We show that the category of cancellative conjugation semigroups is weakly Mal’tsev and give a characterization of all admissible diagrams there. In the category of cancellative conjugation monoids we describe, for Schreier split epimorphisms with codomain B and kernel X , all morphisms $$h:X\rightarrow B$$ h : X → B which induce a reflexive graph, an internal category or an internal groupoid. We describe Schreier split epimorphisms in terms of external actions and consider the notions of precrossed semimodule, crossed semimodule and crossed module in the context of cancellative conjugation monoids. In this category we prove that a relative version of the so-called “Smith is Huq” condition for Schreier split epimorphisms holds as well as other relative conditions.

中文翻译：

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取消共轭半群和幺半群

我们证明了取消共轭半群的范畴是弱 Mal'tsev 并给出了那里所有可容许图的表征。在我们描述的取消共轭幺半群的范畴中，对于具有 codomain B 和核 X 的 Schreier 分裂表同态，所有态射 $$h:X\rightarrow B$$h : X → B 诱导自反图、内部范畴或内部群像。我们根据外部作用描述 Schreier 分裂表同态，并在取消共轭幺半群的背景下考虑预交叉半模、交叉半模和交叉模的概念。在这一类中，我们证明 Schreier 分裂同胚的所谓“Smith is Huq”条件的相对版本以及其他相对条件都成立。