An equivalence relation can be constructed from a given (homogeneous, binary) relation in two steps: first, construct the smallest reflexive and transitive relation containing the given relation (the "star" of the relation) and, second, construct the largest symmetric relation that is included in the result of the first step. The fact that the final result is also reflexive and transitive (as well as symmetric), and thus an equivalence relation, is not immediately obvious, although straightforward to prove. Rather than prove that the defining properties of reflexivity and transitivity are satisfied, we establish reflexivity and transitivity \emph{constructively} by exhibiting a particular starth root -- in a way that emphasises the creative process in its construction. The constructed starth root is fundamental to algorithms that determinethe strongly connected components of a graph as well as the decomposition of a graph into its strongly connected components together with an acyclic graph connecting such components.

翻译：给定一个（齐次、二元）关系，可通过两个步骤构造等价关系：首先，构造包含给定关系的最小自反传递关系（该关系的"星"闭包），其次，在第一步结果中构造包含在内的最大对称关系。最终结果同样具有自反性与传递性（以及对称性），从而构成等价关系，这一事实虽可直接证明却并非显然。我们并未通过验证自反性和传递性的定义属性来证明，而是通过展示特定的星根来构造性地确立自反性与传递性——这种方式突显了构造过程中的创造性。所构建的星根对于确定图的强连通分量算法以及将图分解为强连通分量及其连接这些分量的无环图具有基础性意义。