Comments on A Neighborhood of Infinity: Monads from Algebra and the the Gray Code from Gro...

You lost me at "For example, suppose S = {x,y}. T...

2009-04-15T05:09:00.000-07:00

You lost me at

"For example, suppose S = {x,y}. Then we know that e, x and y are all in FS. We also know that x·x, x·y, y·x and y·y are all in S. Importantly we know that all 4 of these elements are distinct because we know there can be no equations connecting them beyond those that define a monoid."

Earlier, you wrote that "·" has the type ·:M×M→M. I understood this to mean that either x or y must be the "e" element, and x·y must be either x or y.

Saizan,

Dude! You're hired! Can you check a...

2007-07-03T19:55:00.000-07:00

Saizan,

Dude! You're hired! Can you check all my posts in future? :-)

I'll post a fixed version soon.

i mean that the second clause should be:
g (Rig...

2007-07-03T19:38:00.000-07:00

i mean that the second clause should be:
g (Right z) = let G u = f z
in reverse (map (either Right Left) u)
since, as you say, (xy)^-1=y^-1x^-1
also without this and using structural equality:
m >>= return /= m if m == G [Left a,Right b]

..or maybe I just didn't get this at all..

Saizan,

I notice there's an indentation pro...

2007-07-03T14:54:00.000-07:00

Saizan,

I notice there's an indentation problem (damn spaces keep disappearing on me) but I'm not sure what you're describing. If you mean what I think you mean then the code wouldn't type check, but it does.

You've forgotten to inverse the Either contructors...

2007-07-03T14:20:00.000-07:00

You've forgotten to inverse the Either contructors in the second clause of g when implementing (>>=), i think.

Of course, the "monad" in each case is the monad g...

2007-07-01T22:54:00.000-07:00

Of course, the "monad" in each case is the monad generated by the adjunction between the "free algebra generated by" functor and its forgetful converse.

Greg,

Pierce's "Basic Category Theory for C...

2007-07-01T21:52:00.000-07:00

Greg,

Pierce's "Basic Category Theory for Computer Scientists" is good on the category theory side. Talks about free monoids and adjoints, though not monads.

It appears that the first link (Saizan's comment) ...

2007-07-01T20:25:00.000-07:00

It appears that the first link (Saizan's comment) is broken. Also, can anyone recommend a good introductory "Abstract Algebra for Computer Scientists" text?

I'm being dense and confusing >>= and μ....

2007-07-01T16:14:00.000-07:00

I'm being dense and confusing >>= and μ. You're right. I meant concat.

Normally when I've seen mu it's been "join", not "...

2007-07-01T16:13:00.000-07:00

Normally when I've seen mu it's been "join", not ">>=". Indeed I can't see how to fit a functional argument into a F(FS) -> FS transformation. ?

stefanor,

Thanks. What I really meant was ...

2007-07-01T16:08:00.000-07:00

stefanor,

Thanks. What I really meant was concatMap.

I think you meant just 'concat' for mu. 'concat ....

2007-07-01T15:54:00.000-07:00

I think you meant just 'concat' for mu. 'concat . map' isn't even typable as an expression.

An interesting comment on lambda the ultimate about free monads.

It is probably not right to say this but I see free algebras as the static view and the monads as the dynamic view.

Oh, so that's what free things are. Abou...

2007-07-01T06:32:00.000-07:00

Oh, so that's what free things are. About a dozen things from my abstract algebra class some years ago just clicked. What a time for my copy of Mac Lane to be packed away...