In the paper we investigate topological properties of a topological
Brandt λ0-extension B0λ(S) of a semitopological monoid
S with zero. In particular we prove that for every Tychonoff
pseudocompact (resp., Hausdorff countably compact, Hausdorff
compact) semitopological monoid S with zero there exists
a unique semiregular pseudocompact (resp., Hausdorff
countably compact, Hausdorff compact) extension B0λ(S) of S
and establish their Stone-Cˇ ech and Bohr compactifications.
We also describe a category whose objects are ingredients in
the constructions of pseudocompact (resp., countably compact,
sequentially compact, compact) topological Brandt λ0-
extensions of pseudocompact (resp., countably compact, sequentially
compact, compact) semitopological monoids with
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