  Weakly recognizing morphisms from free semigroups onto finite semigroups are
a classical way for defining the class of omega-regular languages, i.e., a set
of infinite words is weakly recognizable by such a morphism if and only if it
is accepted by some B\"uchi automaton. We consider the descriptional complexity
of various constructions for weakly recognizing morphisms. This includes the
conversion from and to B\"uchi automata, the conversion into strongly
recognizing morphisms, and complementation. For some problems, we are able to
give more precise bounds in the case of binary alphabets or simple semigroups.
