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.