AI 中文总结
研究幺半群的施赖埃尔扩张，通过证明幺半群范畴相关性质及利用施赖埃尔扩张类的\(S - LACC\)定义 wreath 积，建立非分裂施赖埃尔扩张的卡卢日宁 - 克拉斯纳嵌入定理，采用特殊证明方法。
AI 中文摘要
我们研究幺半群的施赖埃尔扩张并建立施赖埃尔扩张的卡卢日宁 - 克拉斯纳嵌入定理。首先，证明幺半群范畴不是局部代数笛卡尔闭（LACC）的，且幺半群在幺半群范畴中代数可指数化当且仅当它是戴德金有限幺半群。其次，回忆幺半群扩张范畴对于施赖埃尔扩张类\(S\)是\(S - LACC\)，定义了任意两个幺半群的 wreath 积\(A\wr B\)。最后，证明了非分裂施赖埃尔扩张的卡卢日宁 - 克拉斯纳嵌入定理，即给定任意幺半群的施赖埃尔扩张\(A\hookrightarrow G\twoheadrightarrow B\)，存在单同态\(\phi_G\colon G\hookrightarrow A\wr B\)，它是扩张态射的一部分。证明通过用施赖埃尔性质提供的替代品替换需要逆元的共轭，改编了经典群论论证。
英文摘要
We study Schreier extensions of monoids and establish a Kaluzhnin--Krasner embedding theorem for Schreier extensions. First, we prove that the category of monoids is not locally algebraically cartesian closed (LACC) and that a monoid is algebraically exponentiable in the category of monoids if and only if it is a Dedekind-finite monoid. Second, we recall that the category of extensions of monoids is $S$-LACC with $S$ the class of Schreier extensions, which defines a wreath product $A \wr B$ for any two monoids. Finally, we prove a Kaluzhnin-Krasner embedding theorem for Schreier extensions that are not necessarily split, i.e. given any Schreier extension $A \hookrightarrow G \twoheadrightarrow B$ of monoids, there is a monomorphism $ϕ_G \colon G \hookrightarrow A \wr B$, which is part of a morphism of extensions. The proof adapts the classical group-theoretic argument by replacing conjugation, which requires inverses, with a substitute made available by the Schreier property, namely, the unique factorization of elements in the fibers of the projection $p \colon G \twoheadrightarrow B$.
