User blog comment:P進大好きbot/Please Help me on study of Pair Sequence System (2-rowed Bashicu Matrix System)/@comment-35870936-20180813215517/@comment-35870936-20180814000520

@P進大好きbot

You said this in your comment:

\begin{eqnarray*} & & \psi_0(\psi_1(\psi_2(0))) \leq \psi_0(\psi_1(\psi_2(\psi_3(0)))) \leq \psi_0(\psi_1(\psi_3(0))) \\ & \leq & \psi_0(\psi_2(0)) = \psi_0(\psi_1(\psi_2(0))) \end{eqnarray*}

However, this isn't correct. We have: \(\psi_0(\psi_1(\psi_2(0))) = \psi_0(\psi_2(0)) \leq \psi_0(\psi_1(\psi_2(\psi_3(0)))) = \psi_0(\psi_1(\psi_3(0))) = \psi_0(\psi_3(0))\) There is nothing special about the \(\psi_3\) function.

We also have \begin{eqnarray*} \psi_0(\psi_1(0)) & = & \psi_0(\psi_1(0)) \\ \psi_0(\psi_1(\psi_2(0))) & = & \psi_0(\psi_2(0)) \\ \psi_0(\psi_1(\psi_2(\psi_3(0)))) & = & \psi_0(\psi_3(0)) \\ \psi_0(\psi_1(\psi_2(\psi_3(\psi_4(0))))) & = & \psi_0(\psi_4(0)). \end{eqnarray*}

Also, note that in UNOCF, \(\psi_x(0)\) just equals \(\Omega_x\).