Congruence
2020 James B. Wilson
Colorado State University
Homomorphisms
Quotients
Algebra is the study of equations, e.g.
\[x^2+1=0\]
Abstract algebra is when we see every symbol in an equation as a variable, including +, and =.
We are engaged in replacing "=".
Theme
Objective
- We found congruence is what can replace "=".
- Homomorphisms give congruences.
- We now define Quotients to produce homomorphisms.
Problem
Want an equivalence but don't have a function.
Consider the following relation of positive integers
\[(a,b)\equiv (c,d) \Longleftrightarrow ad=bc\]
Recognize this is \(a/b=c/d\) but written without fractions to focus on the problem.
Yes: simply "reduce the fraction", e.g. \((2,4)\mapsto (1,2)\); \((21,27)\mapsto (7,9)\).
Is this the equivalence given by a function, ie.:
\[x\equiv y \pmod{f}\Longleftrightarrow f(x)=f(y)?\]
Consider \[\mathbb{Z}[\sqrt{-5}]=\{a+b\sqrt{-5}\mid a,b\in \mathbb{Z}\}\]
Can you "reduce" \(\frac{9}{3}\)?
Easy: \[\frac{9}{3}= \frac{3\cdot 3}{3\cdot 1}=\frac{3}{1}.\]
Not so easy but equally true:
\[\frac{9}{3}=\frac{(4+5)}{3}=\frac{2^2-\sqrt{-5}^2}{3}=\frac{(2+\sqrt{-5})(2-\sqrt{-5})}{3}\]
which does not reduce.
Fact: \(\mathbb{Z}[\sqrt{-5}]\) does not have unique factorization.
Consequence
Some "equals" do not seem to be based on functions, e.g. fractions over \(\mathbb{Z}[\sqrt{-5}]\) do not have unique reduced forms.
Equivalence
Function
Partition
Recall Disjoint Unions
Given a function \(P:I\to type\)
\[\bigsqcup P:=\bigsqcup_{i:I} P_i.\]
More specifically you have intro rules to give you \[\iota_i:P_i\to \bigsqcup P\] and elimination rule to give you \[\sqcup f:\bigsqcup P\to R\] given \((i:I)\mapsto (f_i:P_i\to R)\) and such that
\[\begin{array}{ccl} P_i &\overset{\iota_i}{\longrightarrow} & \bigsqcup P \\ & {_{f_i}\searrow} & \downarrow \sqcup f\\ & & R\end{array}\]
Partition
A partition of a type (or set) \(A\) is an invertible pair of functions
\[f:A\to \bigsqcup P\qquad g:\bigsqcup P\to A.\]
together with a evidence for \(f\circ g=1\) and \(g\circ f=1\), (see bottom slide for details.)
The classes are \(g_i:P_i\to A\) where \[\begin{array}{ccl} X & \overset{\iota_i}{\longrightarrow} & \bigsqcup P\\ & g_i \searrow & \downarrow g\\ & & A\end{array}\]
\(f\)
\(g\)
\(A\)
\(\bigsqcup_{i:I}P_i\)
\(A=\bigsqcup_{X:P} X\)
click box
Informally we "identify" these types hiding \((f,g)\) in notation
Actually a new breed of Type Theory called Homotopy Type Theory (HoTT) can formally identify these.
Treat \(a:X_i\) as \(a:A\) where \(a=g(x_i)\) for some \(x_i:P_i\).
So we can write \(A=\bigcup_{X:P}X\), though "=" here means to use \(f,g\) under the covers to translate.
After "identifying" get an alternative Perspective
A partition of a set \(A\) is a set \(\mathcal{P}\) of subsets where
- \(A=\bigcup_{X\in P} X\)
- For \(X,Y\in \mathcal{P}\), If \(X\cap Y\neq \emptyset\) then \(X=Y\).
Claim. Every partition \((f,g):A\leftrightarrow \bigsqcup_{i:I} P_i\) of \(A\) determines a function \(\varphi:A\to I\) where
\[a\equiv a'\pmod{\varphi}\Leftrightarrow f(a),f(a'):P_{\varphi(a)}.\]
Proof. For \(i:I\) set \(\gamma_i:P_i\to I\) as the constant function
\[\gamma_i:(x:P_i)\mapsto (i:I).\]
By the elimination rule for \(\sqcup P\), there is a function \(\gamma:\bigsqcup P\to I\) where \[\gamma(\iota_i(x)) = \gamma_i(x)=i.\]
Define \(\varphi:A\to P\) as \(x\mapsto \gamma(f(x))\) (where \(f:A\to \bigsqcup P\) is given above.)
Assignment: show the property of \(a\equiv a'\pmod{\varphi}\)\(\Box\)
Consequence
-
Every partition \(A\leftrightarrow \bigsqcup_{i:I}P_i\) of produces a function \(\varphi:A\to I\).
-
Every function \(\varphi:A\to I\) produces an equivalence.
-
So every partition produces an equivalence.
Congruence
Homomorphism
Quotient
Now add algebra!
The Idea
Fix a set (or type) \(A\) with an operation, say \(*:A^2\to A\).
A quotient of \(\langle A,*\rangle\) is a partition such that the operation \(*\) applies on the class of the partition. We rename the classes cosets!
Technical Definition
Fix a set (or type) \(A\) with an operation, say \(*:A^2\to A\).
A quotient of \(\langle A,*\rangle\) is a partition \((f,g):A\leftrightarrow \bigsqcup_{i:I}P_i\), with an operation \(\#:I^2\to I\) where
\[x:P_i, y: P_j\Rightarrow f(g(x)*g(y)): P_{i\# j}.\]
We write this condition as \[P_i*P_j \subset P_{i\# j}.\]
In a quotient \(A\leftrightarrow \bigsqcup P\), the \(P_i\) are called cosets.
Shorthand Definition
Fix a set (or type) \(A\) with an operation, say \(*:A^2\to A\), identified with a quotient \(\bigsqcup_{X:P} X\) then for \(X,Y\in P\)
\[x:X, y:Y\Rightarrow x*y: X*Y.\]
If it exist it is unique
Recall \((f,g):A\leftrightarrow \bigsqcup_{i:I}P_i\) produces a map \(\varphi:A\to I\) where
\[a:A \Rightarrow f(a):P_{\varphi(a)}\]
Hence, \(P_i*P_j \subset P_{i\#j}\) means \[i\#j = \varphi(f(g(x)*g(y))).\]
Example. \(\langle \mathbb{Z},+\rangle\) as a quotient of \(\langle \mathbb{N}^2,+\rangle\).
Define
\[m-n:=\{(s,t)\in\mathbb{N}^2\mid m+t=n+s\}\]
\[\mathbb{Z}:=\{m-n\mid m,n\in\mathbb{N}^2\}.\]
Here \(m-n\) is a coset, like fractions \(m/n\), even though it is tempting to think it is substraction now, wait to do so.
Define
\[m-n:=\{(s,t)\in\mathbb{N}^2\mid m+t=n+s\}\]
\[\mathbb{Z}:=\{m-n\mid m,n\in\mathbb{N}^2\}.\]
Claim 1. \(\mathbb{Z}\) is a partition of \(\mathbb{N}^2\).
Proof. Assignment.
Define \(m-n:=\{(s,t)\in\mathbb{N}^2\mid m+t=n+s\}\),
\(\mathbb{Z}:=\{m-n\mid m,n\in\mathbb{N}^2\}.\)
Claim 1. \(\mathbb{Z}\) is a quotient of \(\langle\mathbb{N}^2,+\rangle\) where
\[(m,n)+(s,t):=(m+s,n+t)\]
is the addition in \(\mathbb{N}^2\).
Proof. Fix \((s,t)\in (m-n)\) and \((s',t')\in (m'-n')\).
\[(s,t)+(s',t')=(s+s',t+t')\]
and \[\begin{aligned}(m+m')+(t+t') & = (m+t)+(m'+t')\\ & = (n+s)+(n'+s')\\ & = (n+n')+(s+s')\end{aligned}\]
Hence, \((m-n)+(m'-n'):=(m+m')-(n+n')\) contains \((s+s',t+t')\) as required. So \(\mathbb{Z}\) is a quotient. \(\Box\)
In General
- A quotient must respect not only binary operators but all operators of an algebra.
- We abbreviate this writing \[[P_{i_1},\ldots,P_{i_n}]\subset P_{\langle i_1,\ldots,i_n\rangle}\]
- The product on the indices \(I\) becomes a new smaller algebra with compatible operations.
- The \(P_i\) are called cosets.
Conclusion
- A quotient is an \(\sigma\)-algebraic structure \(A\) with a partition \(Q\) that respect all operators.
- Quotients are \(\sigma\)-algebraic structures themselves.
- There is always a homomorphism \(A\to Q\) from an algebra to its quotients.
- Quotients are third source for new congruences.