For finite permutation groups, simplicity of the augmentation submodule is equivalent to 2-transitivity over the field of complex numbers. We note that this is not the case for transformation monoids. We characterize the finite transformation monoids whose augmentation submodules are simple for a field 𝔽 (assuming the answer is known for groups, which is the case for ℂ, ℝ, and ℚ) and provide many interesting and natural examples such as endomorphism monoids of connected simplicial complexes, posets, and graphs (the latter with simplicial mappings).

中文翻译：

---

变换幺半群的增强子模块的简单性

对于有限置换群，扩充子模块的简单性相当于复数域上的 2-传递性。我们注意到这不是变换幺半群的情况。我们描述了有限变换幺半群，其增强子模块对于一个域 𝔽 很简单（假设答案对于群是已知的，这是 ℂ、ℝ 和 ℚ 的情况），并提供了许多有趣和自然的例子，例如连接单纯形的内同态幺半群复合体、偏序集和图（后者具有简单映射）。