关于么半群单态的正规性
On the normality of monoid monomorphisms
摘要：
在么半群范畴中，我们刻画了在适当的意义下正规于内部自反关系、预序或等价关系的单态。这种内部关系的零类首先用与它们相关联的方便的句法关系来描述，然后通过与相应的规范化函子相关联的解释来描述。由这些映射所诱导的最大范畴等价提供了由它们的零类所生成的关系范畴与我们建议称之为{正规于}的单态关系范畴之间的等价。这一想法，虽然是横向的文献在该领域，并没有在我们看来，提出和探讨充分的一般性。伴随正规函子的存在允许发展正规单态理论，从而将许多结果从群和原模范畴扩展到幺半群和酉范畴。
英文摘要：
In the category of monoids we characterize monomorphisms that are normal, in an appropriate sense, to internal reflexive relations, preorders or equivalence relations. The zero-classes of such internal relations are first described in terms of convenient syntactic relations associated to them and then through the adjunctions associated with the corresponding normalization functors. The largest categorical equivalences induced by these adjunctions provide an equivalence between the categories of relations generated by their zero-classes and the ones of monomorphisms that we suggest to call {normal with respect to} the internal relations considered. This idea, although being transverse to the literature in the field, has not in our opinion been presented and explored in full generality. The existence of adjoints to the normalization functors permits developing a theory of normal monomorphisms, thus extending many results from groups and protomodular categories to monoids and unital categories.
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