Deviation Of A Poset
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In order-theoretic mathematics, the deviation of a poset is an
ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the leas ...
measuring the complexity of a
poset In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...
. A poset is also known as a partially ordered set. The deviation of a poset is used to define the
Krull dimension In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally ...
of a
module over a ring In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a ''module'' also generalizes the notion of an abelian group, since the ...
as the deviation of its poset of submodules.


Definition

A trivial poset (one in which no two distinct elements are comparable) is declared to have deviation -\infty. A nontrivial poset satisfying the
descending chain condition In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings. These conditions played an important r ...
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 ''a''0 > ''a''1 >... 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 number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...
s (with their standard numerical ordering)


Examples

The poset of positive integers has deviation 0: every descending chain is finite, so the defining condition for deviation is
vacuously true In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied. It is sometimes said that a s ...
. However, its opposite poset has deviation 1. Let ''k'' be an algebraically closed field and consider the poset of ideals of the polynomial ring ''k ' in one variable. Since the deviation of this poset is the Krull dimension of the ring, we know that it should be 1. This corresponds to the fact that ''k ' does not have the descending chain condition (so the deviation is greater than zero), but in any descending chain, consecutive elements are 'close together'. For instance, take the descending chain of ideals (x)\supset (x^2)\supset (x^3)\supset... - this is an infinite descending chain, but for any two consecutive terms, say (x^n) and (x^), there is no infinite descending chain of ideals of ''k ' contained between these terms. Extending this example further, consider the polynomial ring in two variables, ''k ,y', which has Krull dimension 2. Take the descending chain (x)\supset (x^2)\supset(x^3)\supset.... Given any two adjacent terms in this chain, (x^n) and (x^), there is an infinite descending chain (x^ny,x^)\supset(x^ny^2,x^)\supset(x^ny^3,x^)\supset.... So we can find a descending chain such that between any two adjacent terms there is a further infinite descending chain - we can 'nest' descending chains two layers deep. Extending this, it is easy to see that in the polynomial ring in ''n'' variables, it is possible to nest descending chains ''n'' layers deep and no more. This is essentially what it means for the poset of ideals to have deviation ''n''.


References

* {{refend Order theory