On Semitopological Bicyclic Extensions of Linearly Ordered Groups

Abstract

For a linearly ordered group G , we define a subset A ⊆ G to be a shift-set if, for any x, y, z ϵ A with y < x, we get x · y^-1 ··z ϵ A. We describe the natural partial order and solutions of equations on the semigroup B(A) of shifts of positive cones of A . We study topologizations of the semigroup B(A). In particular, we show that, for an arbitrary countable linearly ordered group G and a nonempty shift-set A of G , every Baire shift-continuous T₁-topology τ on B(A) is discrete. We also prove that, for any linearly nondensely ordered group G and a nonempty shift-set A of G , every shift-continuous Hausdorff topology τ on the semigroup B (A) is discrete.