Defining monoids

[Charles Wells <charles@freude.com>]

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
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