You need to check with your Building Inspection Department. Air gaps are local code-required in many places.
If it is not, you can just loop the dw drain hose as high as you can over the top of the dw to achieve a similar result.
Without being able to see what you're describing, it may be an air admittance valve used in lieu of a normal vent pipe. Here's what air admittance valves look like: www.studor.net
If it is a drain hose connection, it should be a "Y" connection on the vertical sink drain tailpiece, or on the side of a disposal, and be open-ended at the top of the "Y".
The dw drainhose just slides onto the end of it and clamps on with ss hose clamps. If it is on a disposal, there is a knock-out plug in the disposal that needs to be tapped out with a screwdriver and hammer before attaching the dw drain hose.
Hope that this helps.