Absolutely.
I do think that the moment you start caring about representations or presentations, as I believe emily has mentioned, some stronger assumptions for certain families of interfaces is worth considering to allow for cool stuff when those assumptions are admissible.
A simpler example of groups with no computable equality: the Sum group on computable reals. x + negate x === 0 for all x, but we can never fully prove it, only demonstrate it for each number of digits. And yet this is a perfectly well behaved abelian group.
On Sun, Dec 13, 2020 at 12:19:18AM -0500, Carter Schonwald wrote:
> Having a decidable equality seems important for reasoning about groups. Or
> computing on representations thereof.
This is of course not always possible. If a group is presented as a
quotient of a free group on a set of generators via some set of
relators, then deciding whether two words are equal is IIRC known to be
generally intractable. One can still perform the group operation, but
there is no known terminating algorithm for constructing a canonical
form for an element, performing equality tests, ...
Of course one might take the view that groups where equality is not
computable are not "useful", but that might rule out some applications.
--
Viktor.
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