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. \(\Box\)