies the Prime-Annihilator Condition. For finite distributive lattices prime ideals and prime annihilators coincide. A lattice is modular iff its lattice of ideals satisfies the Prime- Annihilator Condition.

Reviewer: G.Kalmbach

MSC:

06B05	Structure theory of lattices
06B10	Lattice ideals, congruence relations
06D05	Structure and representation theory of distributive lattices
06C05	Modular lattices, Desarguesian lattices

Keywords:

Prime-Annihilator Condition; intersection of prime annihilators; prime ideals; finite distributive lattices; lattice of ideals

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