# What does the ratio of a sets mean?

I'm reading a mathematical physics book and they define a couple sets like this:

$\epsilon \mathbb{Z}^d/L\mathbb{Z}^d$ and $\mathbb{R}^d/L\mathbb{Z}^d$

This is in the context of lattice field theories so I'm assuming it has to do with a set of lattice points with spacing $\epsilon$ and size $L^d$ but I'm intrested in more precely what the above expressions mean.

• I think that it denotes the quotient of groups. – Crostul Apr 11 '16 at 23:04

## 2 Answers

An equivalence relation is a binary relation $\sim$ that works similar to equality, except nonidentical things can be related. It satisfies $a\sim a$ (reflexivity), $a\sim b\iff b\sim a$ (symmetric) and $a\sim b\sim c\implies a\sim c$ (transitivity). Equivalence relations correspond to partitions of a set (ways of distributing its elements into groups): the $a\sim b$ essentially means $a$ and $b$ are in the same group. Examples (for, say, people) include "are the same height," "have the same hair color," "are the same age."

If we have an equivalence relation $\sim$ defined on a set $X$, then the notation $X/\sim$ refers to the collection of all equivalence classes, or in other words the thing you get when you pretend two things that are in the same class are in fact the same thing. For instance, if we have a seating arrangement of people with $m$ rows and $n$ columns, if $\sim$ is the relation "is in the same row as," then $X/\sim$ is essentially the set of rows.

When we talk about a vector space $V$, and a subset $L$ closed under the operations of addition and subtraction, the quotient $V/L$ means the quotient set obtained by using the equivalence relation "differs by an element of $L$." That is, $v\sim w$ when $v-w\in L$. So for example, suppose $V=\mathbb{R}^2$ and $L=\{0\}\times\mathbb{R}$ (the vertical axis in the plane). This equivalence relation means any two things in the same vertical line are the same thing in the quotient set, so $V/L$ is essentially the collection of all vertical lines in $L$. Notice that these may be parametrized by elements of $\mathbb{R}\times \{0\}$.

In fact, $V/L$ inherits its addition operation from $V$. If $V$ has a topology on it, then we can define a corresponding quotient topology on $V/L$.

When $L$ is a discrete subset (meaning, a collection of isolated points, or equivalently no sequence of things in $L$ converges to something not in $L$) of a Euclidean space, we call it a lattice. Then there is a corresponding quotient space. In particular, consider $\mathbb{R}/\mathbb{Z}$. Since $0\sim1$, if we start at the point $0$ and move to the right, once we reach $1$ we will be back to where we started in $\mathbb{R}/\mathbb{Z}$ since $0\sim 1$, so $\mathbb{R}/\mathbb{Z}\cong\mathbb{S}^1$ is essentially the one-dimensional circle.

Then $\mathbb{R}^2/\mathbb{Z}^2$ works similarly: use the points inside the unit square, and then invoke "thinking with portals." If you traverse the right edge of the square, you end up on the left edge at the same height, and similarly for the top and bottom edge of the square. If you literally take a square piece of rubber, glue the two left/right edges together you get a hollow cylinder, and then if you glue the two circles together you get a two-dimensional torus. Indeed $\mathbb{R}^2/\mathbb{Z}^2\cong\mathbb{T}^2=\mathbb{S}^1\times\mathbb{S}^1$.

This is called the quotient of the groups and are how we define equivalence classes within a group, with $G/H$ representing equivalence classes of elements of $G$, where two elements are equivalent if their difference is in $H$. You can read the wikipedia page here. In your context, you are likely considering the corners of the lattice to be equivalent, and only looking at relative position within the lattice.