Cupping in the computably enumerable degrees - Tran, Hong Hanh (Nanyang Technological University, 2023)
This thesis is mainly concerned with the cupping property in the computably enumerable (c.e.) degrees. In particular, we study major sub-degrees, n-cuppable degrees and the quotient structure R/Ncup. In the first part, we present a direct construction of a cuppable high c.e. h with a low major sub-degree l such that h>= a for a given c.e. degree a. In the second part, we generalize the technique of Bie and Wu, used in the construction of a minimal pair in R/Ncup which is also a minimal pair in M/Ncup, to construct three incomplete c.e. degrees which are 2-cuppable but not 3-cuppable. This result will be directly generalized to arbitrary n>3 c.e. degrees. Consequently, for any n>0, there are n degrees which are (n-1)-cuppable but not n-cuppable. In the third part, using Bie and Wu's technique, we prove a claim by Li, Wu and Yang that the diamond lattice can be embedded in R/Ncup preserving 0 and 1.