In [15], Macanuola and Umar studied the algebraic properties of the upper C+(a; b) =
{qipj ∈ C (p; q): i 6 j} and lower C−(a; b) = {qipj ∈ C (p; q): i > j} semigroups of the bicyclic monoid C (a; b).Let C+(p; q)0 and C−(p; q)0 be the semigroups C+(a; b) and C−(a; b) with an associated zero. It is known [7] that on a bicyclic semigroup with an associated zero C (p; q)0, every Hausdorff locally compact translationally continuous topology is either compact or discrete.This paper describes all locally compact translationally continuous Hausdorff topologies on the additive semigroup of nonnegative integers with associated zero (!; +)0 and on the semigroups C+(p; q)0 and C−(p; q)0. In particular, it is proved that on the semigroups C+(p; q)0 and C−(p; q)0 there exists a continuum of various Hausdorff locally compact translationally continuous topologies up to topological isomorphism, and there are exactly three such topologies on the semigroups C+(p; q)0 and C−(p; q)0 up to homeomorphism.

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