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.

翻译：我们考虑圈积群 $\mathbb{Z} \wr \mathbb{Z}$ 中的半群算法问题。本文聚焦于 Choffrut 与 Karhumäki（2005）提出的两个判定问题：$\mathbb{Z} \wr \mathbb{Z}$ 中有限生成子半群的恒等元问题（半群是否包含单位元？）与群问题（半群是否构成群？）。我们证明这两个问题均为可判定的。该结果补充了 Lohrey、Steinberg 与 Zetzsche（ICALP 2013）所证明的 $\mathbb{Z} \wr \mathbb{Z}$ 中半群成员问题（半群是否包含给定元素？）的不可判定性，并为解决一般可解群中的半群算法问题迈出了重要一步。