A split extension of monoids with kernel k: N -> G, cokernel e: G -> H and splitting s: H -> G is weakly Schreier if each element g in G can be written g = k(n)se(g) for some n in N. The characterization of weakly Schreier extensions allows them to be viewed as something akin to a weak semidirect product. The motivating examples of such extensions are the Artin glueings of topological spaces and, of course, the Schreier extensions of monoids which they generalise. In this paper we show that the lambda-semidirect products of inverse monoids are also examples of weakly Schreier extensions. The characterization of weakly Schreier extensions sheds some light on the structure of lambda-semidirect products. The set of weakly Schreier extensions between two monoids comes equipped with a natural poset structure, which induces an order on the set of lambda-semidirect products between two inverse monoids. We show that Artin glueings are in fact lambda-semidirect products and inspired by this identify a class of Artin-like lambda-semidirect products. We show that joins exist for this special class of lambda-semidirect product in the aforementioned order.

中文翻译：

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$$\lambda $$-逆幺半群的半直积是弱施瑞尔扩展

具有核 k: N -> G, cokernel e: G -> H 和分裂 s: H -> G 的幺半群的分裂扩展，如果 G 中的每个元素 g 都可以写成 g = k(n)se(g ) 对于 N 中的某些 n。弱 Schreier 扩展的特征允许它们被视为类似于弱半直接积的东西。这种扩展的激励示例是拓扑空间的 Artin 胶合，当然还有它们概括的幺半群的 Schreier 扩展。在本文中，我们展示了逆幺半群的 lambda-半直积也是弱 Schreier 扩展的例子。弱 Schreier 扩展的表征揭示了 lambda-半直接产物的结构。两个幺半群之间的一组弱 Schreier 扩展配备了一个自然的偏序结构，这在两个逆幺半群之间的 lambda 半直积集合上引入了一个阶次。我们证明 Artin 胶合实际上是 lambda 半直接乘积，并受此启发确定了一类类似 Artin 的 lambda 半直接乘积。我们证明了这种特殊类型的 lambda 半直积的连接存在于上述顺序中。