Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Classification of k-nets   A finite \emph{$k$-net} of order $n$ is an incidence structure consisting of
$k\ge 3$ pairwise disjoint classes of lines, each of size $n$, such that every
point incident with two lines from distinct classes is incident with exactly
one line from each of the $k$ classes. Deleting a line class from a $k$-net,
with $k\ge 4$, gives a \emph{derived} ($k-1$)-net of the same order.
  Finite $k$-nets embedded in a projective plane $PG(2,K)$ coordinatized by a
field $K$ of characteristic $0$ only exist for $k=3,4$, see \cite{knp_k}. In
this paper, we investigate $3$-nets embedded in $PG(2,K)$ whose line classes
are in perspective position with an axis $r$, that is, every point on the line
$r$ incident with a line of the net is incident with exactly one line from each
class. The problem of determining all such $3$-nets remains open whereas we
obtain a complete classification for those coordinatizable by a group. As a
corollary, the (unique) $4$-net of order $3$ embedded in $PG(2,K)$ turns out to
be the only $4$-net embedded in $PG(2,K)$ with a derived $3$-net which can be
coordinatized by a group. Our results hold true in positive characteristic
under the hypothesis that the order of the $k$-net considered is smaller than
the characteristic of $K$.

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The categories of this paper are: cmp-lg
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The categories of this paper are: chem-ph
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The categories of this paper are: eess
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The categories of this paper are: acc-phys
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-th
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The categories of this paper are: bayes-an
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The categories of this paper are: funct-an
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The categories of this paper are: quant-ph
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The categories of this paper are: nlin
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The categories of this paper are: plasm-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: atom-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: nucl-ex
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The categories of this paper are: q-fin