Defining monoids

Defining monoids

[Charles Wells]

In talking about defining monoids, Toby Bartels wrote:

"We could also go straight to trees and define them as the basic operations,
then requiring as axiom that grafting of trees produces the same result
as composing the operations."

This is the mu operation of the corresponding monad.  Every single-sorted 
"idea" in the sense of the recent discussion generates a monad in sets with 
a mu like this.  For each set S there is a set of possible computations TS, 
a mu:TTS to TS, and a "OneIdentity" operation in the sense of Mathematica 
that says the computation consisting of a single node results in that node; 
these subject to the monad laws.  In other words, the phenomenon you noted 
is an instance of a general result.

--Charles Wells

Charles Wells,
Emeritus Professor of Mathematics, Case Western Reserve University
Affiliate Scholar, Oberlin College
Send all mail to:
105 South Cedar St., Oberlin, Ohio 44074, USA.
email: charles@freude.com.
home phone: 440 774 1926.
professional website: http://www.cwru.edu/artsci/math/wells/home.html
personal website: http://www.oberlin.net/~cwells/index.html
genealogical website: 
http://familytreemaker.genealogy.com/users/w/e/l/Charles-Wells/
NE Ohio Sacred Harp website: http://www.oberlin.net/~cwells/sh.htm

Re: Defining monoids

[JAMES STASHEFF]

For those who prefer to see the forests and the trees,
that point of view is prominent in the operad/modad/monoidal interaction.
Much such material will be in our book
on Operads (Markl and Shnider and me)

.oooO   Jim Stasheff		jds@math.unc.edu
(UNC)   Math-UNC		(919)-962-9607
 \ (    Chapel Hill NC		FAX:(919)-962-2568
  \*)   27599-3250

        http://www.math.unc.edu/Faculty/jds

On Thu, 6 Dec 2001, Charles Wells wrote:

> 
> 
> In talking about defining monoids, Toby Bartels wrote:
> 
> "We could also go straight to trees and define them as the basic operations,
> then requiring as axiom that grafting of trees produces the same result
> as composing the operations."
> 
> This is the mu operation of the corresponding monad.  Every single-sorted 
> "idea" in the sense of the recent discussion generates a monad in sets with 
> a mu like this.  For each set S there is a set of possible computations TS, 
> a mu:TTS to TS, and a "OneIdentity" operation in the sense of Mathematica 
> that says the computation consisting of a single node results in that node; 
> these subject to the monad laws.  In other words, the phenomenon you noted 
> is an instance of a general result.
> 
> --Charles Wells
> 
> Charles Wells,
> Emeritus Professor of Mathematics, Case Western Reserve University
> Affiliate Scholar, Oberlin College
> Send all mail to:
> 105 South Cedar St., Oberlin, Ohio 44074, USA.
> email: charles@freude.com.
> home phone: 440 774 1926.
> professional website: http://www.cwru.edu/artsci/math/wells/home.html
> personal website: http://www.oberlin.net/~cwells/index.html
> genealogical website: 
> http://familytreemaker.genealogy.com/users/w/e/l/Charles-Wells/
> NE Ohio Sacred Harp website: http://www.oberlin.net/~cwells/sh.htm
> 
> 
> 
> 
>