In this paper we study the notion of first-order part of a computational problem, first introduced by Dzhafarov, Solomon, and Yokoyama, which captures the "strongest computational problem with codomain $\mathbb{N}$ that is Weihrauch reducible to $f$". This operator is very useful to prove separation results, especially at the higher levels of the Weihrauch lattice. We explore the first-order part in relation with several other operators already known in the literature. We also introduce a new operator, called unbounded finite parallelization, which plays an important role in characterizing the first-order part of parallelizable problems. We show how the obtained results can be used to explicitly characterize the first-order part of several known problems.

翻译：本文研究计算问题的初等部分这一概念，该概念由Dzhafarov、Solomon和Yokoyama首次引入，它刻画了"在Weihrauch可归约于$f$的条件下，值域为$\mathbb{N}$的最强计算问题"。该算子对于证明分离结果非常有用，尤其是在Weihrauch格的高层级中。我们探讨了初等部分与文献中已知的若干其他算子之间的关系。此外，我们引入了一个名为无界有限并行化的新算子，它在刻画可并行化问题的初等部分中扮演着重要角色。我们展示了所得结果如何用于显式刻画若干已知问题的初等部分。