Inspired by the definition of the Buchberger graph of a monomial ideal, we study proper divisibility of monomials as a partial order in N^n, from a combinatorial and topological point of view. From this order relation we obtain a new family of posets, that we call posets of proper divisibility. Surprisingly, the order complexes of these posets are homologically non-trivial. We prove that these posets are dual CL-shellable, we completely describe their homology (with coefficients in Z) and we compute their Euler characteristic using generating functions. Moreover this relation gives a new interesting example of a dual CL-shellable poset which is not CL-shellable.

This is a joint work with Davide Bolognini, Emanuele Ventura and Volkmar Welker.