Possible Degrees in Recursive Copies

Let A be a recursive structure, and let R be a recursive relation on A. Harizanov ( 1991) isolated a syntactical condition which (with additional effectiveness conditions) is necessary and sufficient for A to have recursive copies in which the image of R is r.e. of arbitrary te. degree. We had conjectured that a certain extension of Harizanov’s syntactical condition would (with some effectiveness conditions) be necessary and sufficient for A to have recursive copies in which the image of R is Pa of arbitrary z”, degree, but this is not the case. Here we give examples illustrating some restrictions on the possible Pa degrees. In these examples, the image of R cannot be z”, of degree d unless d possesses an “a-table” (a sequence of sets in which each one is r.e. in and above the earlier ones).