Generalized Domination.

Abstract

This thesis develops the theory of the everywhere domination relation between functions from one infinite cardinal to another. When the domain of the functions is the cardinal of the continuum and the range is the set of natural numbers, we may restrict our attention to nicely definable functions from R to N. When we consider a class of such functions which contains all Baire class one functions, it becomes possible to encode information into these functions which can be decoded from any dominator. Specifically, we show that there is a generalized Galois-Tukey connection from the appropriate domination relation to a classical ordering studied in recursion theory. The proof techniques are developed to prove new implications regarding the distributivity of complete Boolean algebras. Next, we investigate a more technical relation relevant to the study of Borel equivalence relations on R with countable equivalence classes. We show than an analogous generalized Galois-Tukey connection exists between this relation and another ordering studied in recursion theory.