Categorically, while different algebraic structures may not be isomorphic, they're likely all capable (under decoration) of carrying the same information.
For instance, in the Boom Hierarchy* (sets,bags,lists,trees) we can represent the information carried by any of the structures by any of the others (exercise!) despite the fact that operations on that information may be made easier or more difficult by the presence or lack of laws (idempotency,commutativity,associativity) that are the linear delta-edges between those types-as-vertices themselves.
[IIRC, magmas would be mobiles, and I'd bet we can extend the above to them as well.]
* I know I repeat these often, but hey, I'm not alone: gopher-the-language was inspired by a "go" lemma-function in someone's sol'n to samefringe.
How much work is "likely" doing..
I'd been at peace if you mentioned the triples, but it's too late now..
"likely" is not doing much work for the domain I was thinking of: structures which are suitable for making into collection data structures, in that their carrier sets are infinite because their ops don't lose information about the values (the hyle if not the morphe) given by their arguments.
Of course, absent those restrictions, it's doing way more work than it can possibly handle.
wasn't it our own Carl Sagan who suggested looking into \pi for alien messages (but/and he was perhaps under the influence)
But not the references inside apple boxen
Could be, although I could also imagine the BHA was intending that as a smoke test: if we were to find a putative alien message, we'd want to make sure the same procedure doesn't find other messages in the first N digits of \pi or related constants.
(shouldn't all messages be in \pi sooner or [much, much...] later?)
[does the def'n of "message" include finite length?]
I'd imagine, though, the KST-patched protocol would be to look at how the certain later digits of \pi vary "in realtime" based on where certain earlier digits (of \hbar) tell us to look.. that'd indicate sophisticated control of remote curvature :)
>Sagan's friend physicist Kip Thorne gave Sagan ideas on the nature of wormholes when Sagan was developing the outline of the novel.