restriction(A, I)
restriction(A, F)
A ^ F
restriction F
The restriction of an arrangement ${\mathcal A}$ to a subspace $X$ is the (multi)arrangement with hyperplanes $H_i\cap X$, where $H\in {\mathcal A}$ but $H\not\supseteq X$. The subspace $X$ may be defined by a ring element or an ideal.
If an index or list (or set) of hyperplanes $S$ is given, then $X=\bigcap_{i\in S}H_i$. In this case, the restriction is a realization of the matroid contraction $M/S$, where $M$ denotes the matroid of ${\mathcal A}$.
In general, the restriction is denoted ${\mathcal A}^X$. Its ambient space is $X$.
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Unfortunately, the term ``restriction'' is used in conflicting senses in arrangements versus matroids literature. In the latter terminology, ``restriction'' to $S$ is a synonym for the deletion of the complement of $S$.