On a locally compact monoid of cofinite partial isometries of ℕ with adjoined zero

Abstract

Let 𝒞ℕ 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 Cscr;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.

Keywords

• partial isometry
• inverse semigroup
• partial bijection
• bicyclic monoid
• discrete
• locally compact
• topological semigroup
• semitopological semigroup
• 20m18
• 20m20
• 20m30
• 22a15
• 54a10
• 54d45