Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A generalization of a theorem of Ern\'{e}   Let $X$ be a finite set, $Z \subseteq X$ and $y \notin X$. Marcel Ern\'{e}
showed in 1981, that the number of posets on $X$ containing $Z$ as an antichain
equals the number of posets $R$ on $X \cup \{ y \}$ in which the points of $Z
\cup \{ y \}$ are exactly the maximal points of $R$. We prove the following
generalization: For every poset $Q$ with carrier $Z$, the number of posets on
$X$ containing $Q$ as an induced sub-poset equals the number of posets $R$ on
$X \cup \{ y \}$ which contain $Q^d + A_y$ as an induced sub-poset and in which
the maximal points of $Q^d + A_y$ are exactly the maximal points of $R$. Here,
$Q^d$ is the dual of $Q$, $A_y$ is the singleton-poset on $y$, and $Q^d + A_y$
denotes the direct sum of $Q^d$ and $A_y$.

35
The categories of this paper are: mtrl-th
19
The categories of this paper are: eess
13
The categories of this paper are: quant-ph
5
The categories of this paper are: gr-qc
36
The categories of this paper are: cmp-lg
12
The categories of this paper are: nucl-th
7
The categories of this paper are: hep-th
16
The categories of this paper are: stat
18
The categories of this paper are: econ
1
The categories of this paper are: cs
23
The categories of this paper are: solv-int
34
The categories of this paper are: chem-ph
4
The categories of this paper are: cond-mat
10
The categories of this paper are: q-bio
28
The categories of this paper are: alg-geom
30
The categories of this paper are: q-alg