User:Jshflynn/Ideas:Concatenation issue

Outline
I believe concatenation is actually two more primitive operations performed one after the other. A binary operation called sequence join and a unary operation called sequence strip. If this has already been thoroughly researched please leave a note on the discussion page.

Sequence Join
Let $\langle a_n \rangle$ and $\langle b_n \rangle$ be two finite sequences.

Then the join of $\langle a_n \rangle$ with $\langle b_n \rangle$ is defined as: $\langle \langle a_n \rangle ,\langle b_n \rangle \rangle$.

Sequence Strip
This is yet to be formalised. So we instead give an ostensive definition:

$(1)$ The strip of $\langle \langle 1, 2 \rangle, \langle 3, 4 \rangle \rangle$ is $\langle 1, 2, 3, 4 \rangle$.

$(2)$ The strip of $\langle \langle \langle 1, 2, 3 \rangle \rangle, \langle 4, 5 \rangle \rangle$ is $\langle \langle 1, 2, 3 \rangle , 4, 5 \rangle$.

$(3)$ As a special case the strip of $\langle 1, 2, 3 \rangle$ is $\langle 1, 2, 3 \rangle$.

$(4)$ The strip of $\langle 1, \langle \langle 2, 3 \rangle \rangle, \langle 4 \rangle \rangle$ is $\langle 1, \langle 2, 3 \rangle , 4 \rangle$.

Putting it together
The concatenation of $\langle 1, 2, 3 \rangle$ and $\langle 4, 5 \rangle$ then is the join of them:


 * $\langle \langle 1, 2, 3 \rangle, \langle 4, 5 \rangle \rangle$

followed by the strip of their join:


 * $\langle 1, 2, 3, 4, 5 \rangle$

Explore
Perhaps this distinction could be used to better explore the idea of ambiguity:

E.g. Given the alphabet $\langle \langle t \rangle, \langle tet \rangle , \langle tte \rangle , \langle tt \rangle \rangle$

And the word $\langle ttetettttett \rangle$

How many ways was it possible to construct this word from the alphabet given?

Perhaps the interplay of these two operations together with domains consisting of nested sequences like this:

$\langle \langle 1, \langle 2 \rangle \rangle, \langle \langle \langle 3, 4 \rangle \rangle \rangle \rangle$

could be quite an interesting area of research...