We develop the theory of transformation semigroups that have degree 2, that is, act by partial functions on a finite set such that the inverse image of points have at most two elements. We show that the graph of fibers of such an action gives a deep connection between semigroup theory and graph theory. It is known that the Krohn–Rhodes complexity of a degree 2 action is at most 2. We show that the monoid of continuous maps on a graph is the translational hull of an appropriate 0-simple semigroup. We show how group mapping semigroups can be considered as regular covers of their right letter mapping image and relate this to their graph of fibers.

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二次变换半群作为图上的连续映射：基础和结构

我们发展了具有 2 次的变换半群的理论，即，通过偏函数作用于有限集，使得点的逆像至多有两个元素。我们表明，这种动作的纤维图在半群论和图论之间提供了深刻的联系。已知 2 次动作的 Krohn-Rhodes 复杂度最多为 2。我们证明了图上连续映射的幺半群是适当的 0-单半群的平移壳。我们展示了如何将组映射半群视为其右字母映射图像的常规覆盖，并将其与它们的纤维图相关联。