Numberings, c.e. oracles, and fixed points

The Arslanov completeness criterion says that a c.e. set A is Turing complete if and only there exists an A-computable function f without fixed points, i.e. a function f such that W f ( x ) ≠ W x for each integer x. Recently, Barendregt and Terwijn proved that the completeness criterion remains true if we replace the Gödel numbering x ↦ W x with an arbitrary precomplete computable numbering. In this paper, we prove criteria for noncomputability and highness of c.e. sets in terms of (pre)complete computable numberings and fixed point properties. We also find some precomplete and weakly precomplete numberings of arbitrary families computable relative to Turing complete and non-computable c.e. oracles respectively.