Embeddings of the 1-3-1 lattice

Abstract

This chapter studies embeddings of the 1–3–1 lattice, proving Theorem 1.6. One of the central and longstanding areas of classical computability theory concerns the structure of the degrees of unsolvability, and particularly the computably enumerable degrees. In the same way that studying symmetries in nature and solutions to equations leads to group theory, studies of the computational content of mathematics lead naturally to the structure of sets of integers under reducibilities. Understanding these structures should lead to insights into relative computability. Notable in these studies is the question of embeddability into the c.e. degrees. There are nondistributive lattices that can be embedded. Ultimately, the embedding of the 1–3–1 lattice was an amazing result, and introduced the “continuous tracing” technique into computability theory.