Toral posets and the binary spectrum property

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