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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.

Ok, I got a copy of Emil Artin‘s Geometric Algebra from the library a couple of days ago, and a careful reading of some of the parts from the first chapter has convinced me even more now that one should indeed learn mathematics from the masters themselves!

The subject employs concepts/theorems/results from set theory, vector spaces, group theory, field theory and so on, and centers around the foundations of affine geometry, the geometry of quadratic forms and the structure of the general linear group. The book also deals with symplectic and orthogonal geometry and also the structure of the symplectic and orthogonal groups.

I intend to blog on this subject for as long as I can, and this will be my first serious project for now. There are some neat things I learned in chapter I which basically deals with preliminary notions. The actual text begins from chapter II, which is titled Affine and Projective Geometry. But, chapter I has some cool techniques too, and I wish to share them in my subsequent posts.

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March 2023