Simple and Immune Relations on Countable Structures ∗

Let A 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 A in which the image of R (¬R, resp.) is simple (immune, resp.) relative to B. We also establish, under certain effectiveness conditions on A and R, a necessary and sufficient condition for the existence of a computable copy B of A in which the image of R (¬R, resp.) is simple (immune, resp.). ∗The first three authors gratefully acknowledge support of the NSF Binational Grant DMS