add Bruno's observations (#1074)

blueprint/src/chapter/677.tex

@@ -21,6 +21,7 @@ \chapter{Equation 677}\label{677-chapter}
   \item (ii) If $x,y \in M$ and $y \op x = x$, then $y = Sx \op x$.  In particular, 255 holds if and only if the equation $y \op x = x$ is solvable for every $x$.
   \item (iii)  For all $x,y \in M$, we have $x = L_y x \op R_y L_y^2 x$.
   \item (iv) For $a,b \in M$, we have $a \diamond b = L_b^{-2} a \op ((L_b^{-1} a \op L_b^{-2} a) \op L_b^{-1} a)$.
+  \item (v) Let $x \in M$.  If $y$ is a fixed point of $L_x S$, then it is a fixed point of $R_x L_x$, which in turn implies that $L_x y$ is a fixed point of $L_x R_x$.  Finally, if $z$ is a fixed point of $L_x R_x$, then $R_x L_x z = x$ (and hence $L_x z = Sx \op x$ by (ii)).
 \end{itemize}
 \end{lemma}
 
@@ -33,6 +34,8 @@ \chapter{Equation 677}\label{677-chapter}
   and the claim then follows by left invertibility.
 
   For (iii), we apply (i) with $x$ replaced by $L_y x$.  For (iv), we rewrite $a \op b$ as $R_b L_b^2 (L_b^{-2} a)$, which by (iii) is equal to $L_{L_b^{-1} a}^{-1} (L_b^{-2} a)$.  The claim then follows from (i).
+
+  Now we prove (v).  If $y$ is a fixed point of $L_x S$, then $L_x L_y y = y$, but from \eqref{677-alt} we have $y = L_x L_y R_x L_x y$, so the first claim follows from left-cancellativity.  The second claim is also clear from left-cancellativity.  Finally, if $z$ is a fixed point of $L_x R_x$, then we have $L_x L_z x = z$, but from \eqref{667-alt} one has $z = L_x L_z R_x L_x z$, giving the claim by right cancellativity.
 \end{proof}
 
 This for instance gives the implication for linear magmas: