...

true = return () (&&) = liftM2 const (||) = mplus false = mzero

I don't think it's by definition yet, but surely by convention, that this is a restricted form of

true = pure () (&&) = (<*) (||) = (<|>) false = empty

All of these functions require only an Applicative or an Alternative, and except for "true", they're all in the library.

True, and after sending I realized I should have used exaclty the definitions you gave. It's just that I don't know whether there exists the concept of "commutative applicative". It is equired for (&&) to be commutative. How could one even state it without the monad bind? The following definitions are from a category theory paper [1] by Anders Kock. import Control.Monad -- tensorial strengths t'' :: (Functor m) => a -> m b -> m (a,b) t'' a mb = fmap (\b -> (a,b)) mb t' :: (Functor m) => m a -> b -> m (a,b) t' ma b = fmap (\a -> (a,b)) ma -- so-called linear extensions in the first and second argument linearext1 :: Functor m => (m c -> c) -> (a -> b -> c) -> m a -> b -> c linearext1 struct f ma b = struct $ (fmap $ uncurry f) $ t' ma b linearext2 :: Functor m => (m c -> c) -> (a -> b -> c) -> a -> m b -> c linearext2 struct f a mb = struct $ (fmap $ uncurry f) $ t'' a mb The first argument to linearext1 and linearext2 is called a "structure map" in the context of [1]. Such structure maps exist when m is a monad. Indeed, we have linearext1 join t'' :: (Monad m) => m a -> m b -> m (a, b) linearext2 join t' :: (Monad m) => m a -> m b -> m (a, b) Now m is called commutative if the above two maps are the same [*]. But join does not exist for applicative functors, it is interdefinable with (>>=). Olaf [1] Anders Kock: "Commutative Monads as a Theory of Distributions". Theory and Applications of Categories, Vol. 26 No. 4 (2012). [*] They are called Fubini maps because a special case is Fubini's theorem about integration by parts.