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Scientific Seminar:
Topological Algebra

Speaker: 
Oleg Gutik

Talk: 
On the monoid of cofinite partial isometries of $\mathbb{N}$ with a bounded finite noise 
When & where:  October 7, 2020, at 16^{40 } in ZOOM^{ } 
Abstract: 
We extend results of the papers [Carl Eberhart and John Selden, On the Closure of the Bicyclic Semigroup, Transactions of the American Mathematical Society, Vol. 144 (Oct., 1969), pp. 115126 ] and [M. O. Bertman and T. T. West, Conditionally Compact Bicyclic Semitopological Semigroups, Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences Vol. 76 (1976), pp. 219226] for the monoid $\mathbf{I}\mathbb{N}_{\infty}^{\textbf{g}[j]}$ of cofinite partial isometries of $\mathbb{N}$ with a bounded finite noise for any positive integer $j$. In particular we show that for any positive integer $j$ every Hausdorff shiftcontinuous topology $\tau$ on $\mathbf{I}\mathbb{N}_{\infty}^{\textbf{g}[j]}$ is discrete and and if $\mathbf{I}\mathbb{N}_{\infty}^{\textbf{\emph{g}}[j]}$ be a proper dense subsemigroup of a Hausdorff semitopological semigroup $S$, then $S\setminus \mathbf{I}\mathbb{N}_{\infty}^{\textbf{\emph{g}}[j]}$ is a closed ideal of $S$, and moreover if $S$ is a topological inverse semigroup then $S\setminus \mathbf{I}\mathbb{N}_{\infty}^{\textbf{\emph{g}}[j]}$ is a topological group.
