Abstract We show that any regular (right) Schreier extension of a monoid M by a monoid A induces an abstract kernel Φ : M → End ⁡ ( A ) Inn ⁡ ( A ) {\Phi\colon M\to\frac{\operatorname{End}(A)}{\operatorname{Inn}(A)}} . If an abstract kernel factors through SEnd ⁡ ( A ) Inn ⁡ ( A ) {\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}} , where SEnd ⁡ ( A ) {\operatorname{SEnd}(A)} is the monoid of surjective endomorphisms of A, then we associate to it an obstruction, which is an element of the third cohomology group of M with coefficients in the abelian group U ⁢ ( Z ⁢ ( A ) ) {U(Z(A))} of invertible elements of the center Z ⁢ ( A ) {Z(A)} of A, on which M acts via Φ. An abstract kernel Φ : M → SEnd ⁡ ( A ) Inn ⁡ ( A ) {\Phi\colon M\to\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}} (resp. Φ : M → Aut ⁡ ( A ) Inn ⁡ ( A ) {\Phi\colon M\to\frac{\operatorname{Aut}(A)}{\operatorname{Inn}(A)}} ) is induced by a regular weakly homogeneous (resp. homogeneous) Schreier extension of M by A if and only if its obstruction is zero. We also show that the set of isomorphism classes of regular weakly homogeneous (resp. homogeneous) Schreier extensions inducing a given abstract kernel Φ : M → SEnd ⁡ ( A ) Inn ⁡ ( A ) {\Phi\colon M\to\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}} (resp. Φ : M → Aut ⁡ ( A ) Inn ⁡ ( A ) {\Phi\colon M\to\frac{\operatorname{Aut}(A)}{\operatorname{Inn}(A)}} ), when it is not empty, is in bijection with the second cohomology group of M with coefficients in U ⁢ ( Z ⁢ ( A ) ) {U(Z(A))} .

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具有非阿贝尔核的幺半群的Schreier扩展的分类

摘要 我们证明了幺半群 A 对幺半群 M 的任何正则（右）施赖尔扩展都会产生一个抽象核 Φ ： M → End ⁡ ( A ) Inn ⁡ ( A ) {\Phi\colon M\to\frac{\operatorname {End}(A)}{\operatorname{Inn}(A)}} . 如果一个抽象核因子通过 SEnd ⁡ ( A ) Inn ⁡ ( A ) {\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}} ，其中 SEnd ⁡ ( A ) {\operatorname {SEnd}(A)} 是 A 的满射自同态的幺半群，然后我们将一个障碍与它联系起来，它是 M 的第三个上同调群的一个元素，系数在阿贝尔群 U ⁢ ( Z ⁢ ( A ) ) {U(Z(A))} A 的中心 Z ⁢ ( A ) {Z(A)} 的可逆元素，M 通过 Φ 作用于该元素。抽象核 Φ : M → SEnd ⁡ ( A ) Inn ⁡ ( A ) {\Phi\colon M\to\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}} (resp . Φ : M → Aut ⁡ ( A ) Inn ⁡ ( A ) {\Phi\colon M\to\frac{\operatorname{Aut}(A)}{\operatorname{Inn}(A)}} ) homogeneous (resp. homogeneous) M 由 A 的 Schreier 扩展当且仅当其阻塞为零。我们还展示了引入给定抽象核 Φ 的规则弱齐次（或齐次）Schreier 扩展的同构类集合： M → SEnd ⁡ ( A ) Inn ⁡ ( A ) {\Phi\colon M\to\frac{ \operatorname{SEnd}(A)}{\operatorname{Inn}(A)}} (resp. Φ : M → Aut ⁡ ( A ) Inn ⁡ ( A ) {\Phi\colon M\to\frac{\operatorname {Aut}(A)}{\operatorname{Inn}(A)}} )，当它不为空时，与系数在 U ⁢ ( Z ⁢ ( A ) ) {U (Z(A))} 。homogeneous) M 由 A 的 Schreier 扩展当且仅当其阻塞为零。我们还展示了引入给定抽象核 Φ 的规则弱齐次（或齐次）Schreier 扩展的同构类集合： M → SEnd ⁡ ( A ) Inn ⁡ ( A ) {\Phi\colon M\to\frac{ \operatorname{SEnd}(A)}{\operatorname{Inn}(A)}} (resp. Φ : M → Aut ⁡ ( A ) Inn ⁡ ( A ) {\Phi\colon M\to\frac{\operatorname {Aut}(A)}{\operatorname{Inn}(A)}} )，当它不为空时，与系数在 U ⁢ ( Z ⁢ ( A ) ) {U (Z(A))} 。homogeneous) M 由 A 的 Schreier 扩展当且仅当其阻塞为零。我们还展示了引入给定抽象核 Φ 的规则弱齐次（或齐次）Schreier 扩展的同构类集合： M → SEnd ⁡ ( A ) Inn ⁡ ( A ) {\Phi\colon M\to\frac{ \operatorname{SEnd}(A)}{\operatorname{Inn}(A)}} (resp. Φ : M → Aut ⁡ ( A ) Inn ⁡ ( A ) {\Phi\colon M\to\frac{\operatorname {Aut}(A)}{\operatorname{Inn}(A)}} )，当它不为空时，与系数在 U ⁢ ( Z ⁢ ( A ) ) {U (Z(A))} 。