Stringing it Out

The invitation is for you to explore the following transformations and comment on what you find:

First

T(x) = y
T(y) = xy

This transformation has the following successive effect (starting with x):

x
y
xy
yxy
xyyxy
...
...

Next

R(x) = xy
R(y) = z
R(z) = xy

This transformation has the following successive effect:

x
xy
xyz
xyzxy
xyzxyxyz
...
...

Then

S(w) = wx
S(x) = y
S(y) = wz
S(z) = y

This transformation has the following successive effect:

w
wx
wxy
wxywz
...
...

So what do you notice about this pattern and can you explain why it is occurring?

Now

Can you suggest an appropriate set of transformations for the following elements that would have the same effect?

T(v) = ?
T(w) = ?
T(x) = ?
T(y) = ?
T(z) = ?