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Definition: Well-order, Well-ordered Set

Poset Version

If a poset $(V,\preceq)$ has the property that each of its non-empty subsets $S\subseteq V$ has a minimum, then it is called a well-ordered set. Moreover, the partial order $”\preceq”$ is called a well-order on $V.$

Strictly-ordered Set Version

If a strictly ordered set $(V,\prec)$ has the property, that each of its non-empty subsets $S\subseteq V$ has a minimal element, it is called a well-ordered set, and the strict order $”\prec”$ is called a well-order on $V.$