We show that the equational theory of the structure $\langle ω^ω: (x,y)\mapsto x+y, x\mapsto ωx \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.