Lattice embeddings into the R.E. Degrees preserving 1

Abstract

Publisher Summary This chapter describes the lattice embeddings into the recursively enumerable (r.e.) degrees preserving. The characterization of the finite lattices embeddable into the r.e. degrees is important to recursion theorists for two reasons: it provides insight into the structure of the r.e. degrees; and it constitutes a crucial step in determining the decidability of the universal existential theory of the partial ordering of the r.e. degrees and of the existential theory of the r.e. degrees in the language of lattices, possibly with constant symbols for the least and/or greatest element. The full characterization of the lattices embeddable into the r.e. degrees remains open. All known lattice embeddings into the r.e. degrees preserve the least element, 0. Preserving the greatest element, 1, turned out to be quite a bit harder. The chapter shows the embeddability of M5 into the r.e. degrees preserving 1, which is harder as the usual proof for embedding requires infinitary traces. The modular nondistributive five-element lattice, M5, can be embedded into the r.e. degrees preserving the greatest element. The nonmodular nondistributive five-element lattice, N5, can be embedded into the r.e. degrees preserving the greatest element.