Surreal substructures

Conway’s field $\mathbf{No}$ of surreal numbers comes both with a natural total order and an additional “simplicity relation” which is also a partial order. Considering $\mathbf{No}$ as a doubly ordered structure for these two orderings, an isomorphic copy of $\mathbf{No}$ inside itself is called a surreal substructure. It turns out that many natural subclasses of $\mathbf{No}$ are actually of this type. In this paper, we study various constructions that give rise to surreal substructures and analyze important examples in greater detail.