What’s a lattice anyway?

22 Dec 2015

In common language, a lattice is something like a regular grid. This picture agrees with some mathematical terminology, where e.g. the product $\mathbb{Z}\times \mathbb{Z}$ is (reasonably enough) called a lattice. This is not the kind of lattice we’ll be talking about here.

Let $P$ be a poset. An element $v\in P$ is called the join of the pair $a,b\in P$ if it is the least upper bound of $a$ and $b$. In this case, we write $v=a\vee b$. Similarly, we could consider the join of any subset $S\subseteq P$.

Examples. The powerset of a given set $X$ is naturally a poset under inclusion of subsets. The join of two subsets $A,B\subseteq X$ is then simply their union $A\cup B$. Another example is to consider the natural numbers ordered by divisibility, i.e. we say $a\leq b$ whenever $a\vert b$ in $\mathbb{N}$. The join $a\vee b$ is then the least common multiple $lcm(a,b)$.

From a categorical perspective, the join is precisely the coproduct (when viewing the poset as a category). In general, a coproduct is ‘only’ unique up to unique isomorphism, but note that in our case this is precisely equality. Dually, the product is then called the meet, which is the greatest lower bound in poset terms. In the above examples, the meet correspond to the intersection and greatest common divisor, respectively. It is (obviously) denoted $a\wedge b$.

The way I remember the terminology is that the meet corresponds to the intersection of sets, i.e. where the sets « meet », while the union contains both, so it « joins » them. This makes sense, I assure you.

Equipped with these concepts, we can now give the following

Definition. A lattice is a poset admitting all finite meets and joins.

Removing the finiteness condition in favor of arbitrary meets and joins, we get the notion of a complete lattice. Both examples above are complete lattices, but not every lattice is complete: under the same notions of meet and join as in the powerset example, the lattice of open subsets of a topological space is not complete, since the arbitrary intersection of opens need not be open.

Incohesive thoughts

• Does the powerset of a poset inherit some kind of order?
• Right-adjoint functors preserve limits, hence meets. What does an adjunction between posets look like?