On locally compact shift continuous topologies on the semigroup B[0,∞) with an adjoined compact ideal

Let [0,∞) be the set of all non-negative real numbers. The set B[0,∞)=[0,∞)× [0,∞) with the following binary operation (a,b)(c,d)=(a+c-min,c\,b+d-min,c\) is a bisimple inverse semigroup.In the paper we study Hausdorff locally compact shift-continuous topologies on the semigroup B[0,∞) with an adjoined compact ideal of the following tree types.The semigroup B[0,∞) with the induced usual topology τᵤ from R², with the topology τL which is generated by the natural partial order on the inverse semigroup B[0,∞), and the discrete topology are denoted by B¹[0,∞), B²[0,∞), and Bᵈ[0,∞), respectively. We show that if S₁I (S₂I) is a Hausdorff locally compact semitopological semigroup B¹[0,∞) (B²[0,∞)) with an adjoined compact ideal I then either I is an open subset of S₁I (S₂I) or the topological space S₁I (S₂I) is compact. As a corollary we obtain that the topological space of a Hausdorff locally compact shift-continuous topology on S¹₀=B¹[0,∞)∪\0\ (resp. S²₀=B²[0,∞)∪\0\) with an adjoined zero 0 is either homeomorphic to the one-point Alexandroff compactification of the topological space B¹[0,∞) (resp. B²[0,∞)) or zero is an isolated point of S¹₀ (resp. S²₀).Also, we proved that if SdI is a Hausdorff locally compact semitopological semigroup Bᵈ[0,∞) with an adjoined compact ideal I then I is an open subset of SdI.