Let $\wedge $, $\vee $ denote the meet and join operations of lattice. A lattice is called distributive if for all $x, y, z,$
$x\wedge \left ( y\vee z \right )= \left ( x\wedge y \right )\vee \left ( x\wedge z \right )$
It is called complete if meet and join exist for every subset. It is called modular if for all $x, y, z$
$z\leq x\Rightarrow x\wedge \left ( y\vee z \right )=\left ( x\wedge y \right )\vee z$
The positive integers under divisibility ordering i.e. $p\leq q$ if $p$ divides $q$ forms a.
- Complete lattice.
- Modular, but not distributive lattice.
- Distributive lattice.
- Lattice but not a complete lattice.
- Under the give ordering positive integers do not form a lattice.