While every endpointed interval I in a linear order J is, considered as a linear order in its own right, trivially Muchnik-reducible to J itself, this fails for Medvedev-reductions. We construct an extreme example of this: a linear order in which no endpointed interval is Medvedev-reducible to any other, even allowing parameters, except when the two intervals have finite difference. We also construct a scattered linear order which has many endpointed intervals Medvedev-incomparable to itself; the only other known construction of such a linear order yields an ordinal of extremely high complexity, whereas this construction produces a low-level-arithmetic example. Additionally, the constructions here are “coarse” in the sense that they lift to other uniform reducibility notions in place of Medvedev reducibility itself.

中文翻译：

---

避免梅德韦杰夫在线性顺序内减少

虽然线性阶J中的每个端点区间I本身被视为线性阶，但平凡地可归约为J就其本身而言，这对于梅德韦杰夫的削减来说是失败的。我们构造一个极端的例子：一个线性顺序，其中没有端点区间可以梅德韦杰夫约简到任何其他区间，甚至允许参数，除非两个区间具有有限差。我们还构建了一个分散的线性顺序，它有许多梅德韦杰夫端点区间——与它本身无法比较；这种线性顺序的唯一已知构造产生了极其复杂的序数，而这种构造产生了低级算术示例。此外，这里的结构是“粗略的”，因为它们提升到其他统一的可归约性概念来代替梅德韦杰夫的可归约性本身。