CW-complex Nagata idealizations

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 bidegree (d1, d2).