In computable structure theory, one considers computable presentations of abstract structures such as graphs or groups, and one thinks of two different computable presentations as being essentially the same if there is a computable isomorphism between them. Because the inverse of a computable function is also computable, the relation of being computably isomorphic is an equivalence relation, and so the only structure on the set of computable presentations is the number of non-equivalent presentations.

Recently there has been increased interest in primitive recursive presentations of structures, and in this setting, the inverse of a primitive recursive function is not necessarily primitive recursive, and so we get a relation of reducibility between structures which induces a partial pre-ordering on the primitive recursive presentations of a structure. Whenever we have a reducibility notion, one of the natural first questions is whether it is dense. We show that it is not dense: There are primitive recursive presentations $\mathcal{A}\cong \mathcal{B}$ of the same abstract structure, such that $\mathcal{A}$ is reducible to $\mathcal{B}$ (there is a primitive recursive isomorphism $\mathcal{A}\to \mathcal{B}$) but $\mathcal{B}$ is not reducible to $\mathcal{A}$ (there is no primitive recursive isomorphism $\mathcal{B}\to \mathcal{A}$), and for any third primitive recursive presentation $\mathcal{M}$ of the same structure, if $\mathcal{A}$ is reducible to $\mathcal{M}$ and $\mathcal{M}$ is reducible to $\mathcal{B}$, then either $\mathcal{M}$ is reducible to $\mathcal{A}$ or $\mathcal{B}$ is reducible to $\mathcal{M}$.

在可计算结构理论中，人们认为诸如图形或组之类的抽象结构的可计算表示形式，而如果两个不同的可计算表示形式之间存在可计算同构，则认为它们实际上是相同的。因为可计算函数的逆也是可计算的，所以可计算同构的关系是等价关系，因此可计算表示集上的唯一结构是非等价表示的数量。

最近，人们对结构的原始递归表示越来越感兴趣，在这种情况下，原始递归函数的逆数不一定是原始递归，因此我们得到了结构之间的可归约性关系，该可归约性引起了结构上的部分预排序。结构的原始递归表示。每当我们有可约性概念时，自然的第一个问题就是它是否稠密。我们证明它不是密集的：有原始的递归表示$\mathcal{一种}\cong \mathcal{乙}$ 具有相同的抽象结构 $\mathcal{一种}$ 可还原为 $\mathcal{乙}$ （有一个原始的递归同构 $\mathcal{一种}\to \mathcal{乙}$） 但 $\mathcal{乙}$ 不可还原为 $\mathcal{一种}$ （没有原始的递归同构 $\mathcal{乙}\to \mathcal{一种}$），以及任何第三种原始递归表示 $\mathcal{中号}$ 具有相同的结构，如果 $\mathcal{一种}$ 可还原为 $\mathcal{中号}$ 和 $\mathcal{中号}$ 可还原为 $\mathcal{乙}$，然后 $\mathcal{中号}$ 可还原为 $\mathcal{一种}$ 或者 $\mathcal{乙}$ 可还原为 $\mathcal{中号}$。