On a locally compact monoid of cofinite partial isometries of $\mathbb{N}$ with adjoined zero

Abstract: Let $\mathscr{C}\mathbb{N}$ be a monoid which is generated by the partial shift $\alpha\colon n\mapsto n+1$ of the set of positive integers $\mathbb{N}$ and its inverse partial shift $\beta\colon n+1\mapsto n$. In this paper we prove that if $S$ is a submonoid of the monoid $\mathbf{I}\mathbb{N}{\infty}$ of all partial cofinite isometries of positive integers which contains $\mathscr{C}_\mathbb{N}$ as a submonoid then every Hausdorff locally compact shift-continuous topology on $S$ with adjoined zero is either compact or discrete. Also we show that the similar statement holds for a locally compact semitopological semigroup $S$ with an adjoined compact ideal.