-Since we have defined this operation in terms of specific forms, we must ensure
-the operation behaves identically with respect to all similar forms. It would
-be a shame if, for example, $0+0=0$ only held for certain values of $0$.
-
-\begin{theorem} \label{thm:sgkadd-welldefined}
- The binary operation $\sgkadd$ on $\mathbb{U}$ is well defined. That is, for
- numbers $x, x', y, z \in \mathbb{U}$ such that $x \sgkadd y = z$ and $x
- \similar x'$, $\exists z' \in \mathbb{U}$ such that $x' \sgkadd y = z'
- \similar z$.
-\end{theorem}
-
-\begin{proof}
- TODO
-\end{proof}
-
-\begin{theorem} \label{thm:sgkadd-identity}
- The number $0 = \surreal{}{}$ is the identity element for the binary
- operation $\sgkadd$ on $\mathbb{U}$. That is, for any number $x \in
- \mathbb{U}$, $x \sgkadd 0 \similar 0 \sgkadd x \similar x$.
- In this behavior, the number $0$ is unique up to similarity.
-\end{theorem}
-
-\begin{proof}
- TODO
-\end{proof}
-
-\begin{theorem} \label{thm:sgkadd-associative}
- For all $x, y, z \in \mathbb{U}$, it holds that
- $$(x \sgkadd y) \sgkadd z \similar x \sgkadd (y \sgkadd z).$$
-\end{theorem}
-
-\begin{proof}
- TODO
-\end{proof}
-
-\begin{defi} \label{defi:inverse}
- For a number $x$, let \emph{negation} be defined as
- $$
- -x = -\surreal{X_L}{X_R} \equiv \surreal{-X_R}{-X_L}
- $$
- where $-A$ means the set of numbers $-a$ for all $a \in A$.
-\end{defi}
-
-\begin{theorem} \label{thm:sgkadd-inverse}
- For every number $x \in \mathbb{U}$, there exists a number $-x \in
- \mathbb{U}$ such that $x \sgkadd -x = 0$.
- In this behavior, the number $-x$ is unique up to similarity.
-\end{theorem}
-
-\begin{proof}
- TODO
-\end{proof}
-
-Putting that all together, $(\mathbb{U},\sgkadd)$ is well defined, closed, and
-respects the three group axioms. It's a group. Let's name it
-$\mathbb{U}_{\sgkadd}$. It's also commutative.
-
-
-\subsection{Conjecture}
-
-$\mathbb{U}_{\sgkadd}$ really is a group.
-
projects / surreal-numbers / commitdiff