When a magnet is cut the atoms within the domains remain aligned. In almost every case the cut would be between two domains, leaving aligned domains in the two halves. If you cut a domain you would create two smaller domains, each with a north pole and a south pole. So no matter where you cut the result is two magnets, each with its own north and south pole. The more domains, the stronger the magnet.
Fortunately for us, the earth generates its own magnetic field, which effectively fends off dangerous particles from the sun.