A split extension of monoids with kernel $k \colon N \to G$, cokernel $e \colon G \to H$ and splitting $s \colon H \to G$ is Schreier if there exists a unique set-theoretic map $q \colon G \to N$ such that for all $g \in G$, $g = kq(g) \cdot se(g)$. Schreier extensions have a complete characterization and have been shown to correspond to monoid actions of $H$ on $N$. If the uniqueness requirement of $q$ is relaxed, the resulting split extension is called weakly Schreier. A natural example of these is the Artin glueings of frames. In this paper we provide a complete characterization of the weakly Schreier extensions of $H$ by $N$, proving them to be equivalent to certain quotients of $N \times H$ paired with a function that behaves like an action with respect to the quotient. Furthermore, we demonstrate the failure of the split short lemma in this setting and provide a full characterization of the morphisms that occur between weakly Schreier extensions. Finally, we use the characterization to construct some classes of examples of weakly Schreier extensions.

中文翻译：

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幺半群的弱 Schreier 扩展的刻画

如果存在唯一的集合论映射 $，则具有内核 $k \colon N \to G$、cokernel $e \colon G \to H$ 和拆分 $s \colon H \to G$ 的幺半群的分裂扩展是 Schreier q \colon G \to N$ 使得对于所有 $g \in G$，$g = kq(g) \cdot se(g)$。Schreier 扩展具有完整的特征，并且已被证明对应于 $H$ 对 $N$ 的幺半群行动。如果放宽 $q$ 的唯一性要求，则生成的拆分扩展称为弱 Schreier。其中一个自然的例子是框架的 Artin 胶合。在本文中，我们通过 $N$ 提供了 $H$ 的弱 Schreier 扩展的完整表征，证明它们等价于 $N \times H$ 的某些商，并与一个函数配对，该函数的行为类似于对商。此外，我们证明了在这种情况下分裂短引理的失败，并提供了弱 Schreier 扩展之间发生的态射的完整表征。最后，我们使用表征来构造一些弱 Schreier 扩展的例子。