The purpose of this paper is
to settle two problems. The first is Sorenson’s conjecture on whether every right
cancellative left amenable semigroup is lefl cancellative. The second, posed by
Argabright and Wilde, is whether every left amenable semigroup satisfies the strong
Følner condition (SFC). We first show that these two problems are equivalent,
then prove that the answer to both questions is no, through analyzing the
semidirect product of semigroups in relation to amenability and cancellation
properties. We conclude by investigating further the properties of semigroups
satisfying SFC, and finally include some analogous results for left measurable
semigroups.
