Varieties Subalgebra closed

• Claim. \(A\) an algebraic structure and \(\{a:A\mid P(a)\}\) a subalgebra.\[A\in\mathfrak{V}(\Phi)\Rightarrow \{a:A\mid P(a)\}\in \mathfrak{V}(\Phi).\]

Varieties Subalgebra closed

• \(A\) a type and \(P:A\to Prop\) then \[\langle a, e\rangle\in \{a:A\mid P(a)\}=\bigsqcup_{a:A} P(a)\]
• If \(A\) is \(\sigma\)-algebraic structure, a subalgebra means for each \([\ldots]\in \sigma\) there is a product \([\ldots]_P:P(a_1)\times \cdots\times P(a_n)\to P([a_1,\ldots, a_n])\) where \[[\langle a_1,e_1\rangle,\ldots,\langle a_n, e_n\rangle]=\langle [a_1,\ldots,a_n]_A,[e_1,\ldots,e_n]_P\rangle.\] We say \(\{a:A\mid P(a)\}\) is closed to \([\ldots]\).
• Claim. \[A\in\mathfrak{V}(\Phi)\Rightarrow \{a:A\mid P(a)\}\in \mathfrak{V}(\Phi).\]

Eg.  \(A=\mathbb{M}_d(\mathbb{R})\) as monoid under composition (matrix mult).

\(GL_d(\mathbb{R}):=\{X\in \mathbb{M}_d(\mathbb{R})\mid X^{-1}\) exists \(\}\).

1. Declare Evidence \(wit(\tilde{X}):P(X)\) where \(X\tilde{X}=I=\tilde{X}X\).
2. Make product of evidence: \[\cdot:P(X)\times P(Y)\to P(XY)\] \[wit(\tilde{X})\cdot wit(\tilde{Y}) := wit(\tilde{Y}\tilde{X})\]
3. Check \(wit(\tilde{Y}\tilde{X}):P(XY)\) by converting evidence \[(X Y)(\tilde{Y}\tilde{X})=X(Y\tilde{Y})\tilde{X}=I=\tilde{Y}(\tilde{X}X)Y=(\tilde{Y}\tilde{X})(XY)\]
4. So \[X,Y\in GL_d(\mathbb{R})\Rightarrow XY\in GL_d(\mathbb{R})\]
5. So \(GL_d(\mathbb{R})\) is immediately a monoid.

Varieties are Quotient closed

• Claim. \(\sim\) a congruence on \(A\) \[A\in\mathfrak{V}(\Phi)\Rightarrow A/_{\sim}\in \mathfrak{V}(\Phi).\]

Varieties are Quotient closed

• \(A\) a type and \(\sim:A^2\to Prop\) then a partition means \[A/_{\sim}=\bigsqcup_{a:A} [a]\qquad [a]=\{b:B\mid a\sim b\}\qquad a\sim b\to [a]=_{A/_{\sim}}[b]\]
• If \(A\) is \(\sigma\)-algebraic structure, a quotient means for each \(\llbracket\ldots\rrbracket\in \sigma\) there is a product \(\llbracket\ldots\rrbracket_{A/_{\sim}}\) where \[\llbracket[a_1],\ldots,[a_n]\rrbracket_{A/_{\sim}}=[\llbracket a_1,\ldots,a_n\rrbracket_A].\] We say \(A/_{\sim}\) admits \(\llbracket\cdots\rrbracket\).
• Claim. \[A\in\mathfrak{V}(\Phi)\Rightarrow A/_{\sim}\in \mathfrak{V}(\Phi).\]

Eg.  \(GL_d(\mathbb{R}):=\{X\in \mathbb{M}_d(\mathbb{R})\mid X^{-1}\) exists \(\}\) is a monoid, \(X\sim Y\Leftrightarrow (\exists s)(X^{-1}Y=sI_d)\)

• \(wit(s):[X_1]=_{GL_d(\mathbb{R})/_{\sim}} [X_2]\) means \(X_1^{-1} X_2=sI_d\).
• Make a product on the evidence \[([X_1]=[X_2])\times ([Y_1]=[Y_2])\to ([X_1Y_1]=[X_2Y_2])\] where \[wit(s)wit(t) := wit(st)\]
• Check that the type of evidence is correct: \[\begin{aligned} (X_1 Y_1)^{-1}(X_2 Y_2) & =Y_1^{-1} X_1^{-1} X_2 Y_2\\ & =Y_1^{-1} (sI_d) Y \\ & =s(tI_d)=(st)I_d.\end{aligned}\]
• So \(PGL_d(\mathbb{R}):=GL_d(\mathbb{R})/_{\sim}\) is immediately a monoid.

Varieties are Cartesian Closed

• Claim. \[A_1,\ldots,A_n\in\mathfrak{V}(\Phi))\Rightarrow A_1\times\cdots\times A_n\in \mathfrak{V}(\Phi).\]
• And more generally for any list \[(i:I) \rightarrow A_i\in\mathfrak{V}(\Phi))\Rightarrow \prod_{i:I} A_i\in \mathfrak{V}(\Phi).\]

If \(A\) and \(B\) are groups then \(A\times B\) is a group with operations

\[(a_1,b_1)*(a_2,b_2)=(a_1*a_2,b_1*b_2)\]

\[(a_1,b_1)^{-1}=(a_1^{-1},b_1^{-1})\] \[(1_A,1_B)=1_{A\times B}\]

And yes, you can replace group with ring, monoid, semigroup, etc. and use as many terms as you like.

An expected construction

Varieties are Cartesian Closed

• \(A:I\to type\) a family of types then a cartesian product means \[(i:I)\mapsto (\vec{a}(i):A_i) : \prod_{i:I} A_i.\] The elements are called tuples, coordinates, dependent functions.
• If \(A_i\) are \(\sigma\)-algebraic structure, the direct product means for each \(\llbracket\ldots\rrbracket\in \sigma\) there is a product \(\llbracket\ldots\rrbracket_{\prod A}\) where \[\llbracket \vec{a}_1,\ldots,\vec{a}_n\rrbracket_{\prod A}(i)=[\vec{a}_1(i),\ldots,\vec{a}_n(i)]_{A_i}.\]
• Claim. \[(i:I \rightarrow A_i\in\mathfrak{V}(\Phi))\Rightarrow \prod_{i:I} A_i\in \mathfrak{V}(\Phi).\]

Proof-\(\varepsilon\)

It is enough to look at \(A\times B\).

• For a polynomial formula \(\Phi(X)\), 
• \[\Phi((a_{1},b_{1}),\ldots,(a_{n},b_{n})) = (\Phi(a_{1},\ldots,a_{n}), \Phi(b_{1},\ldots,b_{n})).\]  Follows by induction on operators.
• Fix a Law \(\Phi_1(X)=\Phi_2(X)\).
• \[\begin{aligned} \Phi_1((a_{1},b_{1}),\ldots,(a_{n},b_{n})) & = (\Phi_1(a_{1},\ldots,a_{n}), \Phi_1(b_{1},\ldots,b_{n}))\\ & = (\Phi_2(a_{1},\ldots,a_{n}), \Phi_2(b_{1},\ldots,b_{n}))\\ &= \Phi_2((a_{1},b_{1}),\ldots,(a_{n},b_{n})).\end{aligned}\]
• So \(A,B\in\mathfrak{V}(\Phi)\Rightarrow A\times B\in\mathfrak{V}(\Phi)\).

Main lemma

You can embed a free algebra in a larger enough cartesian product.

Let \(A\) be an algebra with generators \(X\) and laws \(\Phi\) (i.e. \(A\) has a presentation \(\langle X\mid \Phi\rangle\).

• \(A^X=\{f:X\to A\}=\prod_{x:X} A\)
• \(\hat{A}  = A^{A^X} = \{f:A^X\to A\}=\prod_{f:X\to A}A\)
• \(\pi_x:A^{X}\to A\) where \[\pi_x(f):=f(x)\] where \(f:X\to A\).  Notice \(\pi_x:\hat{A}\).
• Claim. The subalgebra of \(\hat{A}\) generated by \(\{\pi_x\mid x:X\}\) is relatively free in \(\mathfrak{V}(\Phi)\) on \(X\).

Proof of Birkhoff, Neumann Theorem

Given \(X_i\in \mathfrak{X}\) and \(\Phi\) the laws the \(X_i\) have in common then \[\mathfrak{V}(\Phi)\subset \mathfrak{X}\]

Proof.  Let \(CQS\mathfrak{X}\subset \mathfrak{X}\).  We need laws that define \(\mathfrak{X}\).

• For \(A_i\in\mathfrak{X}\), \(A=\prod_{i:I} A_i\) has as laws \(\Phi\) only the laws that the \(A_i\) have in common. \(C\mathfrak{X}\subset \mathfrak{X}\Rightarrow A\in\mathfrak{X}\).
• By Lemma, a large enough power of \(A\) contains a subalgebra \(F[X]\) free in \(\mathfrak{V}(\Phi)\). \(S\mathfrak{X}\subset\mathfrak{X}\Rightarrow F\in\mathfrak{X}\).
• By presentation Theorem: \(B\in\mathfrak{V}(\Phi)\) is isomorphic to quotient of some \(F[X]\).  So \(Q\mathfrak{X}\subset \mathfrak{X}\Rightarrow B\in\mathfrak{X}\).

Proof of Birkhoff, Neumann Theorem

• Given \(X_i\in \mathfrak{X}\) and \(\Phi\) the laws the \(X_i\) have in common then \[\mathfrak{V}(\Phi)\subset \mathfrak{X}\]
• Increase the sample \(X_i\) until \(\Phi\) is the set of laws common to all of \(\mathfrak{X}\), i.e. \[\mathfrak{X}\subset \mathfrak{V}(\Phi).\]
• \(\mathfrak{X}=\mathfrak{V}(\Phi)\).

If you study algebra in the usual way (cartesian products, quotients, and subalgebras) then you are studying a variety: there are equational laws at play.