Well-order
In mathematics, a
well-order relation 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, 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. 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.