1. Notions of Set Theory

Artin introduces a concept that is referred to as the canonical factoring of a map (function). The basic idea is that any function $f$ can be factored into three functions $f_1, f_2$ and $f_3$ in a somewhat unique way:

$f = f_3 f_2 f_1$, where

$f_1$ is onto, $f_2$ is a bijection, and $f_3$ is an injection. The construction of these three functions is done in a canonical, or natural, way that doesn’t require the use of objects outside the domain and/or range of $f$.

Let $S$ be some non-empty set. If $f$ is a function from $S$ into a set $T$, then we write

$f: S \to T$.

Suppose $f: S \to T$ and $g: T \to U$. Then, we can form a composite function $g \circ f: S \to U$ defined by $(g \circ f)(s) = g(f(s))$ for all $s \in S$. The associative law holds trivially for composition of functions.

Further, if $S_0 \subset S$, then the set of all the images of elements of $S_0$, denoted by $f(S_0)$, is called the image of $S_0$. In general, $f(S) \subset T$. We call the function $f$ onto whenever $f(S) = T$.

Now, let us partition the set $S$ into equivalence classes such that $s_1, s_2 \in S$ are in the same equivalence class iff $f(s_1) = f(s_2)$. This partition is called the quotient set and is denoted by $S_f$.

To illustrate, suppose $S = \{ 1, 2, 3, 4\}$ and $T = \{ a, b, c, d\}$. Also, let $f: S \to T$ such that $f(1) = a, f(2) = b, f(3) = b$ and $f(4) = c$. Then, the quotient set, $S_f = \{ \{ 1\}, \{ 2, 3\}, \{ 4\}\}$.

We construct now a function $f_1: S \to S_f$ that maps each $s \in S$ to its equivalence class. It can be verified that $f_1$ is onto. So, taking the above example, we have $f_1(1) = \{ 1 \}$, $f_1(2) = \{ 2, 3\}$, $f_1(3) = \{ 2, 3\}$ and $f_1(4) = \{4\}$.

Next, we construct a function $f_2: S_f \to f(S)$ where each element (which is an equivalence class) of $S_f$ is mapped to a $t \in T$ where each $t$ is the image of the members of the equivalence class. Recall that $s_1, s_2 \in S$ are in the same equivalence class iff $f(s_1) = f(s_2)$. Therefore, $f_2$ is one-to-one and onto. Continuing with our above example, we have $f_2(\{ 1\}) = a$, $f_2(\{ 2, 3\}) = b$ and $f_2(\{ 4\}) = c$.

And, finally, we construct a trivial function $f_3: f(S) \to T$ where $f_3(t) = t$ for each $t \in f(S)$. Note that $f_3$ is not an identity because it maps a subset, $f(S)$, into a possibly larger set, $T$, i.e. $f_3$ is an identity iff $f$ is onto. In general, $f_3$ is one-to-one and into (i.e. an injection.)

Thus, if $f(s) = t$, we note that $f_3 f_2 f_1 (s) = t$ for every $s \in S$.

And, so,

$f = f_3 f_2 f_1$.

Once again, $f_1$ is onto, $f_2$ is a bijection, and $f_3$ is an injection.

It looks like it doesn’t make much sense to factor $f$ the way we did above, but we will explore more of this with respect to group homomorphisms in my next post.