Toral posets and the binary spectrum property

We introduce a family of posets which generate Lie poset subalgebras of $A_{n-1}=\mathfrak{s}\mathfrak{l}(n)$ whose index can be realized topologically. In particular, if $\mathcal{P}$ is such a $\mathit{\text{toral}}\mathit{\text{poset}}$, 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 $\mathit{\text{binary}}$; 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.