## Franklin's Notes

### Lattice

Given a partial ordered set $P$ with elements $x,y,z\in P$:

• If $x\leq z$ and $y\leq z$, we say that $z$ is an upper bound of $x,y$.

• If $z\leq x$ and $z\leq y$, we say that $z$ is a lower bound of $x,y$.

• If $z$ is an upper bound of $x,y$ that is $\leq$ every other upper bound of $x,y$, then we say that $z$ is a least upper bound, LUB, or join of $x$ and $y$.

• If $z$ is an lower bound of $x,y$ that is $\geq$ every other lower bound of $x,y$, then we say that $z$ is a greatest lower bound, GLB, or meet of $x$ and $y$.

If $P$ is such that every pair of elements $x,y\in P$ has both a least upper bound and a greatest lower bound, then we say that $P$ is a lattice. In this case, the join of $x$ and $y$ is denoted $x\lor y$ and their meet is denoted $x\land y$. It can be shown easily that both of these operations are necessarily both commutative and associative. An element $z$ is called join-reducible if it can be written in the form $z=x\lor y$ for some $x,y < z$, and join-irreducible otherwise. It is conventional for the least element of a lattice, if one exists, not to be considered join-irreducible, even though it meets this criterion.

Categorically speaking, a lattice can be thought of as a abstract partial order category with both coproducts and products.