On the third page of the paper, he writes that
If $\alpha\beta\sim\gamma\delta$ and the words $\alpha$ and $\gamma$ have the same length then $$ \alpha=\mu m,\quad \beta=n\nu,\quad \gamma\sim\mu m',\quad\delta\sim n'\nu,\quad mn\sim m'n' $$ where $m,n,m',n'$ are each one of the letters $a,b,c,d,x,y,u,v$.
How is this property easily seen? Malcev starts be starting with the semigroup of all words generated by eight letters $a,b,c,d,x,y,u,v$, where the operation is concatenation. He defines the pairs of two-letter words as "corresponding" $(ax,by)$, $(cx,dy)$ and $(au,bv)$ and says two words $\alpha$ and $\beta$ are equivalent if one can be obtained from the other by changing $ax$ to $by$, $cx$ to $dy$, etc. or vice versa.
He also proves that there are never any overlap ambiguities of what two letters could be replaced, but I don't see why the above quoted property follows so easily. Thank you.