On semitopological  α-bicyclic monoid

Serhii Bardyla


We consider a semitopological α-bicyclic monoid Bα and prove that it is algebraically isomorphic to the semigroup of all order isomorphisms between the principal upper sets of ordinal ωα. We prove that for every ordinal α and every (a,b)Bα if a or b is a non-limit ordinal, then (a,b) is an isolated point in Bα. We show that for every ordinal α<ω+1 every  locally compact semigroup topology on Bα is discrete. On the other hand, we construct an example of a non-discrete locally compact topology τlc on Bω+1 such that Bω+1,τlc is a topological inverse semigroup. This example shows that there is a gap in the proof of Theorem 2.9 [16]  stating that for every ordinal α the semigroup Bα does not admit non-discrete locally compact inverse semigroup topologies.

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