Definition:Rank Function for Relation

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Definition

Let $\struct {S, \RR}$ be a relational structure.

Let $\struct {T, \prec}$ be a strictly well-ordered set.

Let $\operatorname {rk}: S \to T$ be a mapping such that:

$\forall x, y \in S: \paren {x \ne y \text { and } \tuple {x, y} \in \RR} \implies \map {\operatorname {rk} } x \prec \map {\operatorname {rk} } y$


$\operatorname {rk}$ is known as a rank function for $\RR$.


Examples

Arbitrary Example $1$

Let $X = \set {x, y, z}$.

Let $\RR$ be the relation on $X$ defined as:

$\RR = \set {\tuple {x, x}, \tuple {x, y}, \tuple {x, z}, \tuple {y, y}, \tuple {z, z} }$.

Let the mapping $\operatorname {rk}_0: X \to \N$ be defined as:

\(\ds \map {\operatorname {rk}_0} x\) \(=\) \(\ds 0\)
\(\ds \map {\operatorname {rk}_0} y\) \(=\) \(\ds 1\)
\(\ds \map {\operatorname {rk}_0} z\) \(=\) \(\ds 1\)

Then $\operatorname {rk}_0$ is a rank function for $\RR$.


Sum of Powers of Prime Factors

Consider the relational structure $\struct {\Z_{>0}, \divides}$ formed from the strictly positive integers $\Z_{>0}$ under the divisor relation $\divides$.


Let $\operatorname {rk}_1: \Z_{<0} \to \N$ defined as:

$\forall n \in \Z_{<0}: \map {\operatorname {rk}_1} n = \ds \sum_{k \mathop \in \Z_{>0} } \map {i_k} n$

where:

$\ds n = \prod_{k \mathop \in \Z_{>0} } {p_k}^{\map {i_k} n}$

is the prime decomposition of $n$.


That is, $\operatorname {rk}_1$ is the sum of the exponents of the prime divisors of $n$ in the prime decomposition of $n$.


Then $\operatorname {rk}_1$ is a rank function for $\RR$.


Also see

  • Results about rank functions can be found here.


Sources

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