We define a [formal language][1] as follows:

The free language $\mathcal J$ of all strings $J$ (words) in

alphabet : $R, L$

including the empty string, [].

This language comes naturally equipped with the concatenation operation,

$\tag 1 +: (J,J^{'}) \mapsto JJ^{'}$.

A string $M \text{ in } \mathcal J$ is said to be direct (or pure) if it does not contain both of the letters $R$ and $L$.

The strings $RL$ and $LR$ in $\mathcal J$ are called vis-à-vis steps.

If a vis-à-vis step is inserted anywhere into a string $J$, then the new output string is said to be an expansion of $J$.

If a vis-à-vis step can be removed from a string $J$, then the new output string is said to be a contraction of $J$.

Any string can be contracted to a direct string and can itself be obtained by expanding that same direct string. Two strings $J$ and $J^{'}$ are said to be homoJnic if $J$ can be be transformed into $J^{'}$ using expansions and contractions. When this is the case we can write $J == J^{'}$.

The integers $\mathbb Z$ are defined as the formal language of direct strings in $\mathcal J$. Concatenation is a binary operation on the integers.

Example: $RRRRR + LL = RRRRRLL == RRRRL == RRR$ [1]: https://en.wikipedia.org/wiki/Formal_language

We define a [formal language][1] as follows:

The free language $\mathcal J$ of all strings $J$ (words) in

alphabet : $R, L$

including the empty string, [].

This language comes naturally equipped with the concatenation operation,

$\tag 1 +: (J,J^{'}) \mapsto JJ^{'}$.

A string $M \text{ in } \mathcal J$ is said to be direct (or pure) if it does not contain both of the letters $R$ and $L$.

The strings $RL$ and $LR$ in $\mathcal J$ are called vis-à-vis steps.

If a vis-à-vis step is inserted anywhere into a string $J$, then the new output string is said to be an expansion of $J$.

If a vis-à-vis step can be removed from a string $J$, then the new output string is said to be a contraction of $J$.

Any string can be contracted to a direct string and can itself be obtained by expanding that same direct string. Two strings $J$ and $J^{'}$ are said to be homoJnic if $J$ can be be transformed into $J^{'}$ using expansions and contractions. When this is the case we can write $J == J^{'}$.

If $J_\alpha == J_\beta$ and ${J_\alpha}^{'} == {J_\beta}^{'}$ then $J_\alpha {J_\alpha}^{'} == J_\beta {J_\beta}^{'}$.

The addition of integers $\mathbb Z$ can therefore be defined as concatenation that is invariant under homoJnic transformations, each class being represented by a direct strings in $\mathcal J$.

Example: Adding two direct $RRRRR + LL = RRRRRLL == RRRRL == RRR$

Example: Since $RRRRR == RRLRRRR$ and $LL == RLLL$, we have
$\qquad RRLRRRR + RLLL = RRLRRRRRLLL == RRR$