On a representation of lattices (Q1174503)

The authors give a representation of complete lattices. They use the following definition: \(L_{\mathcal A}=\{x\mid x\in L, x=\bigvee\{y\mid y\triangleleft_{\mathcal A} x\}\}\), where \(\triangleleft_{\mathcal A}\) is a binary relation defined on a lattice \(L\) with connection to a set \(\mathcal A\) of covers of \(L\). They ``show that in fact an \(L_{\mathcal A}\) can have any shape whatsoever. More exactly, for any complete lattice \(S\), there is a locale, indeed an atomic Boolean algebra, \(L\) and a system of covers \(\mathcal A\) such that \(S\) is isomorphic to \(L_{\mathcal A}\). Moreover, \(\mathcal A\) can be chosen so that the \(A\in{\mathcal A}\) consist of two elements each. Furthermore, each finite \(S\) is isomorphic to an \(L_{\mathcal A}\) with \({\mathcal A}=\{A,B\}\) consisting of just two covers''.