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 $V.$

| | | | | created: 2018-12-17 23:15:36 | modified: 2020-06-22 18:04:14 | by: bookofproofs | references: [979]

1.Explanation: A Note on Well-ordered Sets 

2.Proposition: Well-ordered Sets are Chains 

3.Proposition: Finite Chains are Well-ordered