A note on the diamond operator

Abstract

We show that if $1 \leq_W F$ and $F \star F \leq_W F$, then $F^\diamond \leq_W F$, where $\star$ and $\diamond$ are the following operations in the Weihrauch lattice: $\star$ is the compositional product, which allows the use of two principles in sequence, while the diamond operator $\diamond$ allows an arbitrary but finite number of uses of the given principle in sequence. This answers a question of Pauly.