For a given inverse semigroup, one can associate an étale groupoid which is called the universal groupoid. Our motivation is studying the relation between inverse semigroups and associated étale groupoids. In this paper, we focus on congruences of inverse semigroups, which is a fundamental concept for considering quotients of inverse semigroups. We prove that a congruence of an inverse semigroup induces a closed invariant set and a normal subgroupoid of the universal groupoid. Then we show that the universal groupoid associated to a quotient inverse semigroup is described by the restriction and quotient of the original universal groupoid. Finally we compute invariant sets and normal subgroupoids induced by special congruences including abelianization.

对于给定的反半群，可以关联一个称为全群的 étale 群。我们的动机是研究反半群和相关的 étale groupoids 之间的关系。在本文中，我们关注反半群的同余，这是考虑反半群商的基本概念。我们证明了一个反半群的同余导致了一个封闭的不变集和一个全群群的正规子群。然后我们证明了与商反半群相关联的全群群是由原始全群群的限制和商来描述的。最后，我们计算由包括阿贝尔化在内的特殊同余引起的不变集和正态子群。