We analyse the connection between the
computability and continuity of functions
in the case of homomorphisms between
topological algebraic structures. Inspired
by the Pour-El and Richards equivalence theorem
between computability and boundedness for closed
linear operators on Banach spaces, we study
the rather general situation of partial homomorphisms
between metric partial universal algebras. First, we
develop a set of basic notions and results that reveal
some of the delicate algebraic, topological and effective
properties of partial algebras. Our main computability concepts
are based on numerations and include those of effective metric
partial algebras and effective partial homomorphisms. We prove a
general equivalence theorem that includes a version of
the Pour-El and Richards
Theorem, and has other applications. Finally, the Pour-El and Richards
axioms for computable sequence structures on Banach spaces are
generalised to computable partial sequence structures on metric
algebras, and we prove their equivalence with our computability
model based on numerations.