@DanielDonnelly sorry for this late answer: $+_K$ is Not associative, but it does have inverses: arxiv.org/abs/2401.08635#

+1 Very nice theorem and presentation!

I will accept Martin‘s answer while looking forward to Ville‘s answer

Thanks @VilleSalo, an answer would be very much appreciated!

Is ${\tt AND}$ , $\land$, a two-to-one cellular automaton? If yes, the argument would not work. If we take $s = {\tt 01\, 001\, 0001 \, 00001}\ldots$ then $s \land \ell_1(s)$ is the constant 0 sequence. A similar argument can be made for ${\tt OR}$, $\vee$. Maybe you can elaborate your argument in an answer - many thanks.

Right - will correct