Generalizing high-school aglebra

Highschool Algebra

• Equation: \(\ell:\equiv (y=x^2+3).\)
• Algebraic Variety: \[Z(\ell)=\{(x,y)\mid \ell(x,y)\}.\]
• Geometry: parabola.

Abstract Algebra

• Law: \(\Phi(x,y):\equiv (x+y=y+z).\)
• Variety of algebras: \[\mathfrak{V}(\Phi)=\{\langle A,+\rangle\mid (\forall x,y:A).\Phi(x,y)\}.\]
• Family: commutative

Highschool Algebra

• Equation: \(\ell:\equiv (z=x^2+y^2).\)
• Algebraic Variety: \[Z(\ell)=\{(x,y,z)\mid \ell(x,y,z)\}.\]
• Geometry: paraboloid.

Abstract Algebra

• Law: \(\Phi(x,y,z):\equiv (x*(y*z)=(x*y)*z).\)
• Variety of algebras: \[\mathfrak{V}(\Phi)=\{\langle A,*\rangle\mid (\forall x,y,z:A).\Phi(x,y,z)\}.\]
• Family: associative

Highschool Algebra

• Equation: \(\ell:\equiv (0=xy).\)
• Algebraic Variety: \[Z(\ell)=\{(x,y)\mid \ell(x,y)\}=\{(0,0)\}.\]
• Geometry: point.

Abstract Algebra

• Law: \(\Phi(x):\equiv (x*1=x=1*x).\)
• Variety of algebras: \[\mathfrak{V}(\Phi)=\{\langle A,*\rangle\mid (\forall x:A).\Phi(x)\}.\]
• Family: unital.

Highschool Algebra

Abstract Algebra

• Law: \[\Phi_1(x) :\equiv (x*1=x=1*x)\] \[\Phi_2(x,y,z):\equiv (x*(y*z)=(x*y)*z).\]
• Variety of algebras: \[\mathfrak{V}(\Phi_1,\Phi_2)=\{\langle A,*\rangle\mid (\forall x,y,z:A).\Phi(x)\wedge\Gamma(x,y,z)\}.\]
• Family: monoid.

Polynomials & Laws

Definition.

Fix a signature \(\sigma\).

A polynomial (or "word" or "formula") in \(\sigma\) is a grammatically correct ("parseable") string of

1. variables
2. symbols in \(\sigma\), or
3. parenthetical groupings.

E.g. For \(\sigma=\{+,0\}\), \(x+y,(x+0)+(y+z)\);

but not \(x^2-x\), the latter requires \(\sigma\) contain `\(-\)' and products.

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

\(x^2+1\)

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

\(x^2\)

\(1\)

\(x\)

\(x\)

\(u^{-1}\)

\(u=x-2\)

\(+\)

\(\times\)

\(\times\)

\(x\)

\(-(1+1)\)

\(-v\)

\(v=1+1\)

\(1\)

\(+\)

\(u\)

\(^{-1}\)

\(v\)

\(-\)

\(+\)

\(\circ\)

\(\circ\)

Operator

Variable

Polynomial/Word

(Meta-language)

A generalize view of "polynomials", "words", and "formulas"

Definition.

Fix a signature \(\sigma\).

• A polynomial in \(\sigma\) is a grammatically correct ("parseable") string of variables & symbols in \(\sigma\), and groupings.
• An (equational) law is \(\Phi(x_1,\ldots)=\Gamma(x_1,\ldots)\) where \(\Phi,\Gamma\) are words/formulas/polynomials.

E.g. For \(\sigma=\{+,0\}\), \(x+y=y+x, x+0=x\);

but not \(x+(-x)=0\), uses symbols outside \(\sigma\).

Varieties of Algebra

Definition.

• Fix a signature \(\sigma\).
• A variety of \(\sigma\)-algebraic structures is the set of all algebraic structures that satisfy a fixed set \(\Phi_*=\{\Phi_1,\Phi_2,\ldots\}\) of laws.
• Denote it \(\mathfrak{V}(\Phi_*)\).  (Fraktur "V")

Do not confuse with more "high-school" concept of "algebraic variety" whose elements are points in space. 

Example: Semigroup

• Fix a signature \(\sigma=\{*\}\).
• Law: associative
• Variety: \(\mathfrak{V}(x*(y*z)=(x*y)*z)\) "Semigroups"

Example: Monoid

• Fix a signature \(\sigma=\{*,1\}\).
• Laws: associative, identity
• Variety: \(\mathfrak{V}(x*(y*z)=(x*y)*z,x*1=x=1*x)\) "Monoid"

Technically \(x*1=x\) and \(1*x=x\) are separate laws.

Example: Groups

• Fix a signature \(\sigma=\{*,^{-1},1\}\).
• Laws: 
  • Asc. \(\Phi_1(x,y,z):\equiv (x*(y*z)=(x*y)*z\)
  • Id. \(\Phi_2(x):\equiv (x*1=x=1*x)\)
  • Inv. \(\Phi_3(x):\equiv (x*x^{-1}=1=x^{-1}*x)\).
• Variety: \(\mathfrak{V}(\Phi_*)\) "Groups"

Example: Abelian groups

• Fix a signature \(\sigma=\{+,-,0\}\).
• Laws: 
  • Group
  • Comm. \(\Phi_4(x,y):\equiv (x+y=y+x)\).
• Variety: \(\mathfrak{V}(\Phi_*)\) "Abelian Groups"

Example: Ring

• Fix a signature \(\sigma=\{*,1,+,-,0\}\).
• Laws: 
  • Asc. +, * \(\Phi_1^{\#}(x,y,z):\equiv (x\# (y\# z)=(x\# y)\# z\)
  • Id. (*,1), (+,0) \(\Phi_2^{\#,\epsilon}(x):\equiv (x\# \epsilon=x)\)
  • Comm. + \(\Phi_3(x,y):\equiv (x+y=y+x)\).
  • Inv. +,-,0 \(\Phi_4(x):=\equiv (x+(-x)=0)\).
  • L.Dist. \(\Phi_5(x,y,z):\equiv (x*(y+z)=x*y+x*z)\).
  • R.Dist. \(\Phi_6(x,y,z):\equiv ((x+y)*z=x*z+y*z)\).
• Variety: \(\mathfrak{V}(\Phi_*)\) "Ring"

Non-Example: Fields

• Fix a signature \(\sigma=\{*,1,+,-,0\}\).
• Laws: 
  • Ring *,1,+,-,0
  • Mult. Inv. \(\Phi_6(x):\equiv (x\neq 0\Rightarrow x*x^{-1}=1)\).
• Not Variety:  clearly last rule is not like the others, much more than an equation.

Fields are clearly important, why kick them out over such a small thing?

Come back to this in next lesson.

Example: Lie ring

• Fix a signature \(\sigma=\{[,],1,+,-,0\}\).
• Laws: \(\Phi_i\)
  • Abelian Group +,-,0
  • \([x,y+z]=[x,y]+[x,z]\)
  • \([x+y,z]=[x,z]+[y,z]\)
  • Jacobi \[[x,[y,z]]=[[x,y],z]+[y,[x,z]]\] \[\frac{d}{dx}(fg)=\frac{df}{dx}g+f\frac{dg}{dx}.\]
• Variety: \(\mathfrak{V}(\Phi_*)\) "Lie Ring"