  We show that the equational theory of the structure $\langle \omega^{\omega}:
(x,y)\mapsto x+y, x\mapsto \omega x \rangle $ is finitely axiomatizable and
give a simple axiom schema when the domain is the set of transfinite ordinals.
We give an algorithm that given a pair of terms $(E,F)$ decides in linear time
with respect of their common length whether or not $E=F$ is a consequence of
the axioms.
