| I've been experimenting with answering that question. An extremely heavily-commented approach to defining equality for things like Double (where there are multiple sensible ways to compare things) will probably be a better introduction to these ideas: https://github.com/mrkgnao/noether/blob/master/library/Noeth... https://github.com/mrkgnao/noether/blob/master/library/Noeth... The rest of the repo implements abstract algebraic structures along similar lines. Here is what monoids look like in this formalism at present. It's a bit involved: first I use a fine-grained separation into highly polymorphic Magma and Neutral classes: https://github.com/mrkgnao/noether/blob/master/library/Noeth... And there are "strategies" for combining those pieces of structure, whose use looks like type instance MonoidS (op :: BinaryNumeric) Rational
= DeriveMonoid_Semigroup_Neutral op Rational
Which means "derive the preferred monoid structure with operation op" (MonoidS op) on Rational from the preferred semigroup structure and preferred neutral element on it with that operation.https://github.com/mrkgnao/noether/blob/master/library/Noeth... You can always use some other monoid structure for the same combination of type and operation (!) if you like. For instance, here's a demo of using the self-action of a ring on itself (i.e. considering a ring as a module over itself) to compute something, even if your preferred choice of action has been declared to be something else (or hasn't been defined at all). https://github.com/mrkgnao/noether/blob/master/library/Noeth... This says "here, use the self-action derived from the multiplicative magma structure on R". |