Deviation of a poset

In order-theoretic mathematics, the deviation of a poset is an ordinal number measuring the complexity of a partially ordered set.

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

Definition

A trivial poset (one in which no two elements are comparable) is declared to have deviation $-\infty$. A nontrivial poset satisfying the descending chain condition is said to have deviation 0. Then, inductively, 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. 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)

Example

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.

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, MR 1811901