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?