The local hardware store identified it ... it's an "Air Admittance Valve", a code-acceptable way to vent a sink when there's no way to run a stack pipe. The valve part is out of sight, up by the sink, above the level of the trao, as it should be.

However, it's usually running vertically, not out of the side like that.
http://www.diyadvice.com/diy/plumbing/kitchen/install-air-admittance-valve/ shows a properly installed one.

The only thing that clerk in the hardware store confirmed was he/she confirmed why they are working as a stock clerk in a hardware store instead of working in the plumbing industry...

THAT IS NOT AN AAV--(Air Admittance valve).

From the top down you have what appears to be an 1-1/4" DWV grade tubular tailpiece into a trap adapter.

The trap adapter is an 1-1/4" compression x 1-1/2" female slip.

Although you cannot see it, inside the bottom of the trap adapter and the top of the Tee there is an 1-1/2" x close nipple.

The tee is an 1-1/2" slip x 1-1/2" slip x 1" FIP Sanitary tee.

From the side inlet of that tee you have a 1" polythelene MIP x barb adapter to which the 1" SD polyethelene pipe is attached.

We don't know what there intended purpose for that poly line might be, but if it runs up to an AAV behind the drawer that is also a couple code violations.

First off, the minimum size line permitted for a vent is 1-1/4".

Second-"Code Prohibits using Poly pipe inside a structure."

Third, an AAV may not be installed in a concealled location.

God only knows what the hose clamp on the bottom of the tee is used for. It appears as if they may be relying on the screw part of that clamp to serve as a pipe hanger by resting it on that small projection in the hole through the floor.

WHERE IS THE TRAP?

If the trap is below the floor then the it exceeds the code maximum of 24" vertical from the drain opening to the water level in the trap.

The gray water line is polybutelene pipe with plastic fittings and knowing the failure rate of that junk, Which is evidenced by the number of class action lawsuites against it, that line is nothing but a ticking time bomb.

Hey ... not my choice of plumber, and this town isn't big enough to support more than a few of them. Medium-range plans (next year or two) are to replace that vanity, so we'll replace that stuff with real copper

I have a copy of the building permit where it was signed off as OK. :) OK?

The trap is there, just above the top of the picture, and the valve is up and to the right, out of the picture. That drain is resting on packed dirt under the house, with a pipe hanger to the stud next to it.

Apparently the plumber responsible for this is well-known enough locally that when he saw the "idiosyncratic" way the trap was arranged (in a picture I have not posted here), the guy in the hardware store said, "I'd tell you who he is so you can avoid him, but he's dead."

Cheap, but dead.