In universal algebra and model theory, one usually requires that the underlying set be non-empty. If the signature has constants in it, then this is no restriction, but otherwise, there is no initial object. For example, under this definition there is no initial object in the category of semigroups. The reason for this restriction is that bad things happen to first-order logic when the underying carrier set is empty. Many standard theorems break down when you allow the empty domain. For example, the following theorem of first-order logic $$ \forall x P(x) \rightarrow \exists x P(x) $$ becomess false.