John Lato wrote:
Here is the current complete documentation for Data.Monoid 'mappend', which happens to be a good example of which I speak:
An associative operation
That could mean anything. There are lots of associative operations.
Yes. In combination with the definition of mempty (the identity for mappend) that's exactly what a monoid is. I'm not (just) being flippant. This particular example actually highlights the limitations of type classes. There are, in fact, too many monoids. Too many to be able to define it based only on the carrier, i.e. the type that mempty belongs to and that mappend operates over. For example every (semi-)ring has at least two of them[1]. For more complicated mathematical structures there can be even more than that. This isn't to say that we should get rid of Monoid, just that using it can be... troublesome. But this has been mentioned already on a few other recent threads. The notion of appending only really applies to "the free monoid on a set A", aka strings of elements drawn from A. It bears highlighting that I said strings (in the mathematical sense) and not (linked) lists. The concatenation involved is mathematical juxtaposition, and the empty element is the empty string (which is subtly different than the empty list which is also used to terminate the datatype's recursion). These monoids are called free because they make no requirements on the underlying set, no requirements other than the monoid laws. All other monoids use some sort of structure in the underlying set in order to simplify expressions (e.g. 1+1 == 2, whereas a+a == aa). Because the free monoid doesn't simplify anything, all it does is append. [1] For example, Natural numbers: (Nat,+,0) (Nat,*,1) Boolean algebra: (Bool,or,False) (Bool,and,True) Bit-vector algebra: (Bits,bitOr,0) (Bits,bitAnd,-1) Any lattice L: (L,join,Bottom) (L,meet,Top) Any total ordering O: (O,max,NegativeInfinity) (O,min,Infinity) Any set universe U: (Power(U),union,EmptySet) (Power(U),intersection,U) ...let alone getting into fun things like the log-semiring, polynomials with natural coefficients, and so on. -- Live well, ~wren