We define a super analog of the classical Pl\"{u}cker embedding of the
Grassmannian into a projective space. Only a very special case was considered
before in the literature. The "super Pl\"{u}cker map" that we introduce takes
the Grassmann supermanifold $G_{r|s}(V)$ to a "weighted projective space"
$P\left(\Lambda^{r|s}(V)\oplus \Lambda^{s|r}(\Pi V)\right)$ with weights
$+1,-1$. Here $\Lambda^{r|s}(V)$ denotes the $r|s$th exterior power of a
superspace $V$. We identify the super analog of Pl\"{u}cker coordinates and
show that our map is a rational embedding. We investigate a super analog of the
Pl\"{u}cker relations. We obtain them for $r|0$ and $n|m$. Also, we consider
another type of relations due to H. Khudaverdian and show that they are
equivalent to (super) Pl\"{u}cker relations for $r|s=2|0$ (this is new even in
the classical case), but in general are only a consequence of the Pl\"{u}cker
relations. We also discuss possible application to super cluster algebras (the
notion only partly known at present).
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