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In recent years, the combinatorial properties of monomials ideals andbinomial ideals have been widely studied. In particular, combinatorialinterpretations of free resolution algorithms have been given in both cases. Inthis present work, we will introduce similar techniques, or modify existingones to obtain two new results. The first is $S[\Lambda]$-resolutions of$\Lambda$-invariant submodules of $k[\mathbb{Z}^n]$ where $\Lambda$ is alattice in $\mathbb{Z}^n$ satisfying some trivial conditions. A consequencewill be the ability to resolve submodules of $k[\mathbb{Z}^n/\Lambda]$, and inparticular ideals $J$ of $S/I_{\Lambda}$, where $I_{\Lambda}$ is the latticeideal of $\Lambda$.  Second, we will provide a detailed account in three dimensions on how to liftthe aforementioned resolutions to resolutions in $k[x,y,z]$ of ideals withmonomial and binomial generators.

A Combinatorial Algorithm to Find the Minimal Free Resolution of an Ideal with Binomial and Monomial Generators