Toral posets and the binary spectrum property

We introduce a family of posets which generate Lie poset subalgebras of $A_{n-1}=\mathfrak{sl}(n)$ whose index can be realized topologically. In particular, if $\mathcal{P}$ is such a \textit{toral poset}, then it has a simplicial realization which is homotopic to a wedge sum of $d$ one-spheres, where $d$ is the index of the corresponding type-A Lie poset algebra $\mathfrak{g}_A(\mathcal{P})$. Moreover, when $\mathfrak{g}_A(\mathcal{P})$ is Frobenius, its spectrum is \textit{binary}; that is, consists of an equal number of 0's and 1's. We also find that all Frobenius, type-A Lie poset algebras corresponding to a poset whose largest totally ordered subset is of cardinality at most three have a binary spectrum.