Well-order

In mathematics, a well-order relation (or well-ordering) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. The set S together with the well-order relation is then called a well-ordered set. The hyphen is frequently omitted in contemporary papers, yielding the spellings wellorder, wellordered, and wellordering.

Every non-empty well-ordered set has a least element. Every element s of a well-ordered set, except a possible greatest element, has a unique successor (next element), namely the least element of the subset of all elements greater than s. There may be elements besides the least element which have no predecessor (see Natural numbers below for an example). In a well-ordered set S, every subset T which has an upper bound has a least upper bound, namely the least element of the subset of all upper bounds of T in S.

If ≤ is a non-strict well-ordering, then < is a strict well-ordering. A relation is a strict well-ordering if and only if it is a well-founded strict total order. The distinction between strict and non-strict well-orders is often ignored since they are easily interconvertible.