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well-ordered set

noun, Mathematics.
a totally ordered set in which every nonempty subset has a smallest element with the property that there is no element in the subset less than this smallest element. Unabridged
Based on the Random House Dictionary, © Random House, Inc. 2017.
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well-ordered set in Technology

A set with a total ordering and no infinite descending chains. A total ordering "<=" satisfies
x <= x
x <= y <= z => x <= z
x <= y <= x => x = y
for all x, y: x <= y or y <= x
In addition, if a set W is well-ordered then all non-empty subsets A of W have a least element, i.e. there exists x in A such that for all y in A, x <= y.
Ordinals are isomorphism classes of well-ordered sets, just as integers are isomorphism classes of finite sets.

The Free On-line Dictionary of Computing, © Denis Howe 2010
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