Lattices
Meet and join (gcd and lcm); bounded and distributive lattices.
The idea
A lattice is a partially ordered set in which every pair of elements has a greatest lower bound and a least upper bound.
A partial order ranks elements by some notion of at most while allowing two elements to be incomparable: sets ordered by inclusion, for example, or the positive integers ordered by divisibility, where neither $4$ nor $6$ divides the other. When comparison is partial, "the larger of $a$ and $b
quot; need not exist, so we ask for less. A lower bound of $a$ and $b$ is an element below both of them; an upper bound is one above both. In a lattice the lower bounds of every pair have a greatest member, written $a \wedge b$ and called the meet, and the upper bounds have a least member, written $a \vee b$ and called the join.Under divisibility, "below" means "divides", so the meet of two numbers is the greatest of their common divisors and the join is the least of their common multiples — their gcd and lcm.
A lattice is bounded when the whole order has a least element and a greatest one — a bottom and a top.
Ways to work on it
- Walkthrough. Meet, join, and bounds in the divisibility lattice.
- Practice. Compute a meet (gcd) or join (lcm).
- Hardest. Verify the distributive law in a lattice.
Not sure where to start? Take the ten-question placement test.