  The purpose of this paper is twofold.
  1. We give combinatorial bounds on the ranks of the groups
$\Tor^{R}_\bullet(k,k)_\bullet$ in the case where $R = k[\Lambda]$ is an affine
semi-group ring, and in the process provide combinatorial proofs for bounds by
Eisenbud, Reeves and Totaro on which Tor groups vanish. In addition, we show
that if the bounds hold for a field $k$ then they hold for $\field[\Lambda]$
and any field $\field$. Moreover, we provide a combinatorial construction for a
free resolution of $\field$ over $\field[\Lambda]$ which achieves these bounds.
  2. We extend the lexicographic discrete Morse function construction of Babson
and Hersh for the determination of the homotopy type and homology of order
complexes of posets to a larger class of facet orderings that includes orders
induced by monomial term orders.
  Since it is known that the order complexes of finite intervals in the poset
of monomials in $k[\Lambda]$ ordered by divisibility in $k[\Lambda ]$ govern
the $\Tor$-groups, the newly developed tools are applicable and serve as the
main ingredients for the proof of the bounds and the construction of the
resolution.
