I have a craftsman DYT 4000 tractor with the 18.5 hp briggs and stratton engine intek plus and wonder if there is an alternator to recharge the battery... I have difficulty keeping it charged and being able to restart the tractor aafter running it for an hour or so.

Hi,
There is a stator underneath the flywheel and some sort of rectifier. Could be as simple as a diode or as complicated as a full blown rectifier They vary from about 3 Amps to about 16 Amps depending on what the manufacturer but from B&S. Numbers or pictures of the starter side of the engine would help.
Peace,
Clarke