Definition
Let \(\mathcal L\) be a finite lattice with unique minimal element \(\hat 0\) whose atoms are \(\mathcal A = \{A_1,...,A_n\}\), and let \(\mathcal G\) be a building set for \(\mathcal L\). For each \(X\in \mathcal L\) let \(\lfloor X\rfloor = \{A\in \mathcal A \mid X \geq A\}\) and define a vector \(v_X\in \mathbb R^n\) by \[(v_X)_i = \begin{cases}1 & \text{if } A_i \in \lfloor X\rfloor \\ 0 & \text{otherwise}\end{cases}\] so that the atoms of \(\mathcal L\) form the basis of \(\mathbb R^n\). Now for each \(S\subseteq \mathcal L\) let \(V(S)\) be the rational polyhedral cone in \(\mathbb R^n\) spanned by the vectors \(v_X\) for \(X\in S\). Finally, let \[\Sigma(\mathcal L, \mathcal G) = \{V(S) ~ \mid ~ S\in \mathcal N(\mathcal L, \mathcal G)\}\] be the fan consisting of all cones associated to nested sets in \(\mathcal L\). Note that the rays of \(\Sigma(\mathcal L, \mathcal G)\) are in bijection with the vertices of \(\mathcal N(\mathcal L, \mathcal G)\) (and therefore the elements of \(\mathcal G\)), the \(2\)-dimensional facets of \(\Sigma(\mathcal L, \mathcal G)\) are in bijection with the edges of \(\mathcal N(\mathcal L, \mathcal G)\), etc.
The toric variety associated to \(\mathcal L, \mathcal G\) is \(X_{\Sigma(\mathcal L, \mathcal G)}.\)