M\"obius Polynomials and Splitting Algebras of Direct Products of Posets

In this paper, we will study the M\"obius polynomial, an invariant of ranked posets that arises in the study of splitting algebras. We will present a formula for the M\"obius polynomial of the direct product of posets in terms of the M\"obius polynomials of the factors. We will then use this formula to calculate Hilbert series and graded trace generating functions associated to the splitting algebras of the Boolean algebra and the poset of factors of a natural number n.