Subsemigroups of an amenable group

The following example is presented. $G$ is a nondiscrete locally compact amenable group. $H$ is a subgroup of $G$ with zero Haar measure such that if $g \in G \sim H$ then (i) there is a subsemigroup $S = \bigcup \nolimits _{n = 0}^\infty {{I_n}}$ of $G$ where the ${I_n}’s$ are open pairwise disjoint ideals of $S$ (all right- or all left-ideals) and (ii) $g \in {I_0}$.