Examples of semicomputable sets of real and complex numbers

We investigate the concept of semicomputability of relations on abstract structures. We consider three possible definitions of this concept, which all reduce to the classical notion of recursive enumerability over the natural numbers. By working in the algebra of the reals, with and without order, we find examples of sets which distinguish between these three notions. We also find interesting examples of sets of real and complex numbers which are semicomputable but not computable.