A representation theorem relates different mathematical structures by providing an isomorphism between them: that is, a one-to-one correspondence preserving their original properties. Establishing that the two structures substantially behave in the same way, representation theorems typically provide insight and generate powerful techniques to study the involved structures, by cross-fertilising between the methodologies existing for each of the respective branches of mathematics. When the related structures have no obvious a priori connection, however, such results can be, by their own nature, elusive. Here, we show how data-mining across distinct web sources (including the Online Encyclopedia of Integer Sequences, OEIS), was crucial in the discovery of two original representation theorems relating event structures (mathematical structures commonly used to represent concurrent discrete systems) to families of sets (endowed with elementary disjointness and subset relations) and to full graphs, respectively. The latter originally emerged in the apparently unrelated field of bioinformatics. As expected, our representation theorems are powerful, allowing to capitalise on existing theorems about full graphs to immediately conclude new facts about event structures. Our contribution is twofold: on one hand, we illustrate our novel method to mine the web, resulting in thousands of candidate connections between distinct mathematical realms; on the other hand, we explore one of these connections to obtain our new representation theorems. We hope this paper can encourage people with relevant expertise to scrutinize these candidate connections. We anticipate that, building on the ideas presented here, further connections can be unearthed, by refining the mining techniques and by extending the mined repositories.

翻译：表示定理通过提供两个数学结构之间的同构（即保持各自原始性质的一一对应关系），将不同数学结构联系起来。这类定理揭示了这两个结构在本质上具有相同的行为方式，通常能通过交叉融合数学各分支已有的研究方法，为研究相关结构提供深刻洞见并产生强大技术手段。然而，当相关结构之间缺乏明显的先验联系时，这类结果因其本质特性往往难以获得。本文展示了如何通过跨不同网络资源（包括整数序列在线百科全书OEIS）的数据挖掘，成功发现了两个原创性表示定理：分别将事件结构（用于表示并发离散系统的常见数学结构）与满足基本不相交关系和子集关系的集合族，以及与完全图建立联系。后者最初源于看似无关的生物信息学领域。正如预期，我们的表示定理具有强大效力，使得利用完全图现有定理即可直接推导出关于事件结构的新结论。本文贡献体现在两方面：一方面，我们阐述了这种新颖的网络挖掘方法，成功挖掘出数学不同领域之间的数千个潜在联系；另一方面，我们深入探索了其中一个联系，从而获得了新的表示定理。我们期待本文能够激励相关领域的专家对这些候选联系进行深入审视。基于本文提出的思路，通过改进挖掘技术并扩展挖掘数据源，我们预测未来能够发现更多数学领域间的关联。