Hi Neil, You wrote:
[...] Is there a name for this property, which I'm numbering 1, (where (%) :: a -> b -> b; i.e. the operator is potentially, but not necessarily, asymmetrically typed):
1: a % (b % c) = b % (a % c)
I don't know any snappy names for this, but the following might help to reveal some structure. Pick some specific (but arbitrary) types: (%) :: A -> B -> B And some values: x, y :: A z :: B f, g :: B -> B f = (x%) g = (y%) Then: x % (y % z) == f (g z) == (f . g) z y % (x % z) == g (f z) == (g . f) z So (%) has property 1 iff the sub-monoid of Endo [1], which is generated by Endo (x%) forall x :: A, is commutative. Property 3 is the same, but with a larger generator set. Note that in your examples, the sub-monoid generated by insert+union is just the same as that generated by insert alone (assuming no infinite sets). This particular sub-monoid also happens to be a bounded join-semilattice (isomorphic to the finite subsets of A), which also makes it idempotent. Regards, Matthew [1]http://haskell.org/ghc/docs/latest/html/libraries/base/src/Data-Monoid.html#...