Tukey Reduction among Analytic Directed Orders

This is a survey of recent work on the structure of Tukey reductions among analytic σ-ideals of compact subsets of compact metric spaces and analytic P-ideals of sets of natural numbers. An attempt is made to organize the results into a unified whole. This organization makes it possible to identify natural unresolved problems. Some new joint results with Stevo Todorcevic are announced. This paper is a survey of certain results around Tukey reducibility. It is not a comprehensive survey and I will concentrate only on what seems to be the theoretical core of the structure of Tukey reduction among definable directed orders. Consequently, I will only touch very lightly on non-definable directed orders and on various applications of Tukey reduction. Discussions of these topics can be found, for example, in [3], [4], [21], and the literature cited in these papers. I will start with defining Tukey reduction. I will then describe convenient domains for the study of Tukey reduction among definable directed orders, starting with the broadest one and ending with six concrete representative examples. Then I will move on to describe what is known and unknown about the structure of Tukey reductions within these domains. The reader can find a diagram illustrating some of the material surveyed in this paper in Figure 1 in Section 4.