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-\b, c\, b+d-\b, 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 ᵤ from R², with the topology L 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 B^d[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 S₃I is a Hausdorff locally compact semitopological semigroup B^d[₀, ) with an adjoined compact ideal I then I is an open subset of S₃I.