Cupping and jump classes in the computably enumerable degrees

We show that there is a cuppable c.e. degree, all of whose cupping partners are high. In particular, not all cuppable degrees are -cuppable, or indeed cuppable for any n, refuting a conjecture by Li. On the other hand, we show that one cannot improve highness to superhighness. We also show that the -cuppable degrees coincide with the array computable-cuppable degrees, giving a full understanding of the latter class.Ministry of Education (MOE)Accepted versionThe first author is partially supported by the Marsden Fund of New Zealand. The second author is partially supported by grant MOE2015-T2- 2-055. The third author is supported by Singapore Ministry of Education Tier 2 grant MOE2016-T2-1-083 (M4020333); NTU Tier 1 grants RG32/16 (M4011672) and RG111/19 (M4012245)