Strengthening prompt simplicity

We introduce a natural strengthening of prompt simplicity which we call strong promptness, and study its relationship with existing lowness classes. This notion provides a ≤wtt version of superlow cuppability. We show that every strongly prompt c.e. set is superlow cuppable. Unfortunately, strong promptness is not a Turing degree notion, and so cannot characterize the sets which are superlow cuppable. However, it is a wtt-degree notion, and we show that it characterizes the degrees which satisfy a wtt-degree notion very close to the definition of superlow cuppability. Further, we study the strongly prompt c.e. sets in the context of other notions related promptness, superlowness, and cupping. In particular, we show that every benign cost function has a strongly prompt set which obeys it, providing an analogue to the known result that every cost function with the limit condition has a prompt set which obeys it. We also study the effect that lowness properties have on the behaviour of a set under the join operator. In particular we construct an array noncomputable c.e. set whose join with every low c.e. set is low.