Boolean algebras derived from a quotient of a distributive lattice
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 \(x\vee y\in\mathcal{K}^d_I\).
scientific article; zbMATH DE number 7596046
scientific article