  We prove that two semigroups with local units are Morita equivalent if and
only if they have a joint enlargement. This approach to Morita theory provides
a natural framework for understanding McAlister's theory of the local structure
of regular semigroups. In particular, we prove that a semigroup with local
units is Morita equivalent to an inverse semigroup precisely when it is a
regular locally inverse semigroup.
