Lattice Definition

Question

I see two Lattice definitions in Mathematics.

1. Partial order set with each pair of elements have a least least upper bound and greatest lower bound.

2. Integer linear combinations of vectors.

Is there any relation between two definitions?

Answer 1

there is a shape common to the two definitions, see LATTICE for example as well as http://en.wikipedia.org/wiki/Lath

The example of partial order that fits really well is a diagram of the divisors of $$ n = p^A q^B,$$ where $p,q$ are distinct primes and $A,B$ are relatively large. One may arrange the divisors in a rectangular pattern, with $1$ at the lower left corner and $n $ at the upper right corner. Every number $p^i q^j$ is placed at coordinates $i,j$ as in the $x,y$ plane, and each point (divisor) in the interior is joined by horizontal and vertical edges, indicating that $m \geq m/p$ and $m \geq m/q,$ also $mp \geq m$ and $mq \geq m.$ Here $s \geq t$ just means $t$ is a divisor of $s.$ So you have a partial order, but fairly well behaved and predictable. It is partial as, for example, $p$ and $q$ are not comparable, as neither divides the other.

Integer lattices have the same picture in two dimensions, just infinite in all directions. In this definition, having the edges slanted actually makes a difference.