Decidability and definability results related to the elementary theory of ordinal multiplication (Q2773366)

For \(\alpha\) an ordinal and \(\times\) ordinal multiplication, the author proves that the elementary theory of \(\langle\alpha;\times\rangle\) is decidable if and only if \(\alpha<\omega^\omega\). The undecidability portion is obtained by interpreting the theory of the free monoid on two generators. The author also considers definability in \(\langle\alpha;\times\rangle\). For example, he shows that for \(\alpha\geq\omega\) neither \(<\) nor \(+\) are definable in \(\langle\alpha;\times\rangle\). Further, for the relations of left and right divisibility, \(|_l\) and \(|_r\), the author shows that for every ordinal \(\xi\) the theories of \(\langle\omega^{\omega^\xi},|_l\rangle\) and \(\langle\omega^{\omega^\xi},|_r\rangle\) are decidable.