论文标题
分级半群
Graded Semigroups
论文作者
论文摘要
我们系统地发展了一个分级半群的理论，即由G组分割的半群，以与S上的乘法相吻合。我们定义了Smash产品S＃G，并表明当S具有本地单位时，S＃G-Mod类别S s＃G-Mod与S＃G-MAD相比是S s＃G-Mod与Set set seb ser的类别相同。我们还表明，当s是一个反向半群时，当且仅当s-gr自然等于s_1-mod时，它会得到强烈的分级，其中s_1是与G的身份元素1相对应的s的分区。这些结果类似于COHEN/MONTGOMERY的众所周知的norems and Montgomery和Dade的dade定理。此外，我们表明，莫里塔等价的分级意味着莫里塔对具有本地单位的半群，表明了由半群的分级编码的大量信息。我们还提供了一个分级的幻影定理，提供了许多天然分级半群的例子，并探索了分级半群，分级环和分级类固定之间的连接。特别是，我们介绍了分级的REES矩阵半群，并将它们与粉碎产品半群相关联。我们特别注意分级图反向半群，并表征那些产生强烈分级的Leavitt路径代数的分数。
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.
