I have a simplicial complex, built out of hyper-tetrahedra (5-cells) with the topology of $S_{4}$ and I would like to assign an ordering to it's vertices (some couple thousand), so that I can apply a boundary operator and co-boundary operator on it.

I have been trying to understand how to do this with a cubic lattice, but moving to a triangulated surface has thrown me, and I can't quite grasp how an ordering for the simplicies falls out from an arbitrary ordering of all the vertices.

Wikipedia says something like:

One standard way to do this is to choose an ordering of all the vertices and give each simplex the orientation corresponding to the induced ordering of its vertices.

I guess I'm not getting the "induced" part. Once I have an induced ordering for each 5-cell, I can start to apply the boundary operator and I will know the respective signs for each of the faces, but how does one determine the canonical labeling for each simplex in the complex?

Thanks

EDIT : After the first answer it was revealed to me that there are two working uses of the word "orientation" being used in homology texts. One has to do with simply labeling the complex, the other has to do with manifold orientability (which is what I was interested in).

My question should be re-phrased. How does one orient a simplicial complex so that when a couple of simplicies are acted on by the boundary operator, or co-boundary operator, the necessary sub- or super-simplices that show up appear with the correct compatible signs between the simplicies.

Thanks, and let me know if I can clarify further.