标题： 具有三重性的几何结构来自线性空间 显示英文标题

标题： Geometries with trialities arising from linear spaces

摘要： 对合是一种类似于超对称的结构，它以长度为三的循环方式交换关联几何中元素的类型。 尽管具有对合的几何展现出迷人的行为，但它们的构造具有挑战性，因此在文献中较为罕见。 为了更深入地理解对合，拥有大量不同的例子至关重要。 在本文中，我们介绍了一种构建具有对合的多种三阶关联系统的通用方法。 具体而言，对于任何二阶关联系统$\Gamma$，我们定义其三角复形$\Delta(\Gamma)$，一个三阶关联系统，其元素由$\Gamma$的旗（相邻元素对）的三个副本组成。 这个三角复形总是具有一个对合，该对合循环置换这三个副本。 随后，我们详细探讨了当$\Gamma$是线性空间时三角复形的性质，包括旗传递性、对合的存在性以及连通性属性。 作为我们工作的结果，此构造产生了第一个具有对合但没有对合的厚的、旗传递且残余连通的几何无限族。 显示英文摘要

摘要： A triality is a sort of super-symmetry that exchanges the types of the elements of an incidence geometry in cycles of length three. Although geometries with trialities exhibit fascinating behaviors, their construction is challenging, making them rare in the literature. To understand trialities more deeply, it is crucial to have a wide variety of examples at hand. In this article, we introduce a general method for constructing various rank-three incidence systems with trialities. Specifically, for any rank two incidence system $\Gamma$, we define its triangle complex $\Delta(\Gamma)$, a rank three incidence system whose elements consist of three copies of the flags (pairs of incident elements) of $\Gamma$. This triangle complex always admits a triality that cyclically permutes the three copies. We then explore in detail the properties of the triangle complex when $\Gamma$ is a linear space, including flag-transitivity, the existence of dualities, and connectivity properties. As a consequence of our work, this construction yields the first infinite family of thick, flag-transitive and residually connected geometries with trialities but no dualities.