Lattice (Ordered Set)/Examples/Divisor Relation

Examples of Lattice in context of Ordered Sets

Let $\struct {\Z_{>0}, \divides}$ denote the order structure consisting of the (strictly) positive integers under the divisor relation $\divides$.

Then $\struct {\Z_{>0}, \divides}$ is a lattice.


Proof

Let $a, b \in \Z_{>0}$.

Let $\gcd \set {a, b}$ denote the greatest common divisor of $a$ and $b$.

Let $\lcm \set {a, b}$ denote the lowest common multiple of $a$ and $b$.

By definition:

the greatest common divisor of $a$ and $b$ is the infimum of $a$ and $b$ with respect to the divisor relation.
the lowest common multiple of $a$ and $b$ is the supremum of $a$ and $b$ with respect to the divisor relation.

The result follows.

$\blacksquare$


Sources

  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): partial order
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): partial order
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