What is a lattice in linear algebra?

Question

In a lecture on linear algebra the prof sort of threw out a term in the middle of the lecture but did not bother to explain it.

The claim was that given $V, U \subset X$ two subspaces, then $V+W$ and $V \cap W$ implies family of subspace forms a lattice.

I referred to my linear algebra book but did not even find the word lattice in the reference. (Maybe it is too elementary). Wikipedia on the other hand is too advanced for me https://en.wikipedia.org/wiki/Lattice_(order)

Can someone please elaborate on what it means for $V,U$ to imply a lattice structure?

Answer 1

In the present case, a lattice is a notion that is not defined in linear algebra, but for ordered sets:

A (partially) ordered set $L$ is a lattice if every pair of elements has a least upper bound and a greatest lower bound.

Here, the set of subspaces is ordered by inclusion, and if $U,V$ are subspaces of $X$, then $$\sup(U,V)=U+V,\quad\inf(U,V)=U\cap V.$$

Added:

As mentioned by @Tobias Kildetoft in a previous comment, there is another notion of lattice in linear algebra: if $A$ is an integral domain with field of fractions $K$, $V$ a finite dimensional $K$-vector space, a lattice $L$ in $V$ is a free $A$-module of rank $\dim_KV$. This notion is especially useful in Algebraic Number Theory.

Answer 2

It means any two element has a unique join (like a least common multiple) and a unique meet(like a greatest common divisor). Think of divisors as subsets of an element and multiples as sets that contain the element as a subset.