Is it possible to perform a split face on just one side of a single surface?

For example, a surface sphere only ending to create a split image on one side, not on the other side as well, would it be possible.

I know one solution is to split the surface into two down the middle but sometimes I don't want the surface to split in two.

Also is it possible to remove or hid split line individually and I don't just mean turning off lines altogether.