AI 中文总结
该研究刻画了广群上具有弱分解性质的广义高阶秩图的乘积系统，证明了相关代数结构的等价性，并探讨了R-条件与Zappa-Szep乘积的关联。
AI 中文摘要
我们对具有弱分解性质的广义高阶秩图给出了乘积系统的刻画。对于这类具有零次广群G的图，我们证明弱分解等价于乘法在其齐次G-G双集的平衡乘积之间诱导出协调双射。因此，具有固定零次广群的广义高阶秩图等价于N^k上广群双集的正规化乘积系统。对于可数左可消去图，这些双集可典范线性化为C*(G)-对应关系的乘积系统。有限对齐蕴含紧对齐，且所得的Nica-Toeplitz代数与Spielberg的全范畴代数典范一致。在模G行有限且无源头条件下，对应的Cuntz-Pimsner商是边界广群代数；在左作用单射的情况下，Cuntz-Nica-Pimsner代数也满足该结论。我们还通过严格乘法分裂的存在性刻画了R-条件，并将此类分裂与高阶秩图/广群Zappa-Szep乘积联系起来。一个可消去二阶例子表明严格分裂不一定存在。
英文摘要
We give a product-system description of generalized higher-rank graphs with the weak factorization property. For such a graph with degree-zero groupoid G, we prove that weak factorization is equivalent to multiplication inducing coherent bijections between balanced products of its homogeneous G-G-bisets. Consequently, generalized higher-rank graphs with fixed degree-zero groupoid are equivalent to normalized product systems of groupoid bisets over N^k. For countable left-cancellative graphs, these bisets linearize canonically to product systems of C*(G)-correspondences. Finite alignment implies compact alignment, and the resulting Nica-Toeplitz algebra agrees canonically with Spielberg's full category algebra. Under row-finiteness modulo G and the no-sources condition, the corresponding Cuntz-Pimsner quotient is the boundary groupoid algebra; with injective left actions, the same conclusion holds for the Cuntz-Nica-Pimsner algebra. We also characterize the R-condition by the existence of a strict multiplicative splitting and relate such splittings to higher-rank graph/groupoid Zappa-Szep products. A cancellative rank-two example shows that strict splittings need not exist.
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