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

Abstract

Let C N be a monoid which is generated by the partial shift α : n → n+1 of the set of positive integers N and its inverse partial shift β : n+1 → n. In this paper we prove that if S is a submonoid of the monoid IN ∞ of all partial cofinite isometries of positive integers which contains C 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.