On Fri, Nov 13, 2009 at 08:37:57PM +0300, Eugene Kirpichov wrote:
For every monoid (M, *, u), the dual to it is the monoid (Dual M, \x y -> y * x, u)
The entirely standard name for this is the opposite monoid. The only places I have seen the name "dual monoid" used to mean opposite monoid are in Data.Monoid and subsequently by some Haskellers. A very good reason not to use the name "dual monoid" is that the basic point of duality is contravariance. The prototypical example of duality is the dual V^* of a finite-dimensional vector space V, defined as the vector space of all linear maps from V to the ground field. If f : V -> W is a linear map, then we get an induced linear map f^* : W^* -> V^* by precomposing with f. Note that f^* "goes in the other direction"; the functor -^* is contravariant. Perhaps a more familiar example is from classical Boolean algebra. If P is a proposition, it's sensible to call not-P the dual of P, since an implication P -> Q yields an implication not-Q -> not-P. (This is closely related to the previous example, since not-P is the statement P -> false.) The notion of opposite monoids is not an example of duality. Given a monoid homomorphism f : M -> N, there is an induced opposite homomorphism f^op : M^op -> N^op, which is the same as f on elements. It goes the same direction as f; -^op is a covariant functor. If you're not convinced by these arguments, try googling for "opposite monoid" and "dual monoid" to see the standard usage for yourself. There is no standard meaning for the phrase "dual monoid", but I would venture that it is never used to mean "opposite monoid" in the mathematical literature. (Sorry for the rant.) Regards, Reid Barton