Not an overflow. Technically, it is an "air gap" device.
The purpose is to prevent waste water (from the disposal/sink drain)
from backing up *into* the dishwasher during a drain cloggage,
as an example. The air gap precludes siphon action occurring.
Many cities require their installation.

If not required, an alternative (not quite as positive)
is to create a loop in the dishwasher outlet hose which
rises under the counter as high as possible before going
down to the disposal.

Jim

When we first bought the house, the dishwasher would actually discharge
through the thing on top of the sink. I checked all the hoses and found the
hose that was connected to the disposer was totally clogged. I replaced the
hose and the DW discharged into the disposer as it was supposed to do. This
is why I thought it was an overflow. The hose coming from the DW attaches
to a "Y" at the "air gap" device, and another hose goes from the "Y" to the
disposer. This was the hose that was clogged. The hose from the DW is about
5 or 6 feet long. If I ran the hose up to the height of the sink and then
down to the disposer, would that create the loop/trap that you mentioned?

Les