= Divisor lattice
The positive <divisors> of a <positive integer> $n$, ordered by <divisibility>, form a <lattice>. The <greatest lower bound> of $a,b\mid n$ is $\gcd(a,b)$, their <greatest common divisor>; the <least upper bound> is $\operatorname{lcm}(a,b)$, their <least common multiple>. Both are again <divisors> of $n$.
For example, the positive <divisors> of $12$ are $1,2,3,4,6,12$. The <divisors> $4$ and $6$ are incomparable under <divisibility>, but their <greatest lower bound> is $2$ and their <least upper bound> is $12$. The bottom and top elements of the whole <lattice> are $1$ and $n$, respectively. Under <prime factorization>, the two operations take the coordinatewise minimum and maximum of the <prime exponents>.
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