The main theorem of discrete Morse theory for Morse matchings with finitely many rays

The main theorem of discrete Morse theory states that a finite, regular CW complex X equipped with a discrete Morse function is homotopy equivalent to a CW complex that has one d-cell for each critical cell in X of index d. We prove, using the terminology of discrete Morse matchings, a version of this theorem that works for infinite complexes, provided the Morse matching induces finitely many equivalence classes of rays in the Hasse diagram. We work in the class of h-regular posets, introduced by Minian, which is strictly larger than the class of face posets of regular CW complexes. A homological version of the theorem for cellular posets is also given.