Punctual categoricity and a jump operation in the primitive recursive degrees

Abstract

The paper is devoted to the study of punctual structures such that any isomorphism between any of its punctual copies is primitive recursively reducible to a fixed 0, 1-valued oracle function. To estimate the complexity of such punctual structures and isomorphisms between them we introduce and investigate the weak (0, 1-valued) jump operation. In the paper we establish that there is a rigid punctual structure for which all isomorphisms are low under the weak jump, and at least one of them is not primitive recursive. Also we construct a rigid punctual structure with every isomorphism reducible to the weak jump of the zero function, and with at least one having a high degree.