Consistent enumeration on a poset

[farleyknight]

Hey all,

I've got a copy of Schaum's outline of Discrete Mathematics, and in the section on ordered subsets and lattices, it includes the definition of a consistent enumeration:

Succinctly, given a poset P, there exists f: P -> N, so that if a < b then f(a) < f(b)

http://books.google.com/books?id=6A...meration"&source=gbs_search_s&cad=0#PPA447,M1

However, I had this in my notes that it was not just a function but an injection. Of course, looking at it again, I didn't consider the case of a || b. I don't know where I got this from and now I'm slightly confused. The closest I could find to this definition was a linear extension and topological sorting, which are slightly different.

Does anyone know this topic well enough to dispell my confusion?

Thanks,
- Farley

 

[HallsofIvy]

IF P is linearly ordered, then "if a< b then f(a)< f(b)" implies that f is an injection. However, in a partially ordered set, it may happen that a and b are "not comparable" (i.e. none of a= b, a< b, nor b< a is true) in which case f(a) may equal f(b) and so f is not necessarily an injection.