Informally, understanding an experiment $E$ means introducing coordinates into phase space of $E$ which are in triangular form under the action of the inputs of $E$.

- The parameter $Q(t)$ determines some obviously important aspects ofthe system. That is, there is a deterministic function $M(Q(t))$ whichmaps $Q(t)$ to "measure" some internal state of the system.
- If the values of such a parameter $Q$ is known at time $t_0$ (denoted $Q(t_0)$)and it is also known what inputs are presented to thesystem from time $t$ to time $t + \epsilon$(denoted $I[t_0, t_0 + \epsilon]$), then the new value of $Q$ is adeterministic function of $Q(t_0)$ and $I[t_0, t_0+ \epsilon]$.

In general, conservation principles give a first coordinate of a triangularization. In the main a large part of physics can be viewed as discovering and introducing functions $e$ of the states $q$ of the system such that under action $a$, $e(a \curvearrowright q)$ depends only on $e(q)$ and $a$, andnoton $q$.

- For semidirect products, I refer you tothe cutest way to write semidirect productsLine of investigation to build physical intuition for semidirect products.

We first heuristically indicate the construction involved in going from the group of symmetries to the triangularization, and then precisely write it out in all pedantic detail.Let an experiment be $E \equiv (X, A, \curvearrowright)$. Then we define $\Pi$ is a

- $\Pi: X \rightarrow X$ is a permutation of $X$.
- $\Pi$ commutes with the action of each $a$:$\Pi(a \curvearrowright x) = a \curvearrowright \Pi(x)$.

- We know that the symmetries of a theory $E$ form a group.
- If $E$ is transitive, then each symmetry $\Pi$ is a regular permutation--- If there exists an $x$ such that $\Pi(x_f) = x_f$ (a fixed point), thenthis implies that $\Pi(x) = x$ for
*all*$x$. - Let the action split $X$ into disjoint orbits $O_1, O_2, \dots O_k$ from whomwe choose representatives $x_1 \in O_1, x_2 \in O_2, \dots x_k \in O_k$.Then, if $E$ is transitive, there is
*exactly one*action that sends aparticular $x_i$ to a particular $x_j$. So, on fixing*one component*of an action, we fix*all components*.

$(X, S) \leq (G, G) \wr (\{ O_1, O_2, \dots O_k\}, T)$