Lindenhurst is an incorporated village, and a fence permit is almost certainly required there.
Contact village hall and ask if there is a fence permit for the address.
If a permit is required but a permit was not pulled, then your neighbor is in violation regardless of the property line issue, and officials can order all the work done so far removed just for that reason. The building department will take it from there if no permit exists.
If a permit was pulled, then a survey of the property being fenced is required (in every place I've ever heard of), and the building department will ensure that the fence is in compliance with the lot lines if you have any questions.
It's imperative that the fence not encroach on your property, or else you could lose your rights to that property over time.
Licensed Real Estate Salesperson
New York, NY