We study congruences on the partial automorphism monoid of a finite rank free group action. We determine a decomposition of a congruence on this monoid into a Rees congruence, a congruence on a Brandt semigroup and an idempotent separating congruence. The constituent parts are further described in terms of subgroups of direct and semidirect products of groups. We utilize this description to demonstrate how the number of congruences on the partial automorphism monoid depends on the group and the rank of the action.

中文翻译：

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自由群作用的部分自同构幺半群的同余

我们研究有限秩自由群作用的部分自同构幺半群的同余。我们确定将这个幺半群上的同余分解为 Rees 同余、Brandt 半群上的同余和幂等分离同余。组成部分根据组的直接和半直接产物的子组进一步描述。我们利用这个描述来证明部分自同构幺半群的同余数如何取决于组和动作的等级。