Not to go into too much detail, but I believe the OBDII system continuously checks sensors based on conditions and output status. Faults are reset if they have not re-occurred in x key cycles, or km, or hours of operation. This is also manufacturer specific. Some cars never reset themselves.

For example, an egr solenoid would have a flow switch that verified exhaust flow (or pressure) whenever the solenoid is actuated. This is typically monitored 'continuously', i.e. the computer checks it once every scan. If the solenoid is supposed to be open and the flow is not present in x seconds (xx scans) then the CPU throws the EGR code.

And while I thought the air pump had a sensor, I think Klaus is correct, the OBDII sytem just checks to make sure the downstream oxygen sensor does not read 'rich' after a cold start. If that sensor were to read rich it would indicate that the catalytic converter is still cold, meaning the air pump failed to heat it and reduce emissions.

So, the device Klaus suggested mimics the voltage that a happy downstream O2 sensor would produce. Whenever the OBDII system chooses to look at the voltage it sees a figure in range and it will never throw a code.

It's a very cheap fix, I think I will try it.