Scale-Multiplicative Semigroups and Geometry

对称性是通过群的代数概念进行数学处理的。相反，几何表示在群论中起着至关重要的作用。许多类群，例如物理学中出现的连通群，都有有用的几何表示，但在一般不连通群的情况下，这样的表示是缺乏的。某些不连通的群在代数上与连通的群密切相关，它们确实有一种几何表示，称为“建筑”。这个项目旨在解决一般不连通群缺乏表示的问题，通过扩展建筑物的概念来创建组合结构，这些群在组合结构上充当对称性。

Symmetry is treated mathematically through the algebraic concept of a group. Conversely, geometric representations play a crucial role in group theory. Many classes of groups, such as the connected groups that arise in physics, have useful geometric representations, but such a representation is lacking in the case of general disconnected groups. Certain disconnected groups, closely related in algebraic terms to the connected ones, do have a geometric representation called a 'building'. This project aims to address the lack of a representation for general disconnected groups by extending the notion of a building to create combinatorial structures on which these groups act as symmetries.