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" ?