Definition
Let \(\mathcal L\) be a lattice. A building set for \(\mathcal L\) is a subset \(\mathcal G\subset \mathcal L\setminus \{\hat 0\}\) such that for every \(X \in \mathcal L \setminus \{\hat 0\}\), there is an isomorphism of posets \[\varphi_X:\prod_{i = 1}^k [\hat 0, G_i] \xrightarrow{\sim} [\hat 0, X]\] where \(\max \mathcal G_{\leq X} = \{G_1,...,G_k\}\) is the subset consisting of maximal lower bounds for \(X\) in \(\mathcal G\). Note that, in particular, \(\mathcal L \setminus \{\hat 0\}\) is a building set for \(\mathcal L\).