分次半群
Graded Semigroups
AI总结：
本文系统建立分次半群理论，定义碎积并证明分次集合范畴表示定理，刻画强分次逆半群、分次 Morita 等价及分次图逆半群与 Leavitt 路代数的联系。
摘要：
我们系统地发展了分次半群理论，即半群 S 以一种与 S 上乘法相容的方式被群 G 分划。我们定义了碎积 S#G，并证明当 S 具有局部单位时，容许 S#G-作用的集合所构成的范畴 S#G-Mod 同构于容许适当 S-作用的分次集合所构成的范畴 S-Gr。我们还证明，当 S 是逆半群时，它是强分次的，当且仅当 S-Gr 自然等价于 S_1-Mod，其中 S_1 是 S 中对应于 G 的单位元 1 的分划。这些结果类似于 Cohen/Montgomery 和 Dade 关于分次环的著名定理。此外，我们证明，对于具有局部单位的半群，分次 Morita 等价蕴含 Morita 等价，这表明半群的分次编码了丰富的信息。我们还给出了分次 Vagner-Preston 定理，提供了大量自然出现的分次半群的例子，并探讨了分次半群、分次环和分次群胚之间的联系。特别地，我们引入了分次 Rees 矩阵半群，并将它们与碎积半群联系起来。我们特别关注分次图逆半群，并刻画了那些产生强分次 Leavitt 路代数的分次图逆半群。
英文摘要：
We systematically develop a theory of graded semigroups, that is semigroups S partitioned by groups G, in a manner compatible with the multiplication on S. We define a smash product S#G, and show that when S has local units, the category S#G-Mod of sets admitting an S#G-action is isomorphic to the category S-Gr of graded sets admitting an appropriate S-action. We also show that when S is an inverse semigroup, it is strongly graded if and only if S-Gr is naturally equivalent to S_1-Mod, where S_1 is the partition of S corresponding to the identity element 1 of G. These results are analogous to well-known theorems of Cohen/Montgomery and Dade for graded rings. Moreover, we show that graded Morita equivalence implies Morita equivalence for semigroups with local units, evincing the wealth of information encoded by the grading of a semigroup. We also give a graded Vagner-Preston theorem, provide numerous examples of naturally-occurring graded semigroups, and explore connections between graded semigroups, graded rings, and graded groupoids. In particular, we introduce graded Rees matrix semigroups, and relate them to smash product semigroups. We pay special attention to graded graph inverse semigroups, and characterise those that produce strongly graded Leavitt path algebras.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
