  We consider semigroup algorithmic problems in the wreath product $\mathbb{Z}
\wr \mathbb{Z}$. Our paper focuses on two decision problems introduced by
Choffrut and Karhum\"{a}ki (2005): the Identity Problem (does a semigroup
contain the neutral element?) and the Group Problem (is a semigroup a group?)
for finitely generated sub-semigroups of $\mathbb{Z} \wr \mathbb{Z}$. We show
that both problems are decidable. Our result complements the undecidability of
the Semigroup Membership Problem (does a semigroup contain a given element?) in
$\mathbb{Z} \wr \mathbb{Z}$ shown by Lohrey, Steinberg and Zetzsche (ICALP
2013), and contributes an important step towards solving semigroup algorithmic
problems in general metabelian groups.

Group Theory;Discrete Mathematics