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 $V.$
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 $V.$
Explanations: 1
Corollaries: 1
Definitions: 2 3
Examples: 4
Explanations: 5
Parts: 6
Proofs: 7 8 9 10 11 12 13 14
Propositions: 15 16 17 18 19