Characteristic

$H\leq G$ where for every automorphism $\varphi:G\to G$, $\varphi(H)=H$.

 Functions

Functions $x\mapsto M$

• Evaluate $M[x:=a]$: replace $x$ with $a$.
• Rewrite: convert new string by stated equal symbols.

Typed Functions

Some Functions $x\mapsto M$ obey inference $$\frac{a:A}{M[x:=a] : B}$$

Write $f:A\to B$

$Y^?=Y\sqcup\{\bot\}$.  Partial function means $f:X\to Y^?$.

Partial Equality: $y\asymp y'$ means if $y,y'\in Y$ then $y=y'$.

$G$ is an abstract category if it has a binary partial operation such that 

$$\begin{aligned} g\cdot (h\cdot k) & \asymp (g\cdot h)\cdot k\\ 1_G \cdot g&=\{g\}\\ g\cdot 1_G&=\{g\}\end{aligned}$$

A mix of groupoid and monoid.

Note $1_G$ is all the identities.

Abstract Category

Laws

• $s(x)=t(y)$ if, and only if, $xy\in C$
• $s(t(x))=t(x)$ & $t(s(x))=s(x)$
• $s(xy)=s(s(x)y)$ & $t(xy)=t(xt(y))$
• $xy, yz\in C$ then $x(yz)=(xy)z$
• $xs(x)=x=t(x)x$

Theory

• $s(s(x))=s(x)$ & $t(t(x))=t(x)$$$\begin{aligned}Pf.\qquad s(s(x)) & = s(t(s(x))) \\ &=t(s(x))=s(x)\quad \Box \end{aligned}$$
• $s(xy)=s(y)$ & $t(xy)=t(x)$$$\begin{aligned}Pf.\qquad s(xy) & = s(s(x)y)\\ &=s(t(y)y)=s(y)\qquad \Box\end{aligned}$$
• $xy,yz\in C \Rightarrow x(yz),(xy)z\in  C$$$\begin{aligned}Pf.\qquad (s(x)&=t(y))  \wedge(s(y)=t(z)) \\ & \Rightarrow t(yz)=t(y)=s(x)\,\Box\end{aligned}$$

$P\vdash$

$\vdash P$

A change in notation

Fact

Abuse of Notation

Notation