The concept of a global phase diagram - a diagram in the space of fluid interaction parameters in which regions of similar types of phase behavior are demarcated - has proved very valuable in both theoretical and experimental studies of fluid mixtures. This type of work sheds light on the essential transition mechanisms involved in the change of character of the phase diagram, such as multi-critical points and double points, and other phenomena of great practical importance in supercritical fluid extraction processes. The first step in this process is to understand the changes in critical-line structure which underlies these differences. It is shown in this paper that the so called lattice gas equation-of-state, when modified for chain-like molecules, is extremely sensitive to small modifications in the chain length parameter. The cause of this effect is traced by analyzing the complete algebraic solution of the polynomial representing the critical lines.