Deviation of a poset

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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[edit]

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:

Example[edit]

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[edit]