 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 $\mathrm{lcm}(a,b)$, 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.