We introduce a construction which allows us to identify the elements of the skeletons of a CW-complex $P(m)$ and the monomials in m variables. From this, we infer that there is a bijection between finite CW-subcomplexes of $P(m)$, which are quotients of finite simplicial complexes, and certain bigraded standard Artinian Gorenstein algebras, generalizing previous constructions of Faridi and ourselves.

We apply this to a generalization of Nagata idealization for level algebras. These algebras are standard graded Artinian algebras whose Macaulay dual generator is given explicitly as a bigraded polynomial of bidegree $(1, d)$. We consider the algebra associated to polynomials of the same type of bidegree $(d_{1}, d_{2})$.