The word-quotient model $F_{\mathrm{word}}(X):=W(X)/{\sim}$ with multiplication induced by concatenation

Let $W(X)$ be the set of all finite words on $X\sqcup X^{-1}$, including the empty word $\varepsilon$, and let $\sim$ be free equivalence as in Words in an alphabet with formal inverses, elementary cancellation, and reduced words. By Free equivalence is an equivalence relation and concatenation respects it, this is an equivalence relation and concatenation respects it.

Throughout, $a^{-1}$ denotes the partner of a formal letter $a\in X\sqcup X^{-1}$ under the pairing that matches each $x\in X$ with $x^{-1}\in X^{-1}$, so $(x^{-1})^{-1}=x$ and an elementary cancellation deletes an adjacent pair $aa^{-1}$ for any formal letter $a$.

The word-quotient model on $X$ is the quotient set

$F_{\mathrm{word}}(X):=W(X)/{\sim}.$

The class of a word $w$ is denoted $[w]$. Define

$[w][v]:=[wv],\qquad 1:=[\varepsilon],$

and define the generator map $i_{\mathrm{word}}:X\to F_{\mathrm{word}}(X)$ by $i_{\mathrm{word}}(x)=[x]$. The congruence property makes the displayed product independent of the representatives.