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

Publication date: 30 December 2021

Abstract: Let

m a t h s c r C_{m} a t h b b N

be a monoid which is generated by the partial shift

a l p h a c o l o n n m a p s t o n + 1

of the set of positive integers

m a t h b b N

and its inverse partial shift

. In this paper we prove that if

S

is a submonoid of the monoid

m a t h b f I m a t h b b N_{i n f t y}

of all partial cofinite isometries of positive integers which contains

m a t h s c r C_{m} a t h b b 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.

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