We introduce the general notions of an index and a core of a relation. We postulate a limited form of the axiom of choice -- specifically that all partial equivalence relations have an index -- and explore the consequences of adding the axiom to standard axiom systems for point-free reasoning. Examples of the theorems we prove are that a core/index of a difunction is a bijection, and that the so-called ``all or nothing'' axiom used to facilitate pointwise reasoning is derivable from our axiom of choice.

翻译：引入关系的一般性指标与核心概念。我们假设一种有限形式的选择公理——即所有部分等价关系均存在指标——并探究将该公理加入无点推理标准公理系统后的推论。本文证明的定理包括：双函子的核心/指标构成双射，以及用于促进逐点推理的所谓"全有或全无"公理可从我们提出的选择公理中推导得出。