We characterize Δ20-categoricity in Boolean algebras and linear orderings under some extra effectiveness conditions. We begin with a study of the relativized notion in these structures.

Let ???? be a computable structure and let R be a new relation on its domain. We establish a necessary and sufficient condition for the existence of a copy ℬ of ???? in which the image of R (¬R, resp.) is simple (immune, resp.) relative to ℬ. We also establish, under certain effectiveness conditions on ???? and R, a necessary and sufficient condition for the existence of a computable copy ℬ of ???? in which the image of R (¬R, resp.) (...) is simple (immune, resp.). (shrink)

In many classes of structures, each computable structure has computable dimension 1 or $\omega$. Nevertheless, Goncharov showed that for each $n < \omega$, there exists a computable structure with computable dimension $n$. In this paper we show that, under one natural definition of relativized computable dimension, no computable structure has finite relativized computable dimension greater than 1.

We establish that the isomorphism problem for the classes of computable nilpotent rings, distributive lattices, nilpotent groups, and nilpotent semigroups is \-complete, which is as complicated as possible. The method we use is based on uniform effective interpretations of computable binary relations into computable structures from the corresponding algebraic classes.