See [Chow rings of toric varieties defined by atomic lattices, Definition 2.3].
Definition
Let \(\mathcal L\) be a finite lattice, \( \mathcal A = \{A_1,...,A_n\} \) its set of atoms, \(\mathcal G\) a building set and \(\mathcal N(\mathcal L, \mathcal G)\) the nested complex associated to \(\mathcal G\). The building algebra of \(\mathcal L\) associated to \(\mathcal G\) is defined \[D(\mathcal L, \mathcal G) = \mathbb Z[\{x_G\}_{G\in \mathcal G}]/(I_1 + I_2)\] where \(I_1\) is generated by elements corresponding to non-nested subsets of \(\mathcal G\), \[I_1 = \left\langle\prod_{i=1}^t x_{G_i} ~ \middle | ~ \{G_1,...,G_t\} \not\in \mathcal N(\mathcal L, \mathcal G)\right\rangle\] and \(I_2\) is generated by sums of elements living over a single atom, \[I_2 = \left\langle\sum_{\substack{G\in \mathcal G,~ A\leq G}} x_G ~ \middle | ~ A \in\mathcal A \right\rangle.\]
Note that “building algebra” is what I decided to call this, I don’t know if it has a standard name in the literature. Feichtner and Yuzvinsky, who define it, simply call it “the algebra \(D(\mathcal L, \mathcal G)\)”.