Brzozowski \cite{Brz64:drgx} showed
how to construct deterministic automata
from regular expressions by defining an automaton structure
on the set of regular expressions.
Rutten showed in \cite{Rut06:FACS-short} how to construct Mealy machines from
arithmetic bitstream specifications by defining a Mealy machine structure
on the set of arithmetic bitstream expressions.
More recently, Bonchi, Bonsangue, Rutten and
Silva~\cite{BBRS:CONCUR09,BonRutSil08:Kleene-poly-short}
showed that the same ideas can be applied in a much more general setting of
coalgebras for inductively defined functors and
generalised regular expressions.

We refer to the abovementioned constructions as examples of
coalgebraic synthesis, since they all rely on the fact that
it is possible to inductively define coalgebraic structure
on a set of algebraic specifications.
Such interaction between syntax (algebra) and behaviour (coalgebra)
can often be captured by a bialgebra for a distributive law
\cite{Bartels:PhD,TuriPlotkin:LICS-GSOS-short}).
Indeed, Jacobs showed in \cite{Jacobs:bialg-dfa-regex-short} that
Brzozowski's construction gives rise to a bialgebra on the set of
regular expressions, and that the well known relationships
between regular expressions, deterministic automata and formal languages
can be viewed in a bialgebraic framework.

In this talk we show that also Rutten's construction of Mealy machines from
arithmetic specifications can be described in terms of a bialgebra.
However, in order to obtain a distributive law, it is necessary to extend the syntax with
an auxilliary operation which can be seen as adding a one-element buffer to Mealy machines.