In this article, we introduce a notion of reducibility for partial functions on the natural numbers, which we call subTuring reducibility. One important aspect is that the subTuring degrees correspond to the structure of the realizability subtoposes of the effective topos. We show that the subTuring degrees (that is, the realizability subtoposes of the effective topos) form a dense non-modular (thus, non-distributive) lattice. We also show that there is a nonzero join-irreducible subTuring degree (which implies that there is a realizability subtopos of the effective topos that cannot be decomposed into two smaller realizability subtoposes).

翻译：本文引入了一种关于自然数上部分函数的可归约性概念，我们称之为亚图灵可归约性。其一个重要方面在于，亚图灵度对应于有效拓扑斯中可实现子拓扑的结构。我们证明了亚图灵度（即有效拓扑斯的可实现子拓扑）构成一个稠密的非模（因而是非分配）格。我们还证明了存在非零的并不可约亚图灵度（这意味着存在一个有效拓扑斯的可实现子拓扑，它不能分解为两个更小的可实现子拓扑）。