From 327b224de9137a516ca272513b70327b97a89af0 Mon Sep 17 00:00:00 2001 From: teorth Date: Sun, 2 Feb 2025 15:08:29 -0800 Subject: [PATCH] add Bruno's observations (#1074) --- blueprint/src/chapter/677.tex | 3 +++ 1 file changed, 3 insertions(+) diff --git a/blueprint/src/chapter/677.tex b/blueprint/src/chapter/677.tex index 202d76534..31331f411 100644 --- a/blueprint/src/chapter/677.tex +++ b/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: