Definition 4.7.1.6. Let $(S, \leq )$ be a linearly ordered set. We say that $(S, \leq )$ is well-ordered if it is well-founded when regarded as a partially ordered set: that is, if every nonempty subset $S_0 \subseteq S$ contains a smallest element. In this case, we will refer to the relation $\leq $ as a well-ordering of the set $S$.
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