Annihilators in modular lattices
type: Item
instance of: scholarly article
title: Annihilators in modular lattices
zbMATH Open document ID: 0613.06004
DOI: 10.1007/BF01224021
author: Brian A. Davey
author: J. Nieminen
published in: Algebra Universalis
publication date: 1986-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 06B05
Mathematics Subject Classification ID: 06B10
Mathematics Subject Classification ID: 06D05
Mathematics Subject Classification ID: 06C05
zbMATH DE Number: 3991561
zbMATH Keywords: Prime-Annihilator Condition
zbMATH Keywords: intersection of prime annihilators
zbMATH Keywords: prime ideals
zbMATH Keywords: finite distributive lattices
zbMATH Keywords: lattice of ideals
reviewed by: Gudrun Kalmbach
MaRDI profile type: Publication
cites work: Hierarchies of Distributive Lattices Satisfying Annihilator Conditions
cites work: Normal lattices
cites work: n-Normal Lattices
cites work: The multiplier extension of a distributive lattice
cites work: Some annihilator conditions on distributive lattices
cites work: Relative annihilators in lattices
full work available at URL: https://doi.org/10.1007/bf01224021
OpenAlex ID: W1972534012
description: scientific article; zbMATH DE number 3991561