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The vertices of simplicial complexes are usually totally ordered so that face maps of each simplex can be defined easily for the purposes of homology. That gives an "oriented" simplicial complex.

But that's not always necessary, since a partial order that restricts to a total order on each simplex would also suffice. In fact, that is called an "ordered" simplicial complex.

One could generalize this even more to consider irreflexive binary relations that restrict to a total order on each simplex; as a consequence these relations will also be antisymmetric. Is there a name for such relations?

(Cross posted from Math Stackexchange)

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You may look into what is called $\Delta$-complexes, and may be the notion of a $\Delta$-complex will suit you. There you give an orientation to each simplex and then glue them by using face maps, which become orientation-reversing or preserving. An orientation-reversing glueing gives you an orientable complex. Beyond that, I have no knowledge of relevant terminology.

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  • $\begingroup$ Thank you for your answer, but I don't see how the face maps induce gluing that is either orientation-reversing or preserving. Could you provide a reference for that? $\endgroup$
    – Herng Yi
    Commented Mar 14, 2016 at 23:17

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