Homomorphisms and Varieties of Algebra

2020 James B. Wilson

Colorado State University

 

Objectives

Prove varieties are closed to Quotients and Homomorphic images.

What happens to polynomials under homomorphisms.

\(\frac{x^2+1}{x-2}\)

\(x^2+1\)

\((x-2)^{-1}\)

\(x^2\)

\(1\)

\(x\)

\(x\)

\(u^{-1}\)

\(u=x-2\)

\(+\)

\(\times\)

\(\times\)

\(x\)

\(-(1+1)\)

\(-v\)

\(v=1+1\)

\(1\)

\(1\)

\(+\)

\(u\)

\(^{-1}\)

\(v\)

\(-\)

\(+\)

\(\circ\)

\(\circ\)

Operator

Variable

Polynomial/Word

(Meta-language)

A generalized polynomial

\(f\left(\frac{x^2+1}{x-2}\right)\)

\(x^2+1\)

\((x-2)^{-1}\)

\(x^2\)

\(1\)

\(x\)

\(x\)

\(u^{-1}\)

\(u=x-2\)

\(+\)

\(\times\)

\(\times\)

\(x\)

\(-(1+1)\)

\(-v\)

\(v=1+1\)

\(1\)

\(1\)

\(+\)

\(u\)

\(^{-1}\)

\(v\)

\(-\)

\(+\)

\(\circ\)

\(\circ\)

Operator

Variable

Polynomial/Word

(Meta-language)

Apply a homomorphism \(f:A\to B\)

\(f\left(\frac{x^2+1}{x-2}\right)\)

\(f(x^2+1)\)

\(f((x-2)^{-1})\)

\(x^2\)

\(1\)

\(x\)

\(x\)

\(u^{-1}\)

\(u=x-2\)

\(+\)

\(\times\)

\(\times\)

\(x\)

\(-(1+1)\)

\(-v\)

\(v=1+1\)

\(1\)

\(1\)

\(+\)

\(u\)

\(^{-1}\)

\(v\)

\(-\)

\(+\)

\(\circ\)

\(\circ\)

Operator

Variable

Polynomial/Word

(Meta-language)

Apply a homomorphism \(f:A\to B\)

\(f\left(\frac{x^2+1}{x-2}\right)\)

\(f(x^2+1)\)

\(f((x-2)^{-1})\)

\(f(x^2)\)

\(f(1)\)

\(x\)

\(x\)

\(f(u^{-1})\)

\(u=x-2\)

\(+\)

\(\times\)

\(\times\)

\(x\)

\(-(1+1)\)

\(-v\)

\(v=1+1\)

\(1\)

\(1\)

\(+\)

\(u\)

\(^{-1}\)

\(v\)

\(-\)

\(+\)

\(\circ\)

\(\circ\)

Operator

Variable

Polynomial/Word

(Meta-language)

Apply a homomorphism \(f:A\to B\)

\(f\left(\frac{x^2+1}{x-2}\right)\)

\(f(x^2+1)\)

\(f((x-2)^{-1})\)

\(f(x^2)\)

\(1_B\)

\(f(x)\)

\(f(x)\)

\(f(u^{-1})\)

\(f(u)=f(x-2)\)

\(+\)

\(\times\)

\(\times\)

\(x\)

\(-(1+1)\)

\(-v\)

\(v=1+1\)

\(1\)

\(1\)

\(+\)

\(f(u)\)

\(^{-1}\)

\(v\)

\(-\)

\(+\)

\(\circ\)

\(\circ\)

Operator

Variable

Polynomial/Word

(Meta-language)

Apply a homomorphism \(f:A\to B\)

\(f\left(\frac{x^2+1}{x-2}\right)\)

\(f(x^2+1)\)

\(f((x-2)^{-1})\)

\(f(x^2)\)

\(1_B\)

\(f(x)\)

\(f(x)\)

\(f(u^{-1})\)

\(f(u)=f(x-2)\)

\(+\)

\(\times\)

\(\times\)

\(f(x)\)

\(f(-(1+1))\)

\(-f(v)\)

\(f(v)=f(1+1)\)

\(1_B\)

\(1_B\)

\(+\)

\(f(u)\)

\(^{-1}\)

\(f(v)\)

\(-\)

\(+\)

\(\circ\)

\(\circ\)

Operator

Variable

Polynomial/Word

(Meta-language)

Apply a homomorphism \(f:A\to B\)

Summary

  • Given a polynomial/word/formula \(\Phi(x_1,x_2,\ldots\)
  • An algebraic structure \(A\) with constants \(a_1,a_2,\ldots:A\)
  • a homomorphism \(f:A\to B\)
  • \(f(\Phi(a_1,a_2,\ldots))=\Phi(f(a_1),f(a_2),\ldots).\)

 

With out all the symbols...

\[f\circ \Phi=\Phi \circ f\]

Theorem.

Fix a signature \(\sigma\).  If \(A\) is in a variety \(\mathfrak{V}(\Phi)\) and \(\varphi:A\to B\) is surjective  homomorphism, then \(B\)

a in \(\mathfrak{V}(\Phi)\).

Proof. 

  • For  \(b_*=b_1,b_2,\ldots:B\), by surjectivity, there are \(a_*=a_1,a_2,\ldots:A\) with \(b_i=f(a_i)\).
  • For any law \(\Phi(xs)=\Gamma(xs)\) of \(\mathfrak{V}\), \(\Phi(a_*)=\Gamma(a_*).\)
  • \[\begin{aligned} \Phi(b_*) & =\Phi( f(a_*)) \\ & = f\circ \Phi(a_*)\\ & = f\circ \Gamma(a_*) \\ & = \Gamma(f(a_*))\\ & =\Gamma(b_*)\end{aligned}\]

Theorem.

Fix a signature \(\sigma\).  If \(A\) is in a variety \(\mathfrak{V}(\Phi)\) and \(\varphi:A\to B\) is surjective  homomorphism, then \(B\)

a in \(\mathfrak{V}(\Phi)\).

Corollary. 

Varieties of algebras are closed under quotients.

Proof. \(A/_{\equiv}\) induces a homomorphims \(f:A\to A/_{\equiv}\) which is surjective. \(\Box\)