Deviation of a poset
A poset is said to have deviation at most α (for an ordinal α) if for every descending chain of elements a0 > a1 >...all but a finite number of the posets of elements between an and an+1 have deviation less than α. The deviation (if it exists) is the minimum value of α for which this is true.
Not every poset has a deviation. The following conditions on a poset are equivalent:
- The poset has a deviation
- The opposite poset has a deviation
- The poset does not contain a subset order-isomorphic to the rational numbers (with their standard numerical ordering)
The poset of positive integers has deviation 0: every descending chain is finite, so the defining condition for deviation is vacuously true. However, its opposite poset has deviation 1.