# Terminology for set systems: “trace” or “projection”?

Although the following question is not in itself mathematical, it is the expertise/breadth of the research community in mathematics that I wish to appeal to, beyond the filtered/trained search results I'd just get from Googling.

Let $$\Omega$$ be a non-empty set and $${\mathcal F}$$ a non-empty subset of the powerset $${\mathcal P}(\Omega)$$. In some work with coauthors we found ourselves frequently needing the following construction: take some $$\Gamma\subseteq\Omega$$ and form the new set $${\mathcal F} \wedge \Gamma := \{ X \cap \Gamma \colon X \in {\mathcal F} \} \subseteq {\mathcal P}(\Gamma).$$ Actually, we then need to take subsets of $${\mathcal F}\wedge \Gamma$$; and then iterate this process.

The terminology we are using in the current draft is to call $${\mathcal F}\wedge \Gamma$$ the projection of $${\mathcal F}$$ onto $$\Gamma$$, since this seems like the intuitive picture/metaphor. On the other hand, while browsing Bollobas's Combinatorics earlier this year, I noticed that in the setting where $$\Omega$$ is finite, various people (Frankl, Bollobas himself, presumably others) have called $${\mathcal F}\wedge \Gamma$$ the trace of $${\mathcal F}$$ on $$\Gamma$$.

Question. Is the terminology used in Bollobas's book now the standard one? If so, have people seen the notion of "subtrace" before?

For sake of comparison, I don't think anyone would refer to a graph minor as anything but a "graph minor", this is now completely standard as far as I can tell. So my question is really about whether traces and subtraces are just as standard, or whether this is a convenient bit of terminology popular within a particular cluster of researchers. I don't think this is an overly subjective question; I am hoping for combinatorists or lattice theorists to weigh in here with references or informed personal observation.

Another question (less important, more subjective): which is preferable, "subprojection" or "subtrace" ?

• For what it's worth, Wikipedia uses "subhypergraph" for your construction: en.wikipedia.org/wiki/Hypergraph#Terminology – Sam Hopkins Nov 22 at 17:41
• @SamHopkins Thanks - this is almost the same although note that my construction allows the empty set to belong to ${\mathcal F}\wedge\Gamma$. Moreover, since in that language I want to delete some vertices and then remove some hyperedges, I seem to have a sub-subhypergraph where the first sub doesn't mean the same as the second sub ... :-/ – Yemon Choi Nov 22 at 17:54
• Right, it does not seem appropriate for your setting, but worth noting. – Sam Hopkins Nov 22 at 17:56