Jump to content

Deviation of a poset

Add links
From Wikipedia, the free encyclopedia
This is an old revision of this page, as edited by R.e.b. (talk | contribs) at 15:23, 21 January 2013 (Expanding article). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

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 as the deviation of the poset of submodules.

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 an and an+1 have deviation less than α.

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 isomorphic to the rational numbers

Example: The poset of positive integers has deviation 0, but 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, MR1811901
Retrieved from "https://en.wikipedia.org/w/index.php?title=Deviation_of_a_poset&oldid=534172898"