Rank Function/Examples

Examples of Rank Functions

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 $\RR$.

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:

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

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 $\RR$.