An oriented simplicial complex is a simplicial complex K equipped with a partial ordering on its vertices that restricts to a linear ordering on each simplex. I am wondering if there is a standard name for simplicial maps $f\colon K\to K$ which preserve the partial order on vertices. I would have liked to call them orientation preserving but I think this is usually used to just mean that the linear ordering on each simplex is preserved up to an even permutation. I had no luck with Google.

Example: If P is a poset, then the order complex (=nerve) of the poset is naturally oriented by the original ordering on P and any order preserving map on P induces a simplicial map preserving orientation in the strong sense I described above. In particular, after a barycentric subdivision one always can get my property.

Also an oriented simplicial complex can be viewed as a simplicial set in a natural way but not all simplicial morphisms translate into morphisms of simplicial sets. The ones I am considering do.

Question: Is there a standard terminology for a simplicial map on an oriented simplicial complex that preserves the ordering on the vertices?