Lattices
A poset which has both least upper bounds and greatest lower bounds is called a lattice. In more detail, let’s say we have a poset 
and
.
- If
and
then
.
and
.
- If
and
then
.
Not every poset has a meet and a join operation, but if these operations do exist they are uniquely specified by these requirements. In fact, we can see this sort of like how we saw that direct products of groups are unique up to isomorphism: if we have two least upper bounds for a pair of elements then they must each be below or equal to the other, so they must be the same.
We can derive the following properties of the operations:
and
.
and
.
and
.
from these we see that there’s a sort of duality between the two operations. In fact, we can see that these provide two commutative semigroup structures that happen to interact in a certain nice way.
Actually, it gets even better. If we have two operations on any set satisfying these properties then we can define a partial order: 
In many cases, a lattice also satisfies 
A lattice may have elements above everything else or below everything else. We call a greatest element of a lattice 
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Ooooooooh, I know this from my Master’s thesis! Yay!
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Could you draw a picture of a non-distributive lattice?
Consider a lattice with five elements:
. The top and bottom elements are 
and 
, respectively; each of 
, 
, and 
live in the middle of those two, but none of them are comparable to each other. Now let’s calculate
So this fails to be distributive.