幺半群的点半双积

摘要：

这里引入幺半群的半双积作为双积（交换阿贝尔群）和半直积（群）的一个普通推广，以探索一类广泛的幺半群扩张。更一般地说，抽象的半双积存在于任何具体的范畴中，在这些范畴中，映射加法是有意义的，从而重新解释了Mac Lane的相对双积。在指出的情况下，他们产生了一类特殊的扩展称为半双积扩展。不是所有的么半群扩张都是半双积扩张，但所有的群扩张都是。建立了幺半群的点半双积范畴与点幺半群作用系统范畴之间的范畴等价性，并由此产生了一个新的作用范畴。与经典扩张理论的主要区别在于，半双积扩张与分裂扩张的处理方式相同，即使截面映射可能不是同态。给出了二元幺半群的14个半双积扩张的列表。

英文摘要：

Semibiproducts of monoids are introduced here as a common generalization to biproducts (of abelian groups) and to semidirect products (of groups) for exploring a wide class of monoid extensions. More generally, abstract semibiproducts exist in any concrete category over sets in which map addition is meaningful thus reinterpreting Mac Lane's relative biproducts. In the pointed case they give rise to a special class of extensions called semibiproduct extensions. Not all monoid extensions are semibiproduct extensions but all group extensions are. A categorical equivalence is established between the category of pointed semibiproducts of monoids and the category of pointed monoid action systems, a new category of actions that emerges from the equivalence. The main difference to classical extension theory is that semibiproduct extensions are treated in the same way as split extensions, even though the section map may fail to be a homomorphism. A list with all the 14 semibiproduct extensions of 2-element monoids is provided.

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