Most manual transmissions do not have solenoids. Automatic transmissions, however, rely on solenoids to facilitate gear changes. Solenoids are electro-hydraulic valves that control one or more gears ...
The transmission in a vehicle must be able to function in challenging circumstances without fail. Arguably the hardest working part of the engine, the transmission doesn't get nearly the recognition ...
You could take your complaint directly to Mazda's customer relations people. You might get a hearing because the warranty had only just expired. We have heard reports of problems with shift solenoids ...