Boolean algebras derived from a quotient of a distributive lattice
type: Item
instance of: scholarly article
title: Boolean algebras derived from a quotient of a distributive lattice
zbMATH Open document ID: 1498.03161
author: Yanyan Li
published in: Bulletin of the Malaysian Mathematical Sciences Society. Second Series
publication date: 2022-10-04T00:00:00Z
review text: A \textit{derivation} on a distributive lattice \(L\) is a function \(d:L\rightarrow L\) such that \(d(x\wedge y)=(d(x)\wedge y)\vee(x\wedge d(y))\); \(d(x\vee y)=d(x)\vee d(y)\). If \(I\) is an ideal of \(L\), then \(\ker_Id=\{x\in L:d(x)\in I\}\); \((a)^d_L=\{x\in L:d(a\wedge x)\in I\}\); \(x\theta^d_Iy\) if and only if \((x)^d_I=(y)^d_I\). \(\mathcal{K}^d_I=\{a\in L:(a)^d_L=\ker_Id\}\). \(\theta^d_I\) is a congruence on \(L\). Now the main theorem of the paper is as follows. If \(I\) is a nontrivial ideal of \(L\), then \(L/\theta^d_L\) is a Boolean algebra if and only if for each \(x\in L\) there is a \(y\in(x)^d_L\) such that \(x\vee y\in\mathcal{K}^d_I\).
Mathematics Subject Classification ID: 03G05
Mathematics Subject Classification ID: 03G10
Mathematics Subject Classification ID: 06D05
zbMATH DE Number: 7596046
zbMATH Keywords: distributive lattice
zbMATH Keywords: Boolean algebra
zbMATH Keywords: congruence
zbMATH Keywords: ideal
zbMATH Keywords: filter
reviewed by: J. Donald Monk
MaRDI profile type: Publication
full work available at URL: s40840-022-01344-7
OpenAlex ID: W4283740398
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DOI: 10.1007/S40840-022-01344-7
description: scientific article; zbMATH DE number 7596046