Deviation of a poset

In mathematics, the deviation of a poset is an ordinal number measuring the complexity of the poset.

The deviation of a poset is used to define the Krull dimension of a module over a ring.

Definition

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 a_n and a_n+1 have deviation less than α.

Not every poset has a deviation: a necessary and sufficient condition for a poset to have a deviation is that it does not contain a subset isomorphic to the rational numbers.

References

• McConnell, J. C.; Robson, J. C. (2001), Noncommutative Noetherian rings, Graduate Studies in Mathematics, vol. 30 (Revised ed.), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2169-5, MR1811901