$\frac{x^2+1}{x-2}$

$x^2+1$

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

$x^2$

$1$

$x$

$x$

$u^{-1}$

$u=x-2$

$+$

$\times$

$\times$

$x$

$-(1+1)$

$-v$

$v=1+1$

$1$

$+$

$u$

$^{-1}$

$v$

$-$

$+$

$\circ$

$\circ$

Operator

Variable

Polynomial/Word

(Meta-language)

A generalized polynomial

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

$x^2+1$

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

$x^2$

$1$

$x$

$x$

$u^{-1}$

$u=x-2$

$+$

$\times$

$\times$

$x$

$-(1+1)$

$-v$

$v=1+1$

$1$

$+$

$u$

$^{-1}$

$v$

$-$

$+$

$\circ$

$\circ$

Operator

Variable

Polynomial/Word

(Meta-language)

Apply a homomorphism $f:A\to B$

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

$f(x^2+1)$

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

$x^2$

$1$

$x$

$x$

$u^{-1}$

$u=x-2$

$+$

$\times$

$\times$

$x$

$-(1+1)$

$-v$

$v=1+1$

$1$

$+$

$u$

$^{-1}$

$v$

$-$

$+$

$\circ$

$\circ$

Operator

Variable

Polynomial/Word

(Meta-language)

Apply a homomorphism $f:A\to B$

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

$f(x^2+1)$

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

$f(x^2)$

$f(1)$

$x$

$x$

$f(u^{-1})$

$u=x-2$

$+$

$\times$

$\times$

$x$

$-(1+1)$

$-v$

$v=1+1$

$1$

$+$

$u$

$^{-1}$

$v$

$-$

$+$

$\circ$

$\circ$

Operator

Variable

Polynomial/Word

(Meta-language)

Apply a homomorphism $f:A\to B$

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

$f(x^2+1)$

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

$f(x^2)$

$1_B$

$f(x)$

$f(x)$

$f(u^{-1})$

$+$

$\times$

$\times$

$x$

$-(1+1)$

$-v$

$v=1+1$

$1$

$+$

$f(u)$

$^{-1}$

$v$

$-$

$+$

$\circ$

$\circ$

Operator

Variable

Polynomial/Word

(Meta-language)

Apply a homomorphism $f:A\to B$

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

$f(x^2+1)$

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

$f(x^2)$

$1_B$

$f(x)$

$f(x)$

$f(u^{-1})$

$+$

$\times$

$\times$

$f(x)$

$f(-(1+1))$

$-f(v)$

$f(v)=f(1+1)$

$1_B$

$+$

$f(u)$

$^{-1}$

$f(v)$

$-$

$+$

$\circ$

$\circ$

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 $\mathfrak{V}(\Phi)$.

Corollary. 

Varieties of algebras are closed under quotients.

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