Morita equivalence of factorizable semigroups

A semigroup is called factorizable if each of its elements can be written as a product. We study equivalences and adjunctions between various categories of acts over a fixed factorizable semigroup. We prove that two factorizable semigroups are Morita equivalent if and only if they are strongly Morita equivalent. We also show that Morita equivalence of finite factorizable semigroups is algorithmically decidable in finite time.