  We describe the problem of Sweedler's duals for bialgebras as essentially
characterizing the domain of the transpose of the multiplication. This domain
is the set of what could be called ``representative linear forms'' which are
the elements of the algebraic dual which are also representative on the
multiplicative semigroup of the algebra. When the algebra is free, this notion
is indeed equivalent to that of rational functions of automata theory. For the
sake of applications, the range of coefficients has been considerably
broadened, i.e. extended to semirings, so that the results could be specialized
to the boolean and multiplicity cases. This requires some caution (use of
``positive formulas'', iteration replacing inversion, stable submodules
replacing finite-rank families for instance). For the theory and its
applications has been created a rational calculus which can, in return, be
applied to harness Sweedler's duals. A new theorem of rational closure and
application to Hopf algebras of use in Physics and Combinatorics is provided.
The concrete use of this ``calculus'' is eventually illustrated on an example.
