The grand-partition-function-zero method is applied to lattice systems of rigid molecules, based on the algebraic technique of Ruelle. Consideration of small collections of lattice molecules, through this approach, provides rigorous delineation of regions of the complex activity plane which are free of zeros of the grand partition function, and hence free of thermodynamic singularities. Two conjectures, as yet unproved, are offered, which greatly reduce the computational effort required in using the technique. A simple proof is provided for the absence of physical phase transitions in monomerdimer systems, and bounds are obtained on the locations of the transitions of other lattice gases.