Divisibility poset

Example

Let $\Z_{>0}$ be the set of positive integers, and write $a \mid b$ when $a$ divides $b$. The relation $\mid$ is a partial order on $\Z_{>0}$. It is reflexive since $a = 1 \cdot a$. It is antisymmetric since $a \mid b$ and $b \mid a$ give $a \le b \le a$ for positive integers. It is transitive since $b = ka$ and $c = lb$ give $c = (kl) a$.

It is a lattice: the meet of $a$ and $b$ is their greatest common divisor, and the join is their least common multiple. It is a distributive lattice. By the fundamental theorem of arithmetic, write $a$, $b$, $c$ as products of primes. For each prime $p$, gcd takes the minimum and lcm takes the maximum of the exponents of $p$. The identity $\min(x, \max(y, z)) = \max(\min(x, y), \min(x, z))$ for integers then gives $\gcd(a, \operatorname{lcm}(b, c)) = \operatorname{lcm}(\gcd(a, b), \gcd(a, c))$.