Some Remarks on Least Moduli

Abstract

Modulus of a computable approximation is a function which returns the number of a stage at which the approximation has already converged for its argument. The least modulus points at the earliest such stage for each of its arguments. We recall and show some properties of least moduli, including their close connection to c.e. degrees, and minimal witnessing functions for FM-representable sets. We observe, for instance, that the non-density theorem for the d.c.e. degrees gives an example of an incomplete degree that has no least moduli below ${\underset{\_}{0}}^{\prime }$ . Using the properties of least moduli themselves, we construct a degree containing no least moduli for itself and having least moduli of incomparable degrees. In particular, the technique used demonstrates an approach of constructing a non-c.e. degree, which is somewhat different from that proposed by Cooper.