The three-component irreversible surface-reaction model A+BC→AC+1/2B2 with infinite reaction rates between nearest-neighbor adspecies is studied by Monte Carlo simulations. For a square lattice the system evolves to a degenerate poisoned state exponentially in time, except for a narrow range of pressures in which a reactive quasi-steady-state exists. The latter poisons very slowly in time. For a hexagonal lattice a true reactive steady state occurs for a range of pressures bordered by continuous and discontinuous transitions to poisoned states. The latter behavior is observed by adding the reaction channel A+B→AB on a square lattice. Some properties of these models are elucidated by analysis of appropriate exact master equations and corresponding pair approximations. Other issues addressed include ‘‘extent of variability’’ of the degenerate poisoned states, the effect of finite reaction rates, and spatial correlations.