Let \(\mathcal {R}\) be an expansion of the ordered real additive group. When \(\mathcal {R}\) is o-minimal, it is known that either \(\mathcal {R}\) defines an ordered field isomorphic to \((\mathbb {R},<,+,\cdot )\) on some open subinterval \(I\subseteq \mathbb {R}\), or \(\mathcal {R}\) is a reduct of an ordered vector space. We say \(\mathcal {R}\) is field-type if it satisfies the former condition. In this paper, we prove a more general result for arbitrary expansions of \((\mathbb {R},<,+)\). In particular, we show that for expansions that do not define dense \(\omega \)-orders (we call these type A expansions), an appropriate version of Zilber’s principle holds. Among other things we conclude that in a type A expansion that is not field-type, every continuous definable function \([0,1]^m \rightarrow \mathbb {R}^n\) is locally affine outside a nowhere dense set.

令\(\mathcal {R}\)是有序实可加群的展开。当\(\mathcal {R}\)是 o 极小时，我们知道\(\mathcal {R}\)定义一个同构于\((\mathbb {R},<,+,\cdot )\)在某个开放子区间\(I\subseteq \mathbb {R}\) 上，或\(\mathcal {R}\)是有序向量空间的约化。如果满足前一个条件，我们说\(\mathcal {R}\)是字段类型。在本文中，我们证明了\((\mathbb {R},<,+)\) 的任意扩展的更一般结果。特别是，我们证明对于没有定义密集\(\omega \)-orders（我们称这些类型为 A 扩展），这是 Zilber 原理的适当版本。除其他外，我们得出结论，在非字段类型的类型 A 扩展中，每个连续可定义函数\([0,1]^m \rightarrow \mathbb {R}^n\)在无处稠密集之外局部仿射.