有界有限噪声下$\mathbb{N}$的余有限部分等距幺半群
On the monoid of cofinite partial isometries of $\mathbb{N}$ with a bounded finite noise
摘要：
本文研究了正整数集$\mathbb{N}$的带有界有限噪声$j$的余有限部分等距幺半群$\mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$的代数性质.对于幺半群$\mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$，我们证明了Eberhart和Selden关于双循环半群在局部紧拓扑逆半群中的闭包的一些经典结果的对应结果.特别地，我们证明了对任意正整数$j$，$\mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$上的每个Hausdorff移位连续拓扑$τ$都是离散的，且若$\mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$是Hausdorff拓扑半群$S$的真稠密子半群，则$S\setminus \mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$是$S$的闭理想，且若$S$是拓扑逆半群，则$S\setminus \mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$是拓扑群.并给出了幺半群$\mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$在局部紧拓扑逆半群中闭包的代数和拓扑结构。
英文摘要：
In the paper we study algebraic properties of the monoid $\mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$ of cofinite partial isometries of the set of positive integers $\mathbb{N}$ with the bounded finite noise $j$. For the monoids $\mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$ we prove counterparts of some classical results of Eberhart and Selden describing the closure of the bicyclic semigroup in a locally compact topological inverse semigroup. In particular we show that for any positive integer $j$ every Hausdorff shift-continuous topology $τ$ on $\mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$ is discrete and if $\mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$ is a proper dense subsemigroup of a Hausdorff semitopological semigroup $S$, then $S\setminus \mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$ is a closed ideal of $S$, and moreover if $S$ is a topological inverse semigroup then $S\setminus \mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$ is a topological group. Also we describe the algebraic and topological structure of the closure of the monoid $\mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$ in a locally compact topological inverse semigroup.
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