Clarke will likely be the only responder but can anyone explain the purpose of the electric fuel solenoid on the bowl of a Sears tractor carb? Is it just to stop the flow of fuel from flooding the carb after you shut down the ignition?

Biggest purpose is "fuel cut", but it likely has the primary purpose to reduce emissions.

In a gravity system, fuel will leak past the fuel valve essentially evaporating gasoline slowly and contributing to increased emissions.

They may have also had to add it, so you have distinct systems, off, idle and power sections of the throttle based on position. You want the off to idle regeime to be abrupt and not gradual to keep emissions in check.

If there is a fuel pump involved, the fuel cut solenoid will help prevent dieseling as well.

In some of the later model carburated cars, you had a fuel cut solenoid and a bowl vent solenoid and of course the charcoal canister.

Hi,
It can and does serve a number of purposes, but did not come into common use until the common use of the OHV engine. An engine without a working ignition ground to shut it off but a working fuel solenoid will run 2-3 seconds before it exhausts the fuel supply in the jets.
It was installed because the leaner running engines are more prone to after-fire in the muffler, which can particularly harm an OHV engine. Engines should be throttled down for a few seconds before turning them off. Kohler single-cylinder Command engines are particularly susceptible to flywheel key shearing if this is not done. They are less likely to cause valve train damage because they use hydraulic lifters. Engines with solid tappets can slowly bow the pushrods.
Peace,
Clarke

Thanks guys, Bob