Looks like from what I can see is that this sink probably backs up quite a bit! IDK
It’s in improperly executed island fixture vent.
It’s rare, but I do see kitchen sinks plumbed as an island when they are not islands. Most often when the kitchen is an add-on in a dbl-wythe brick exterior or seperation wall.
What would the proper venting look like?
That really depends on details of the home and sink location that we are not privy to at this time. Hopefully the OP will return and offer more pertinant details.
Now that the OP posted a photo, it would/should look like the photo that I posted.
That’s what I thought.
Or with an aav.
If allowed in their area, but I prefer the system shown in the photo. No maintenance unless there are problems. AAV’s have a short expected lifespan… 3-5 years. How many people change them before they fail? Zero. Zip. Nada.
What you are looking at is what is called a “LOOP VENT” The illustration above is what you are looking at. Sometimes this is the only way to vent in and island or where there is no other option to get it up a wall. These are except able according to code. I’ve have some plumbers call it a wet vent but that’s a whole other subject.
You won’t find anything called a “loop vent” in the codes.
Where did you get that from?