| Abstract: |
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{b, c}, b + d - min{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 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 SI 1 (SI 2) 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 SI 1 (SI 2) or the topological space SI 1 (SI 2) is compact. As a corollary we obtain that the topological space of a Hausdorff locally compact shift-continuous topology on S1 0 = B1 [0,∞) ∪ {0} (resp. S2 0 = 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 S1 0 (resp. S2 0). Also, we proved that if SId is a Hausdorff locally compact semitopological semigroup Bd [0,∞) with an adjoined compact ideal I then I is an open subset of SId. [ABSTRACT FROM AUTHOR] |
