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.



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> > Question: Is there a standard terminology for a simplicial map on an oriented simplicial complex that preserves the ordering on the vertices?
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Terminology Concerning Oriented Simplicial Complexes
A simplicial complex with partially ordered vertices such that the vertex set of each simplex is a chain of the poset is called an **ordered** simplicial complex. This avoids the confusion with orientability, etc. The terminology is not new, you can find it in [this paper](http://www.maths.ed.ac.uk/~aar/papers/deltars.pdf) from 1969. So, your maps are just *maps of ordered simplicial complexes*. Regarding the paragraph just before your question, if a map $f:K\to K$ of ordered simplicial complexes is *injective* on simplices, then it can be realized as a map of $\Delta$-sets. I think this is covered Friedman's wonderful [notes](http://faculty.tcu.edu/gfriedman/papers/simp.pdf), but it's been a while since I went through those. 


