Partial Results in $\Delta_3^0$-Categoricity in Linear Orderings and Boolean Algebras

The notion of $\Delta_3^0$-categoricity in linear orderings and Boolean algebras is examined. We provide a proof for the fact that there are uncountably many relatively $\Delta_3^0$-categorical linear orderings, and furnish a proof of another fact which suggests that the (unrelatively) $\Delta_3^0$-categorical linear orderings may be very difficult to classify. In stark contrast to these results for linear orderings, a complete classification of the relatively $\Delta_3^0$-categorical Boolean algebras is given.