Definition
Let \(\mathcal L, \mathcal G = \{G_1,...,G_k\}\) be a finite lattice and a building set. We say that \(X,Y\in \mathcal L\) are incomparable if \(X\not\leq Y\) and \(Y\not\leq X\).
We say that a subset \(S \subset \mathcal G\) is nested if, whenever \(X_1,...,X_k \in S\) are pairwise incomparable elements and \(k\geq 2\), the total join is not in \(\mathcal G\): \[X_1 \vee ... \vee X_k \not\in \mathcal G .\]
We denote by \(\mathcal N(\mathcal L, \mathcal G) \subseteq 2^{\mathcal L}\) the nested set complex associated to \(\mathcal G\), so called because it forms an abstract simplicial complex.