标题： 图拟群：拟群和右拟群的广义Cayley图 显示英文标题

标题： Graph quandles: Generalized Cayley graphs of racks and right quasigroups

摘要： 我们解决了 Valeriy Bardakov 关于 racks 的凯莱图以及右拟群的图论实现的两个公开问题。 我们还扩展了 Didier Caucal 对标记凯莱有向图的分类到右拟群及相关代数结构，如quandles。 首先，我们刻画了实现 racks 的图的标记方式。 作为应用，我们从图自同构群的置换表示构造了 rack 理论的（有向）图不变量。 我们描述了如何计算这些不变量，并给出了路径图和循环图的一般结果。 其次，我们证明所有右拟群都可以通过无边图和完全（有向）图来实现。 利用 Schreier （有向）图，我们还刻画了实现右拟群 Q 的 Q 的 Cayley （有向）图。 特别是，所有 racks 都可以通过它们的完整 Cayley （有向）图来实现。 最后，我们给出了右消去半群、右可除半群、右拟群、racks、quandles、对合 racks 和 kei 的标记 Cayley 有向图的图论刻画。 显示英文摘要

摘要： We solve two open problems of Valeriy Bardakov about Cayley graphs of racks and graph-theoretic realizations of right quasigroups. We also extend Didier Caucal's classification of labeled Cayley digraphs to right quasigroups and related algebraic structures like quandles. First, we characterize markings of graphs that realize racks. As an application, we construct rack-theoretic (di)graph invariants from permutation representations of graph automorphism groups. We describe how to compute these invariants with general results for path graphs and cycle graphs. Second, we show that all right quasigroups are realizable by edgeless graphs and complete (di)graphs. Using Schreier (di)graphs, we also characterize Cayley (di)graphs of right quasigroups Q that realize Q. In particular, all racks are realizable by their full Cayley (di)graphs. Finally, we give a graph-theoretic characterization of labeled Cayley digraphs of right-cancellative magmas, right-divisible magmas, right quasigroups, racks, quandles, involutory racks, and kei.