We apply Monte Carlo simulation to explore the adsorption of a positively charged polyelectrolyte on a lipid monolayer membrane, composed of electronically neutral, monovalent anionic and mulvitalent anionic phospholipids. We systematically assess the influence of various factors, including the intrinsic rigidity of the polyelectrolyte chain, the bead charge density of the polyelectrolyte, and the ionic strength of the saline solution, on the interfacial structural properties of the polyelectrolyte/monolayer complex. The enhancement of the polyelectrolyte chain intrinsic rigidity reduces the polyelectrolyte conformational entropy loss and the energy gains in electrostatic interaction, but elevates the segregated anionic lipid demixing entropy loss. This energy-entropy competition results in a nonmonotonic dependence of the polyelectrolyte/monolayer association strength on the degree of chain rigidity. The semiflexible polyelectrolyte, i.e., the one with an intermediate degree of chain rigidity, is shown to associate onto the ternary membane below a higher critical ionic concentration. In this ionic concentration regime, the semiflexible polyelectrolyte binds onto the monolayer more firmly than the pancake-like flexible one and exhibits a stretched conformation. When the chain is very rigid, the polyelectrolyte with bead charge density Z(b) = +1 exhibits a larger tail and tends to dissociate from the membrane, whereas the one with Z(b) = +2 can still bind onto the membrane in a bridge-like conformation. Our results imply that chain intrinsic rigidity serves as an efficient molecular factor for tailoring the adsorption/desorption transition and interfacial structure of the polyelectrolyte/monolayer complex.