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Abstract

The elementary theory of $\langle {\alpha ; \times } \rangle $, where $\alpha $ is an ordinal and $\times $ denotes ordinal multiplication, is decidable if and only if $\alpha < \omega ^{\omega }$. Moreover if $|_r$ and $|_l$ respectively denote the right- and left-hand divisibility relation, we show that Th $\langle {\omega ^{\omega ^{\xi }}; \mid _r} \rangle $ and Th {$\langle {\omega ^{\xi }; \mid _l} \rangle $ are decidable for every ordinal $\xi $. Further related definability results are also presented.