IdentityEither

IdentityEither[F] describes an associative way to combine two values F[A] and F[B] into a value F[Either[A, B]] with an identity value none of type F[Nothing].

Its signature is:

trait AssociativeEither[F[_]] {
  def either[A, B](fa: => F[A], fb: => F[B]): F[Either[A, B]]
}

trait IdentityEither[F[_]] extends AssociativeEither[F] {
  def none: F[Nothing]
}

The none value must be an identity element with respect to the either operator so that, after eliminating cases of Either that can never contain values, the following property holds:

fa <+> none === fa
none <+> fa === fa

This is the same as the law for the Identity abstraction for concrete types but lifted into the context of parameterized types.

To satisfy these properties, rhe none value must must be a value that fails with no useful information.

To see what this means, let's look at the IdentityEither instance for Option.

import zio.prelude._

implicit val OptionIdentityEither: IdentityEither[Option] =
  new IdentityEither[Option] {
    def either[A, B](fa: => Option[A], fb: => Option[B]): Option[Either[A, B]] =
      fa match {
        case None =>
          fb match {
            case None    => None
            case Some(b) => Some(Right(b))
          }
        case Some(a) => Some(Left(a))
      }
    val none: Option[Nothing] =
      None
  }
// OptionIdentityEither: IdentityEither[Option] = repl.MdocSession$MdocApp0$$anon$1@3b6462e9

If we think of an Option[A] as representing either a success with a value of type A or a failure with no useful information, then the implementation of none is None. Combining any Option[A] with None with orElseEither won't change its result.

Now let's compare that to Either to see why we can't define a IdentityEither instance for Either.

We said that the identity value must be a value that fails with no useful information but Either is polymorphic in its error type so we can't define what a value that contains no information would be for an arbitrary type. We don't even know how to construct an value of that type.

def none[E]: Either[E, Nothing] =
  Left(???)

To define an identity value we would have to have the concept of an empty error. For example, if we created a data type that could fail without any error then the empty failure would be the none value.

import zio.Chunk

val noneOption: Either[Option[Nothing], Nothing] =
  Left(None)
// noneOption: Either[Option[Nothing], Nothing] = Left(value = None)

val noneChunk: Either[Chunk[Nothing], Nothing] =
  Left(Chunk.empty)
// noneChunk: Either[Chunk[Nothing], Nothing] = Left(value = IndexedSeq())

Having this none value can be useful in certain situations. For example, it can help us to define a filter operator on a data type in combination with some of the other functional abstractions in ZIO Prelude.

However, supporting a none value requires allowing the possibility of that a value can fail without containing any errors, which is unattractive in many cases.

For example, ZIO and most effect types such as ZManaged and ZStream do not have an IdentityEither instance defined for them because is they fail they always fail with a failure E. Otherwise when we used an error handling operator we would have to deal with the error as well as the possibility that there is a failure but no error at all.

Note that in the case of ZIO a workflow can also die with a cause that can potentially be Cause.empty, but since the associative operator orElseEither does not recover from Die failures that doesn't help us with our problem here.

The other data types that do tend to have IdentityEither instances are collection types that can be empty. For example, the IdentityEither instance for Chunk looks like this.

implicit val ChunkIdentityEither: IdentityEither[Chunk] =
  new IdentityEither[Chunk] {
    def either[A, B](as: => Chunk[A], bs: => Chunk[B]): Chunk[Either[A, B]] =
      as.map(Left(_)) ++ bs.map(Right(_))
    val none: Chunk[Nothing] =
      Chunk.empty
  }
// ChunkIdentityEither: IdentityEither[Chunk] = repl.MdocSession$MdocApp0$$anon$2@2fef95b

Here the identity element is just the empty chunk. If we think of running a collection as producing a set of values and then failing when the collection is empty, none is the collection that doesn't produce any values and fails immediately.

Another more mathematical way to think about it is that if concatenation corresponds to addition then the empty collection corresponds to zero. or this reason, we can also see that collections that cannot be empty, such as NonEmptyChunk from ZIO or NonEmptyList from ZIO Prelude, cannot have IdentityEither instances defined for them.

We can also define none values for contravariant types, though these tend to be less interesting.

For contravariant types, the none value corresponds to a value that can never be run because it has Nothing as its input type and there are no values of type Nothing. There the existence of the none value and our ability to compose it with the orElseEither operator in implies that values can be run independently of each other.

For example, let's look at the IdentityEither instance for Predicate.

trait Predicate[-A] {
  def run(a: A): Boolean
}

object Predicate {
  implicit val PredicateIdentityEither: IdentityEither[Predicate] =
    new IdentityEither[Predicate] {
      def either[A, B](left: => Predicate[A], right: => Predicate[B]): Predicate[Either[A, B]] =
        new Predicate[Either[A, B]] {
          def run(either: Either[A, B]): Boolean =
            either match {
              case Left(a) => left.run(a)
              case Right(b) => right.run(b)
            }
        }
      val none: Predicate[Nothing] =
        new Predicate[Nothing] {
          def run(a: Nothing): Boolean =
            a
    }
  }
}

The none predicate can never be run, but we can still safely compose it with other predicates with the either operator because since there can never be a value of type Nothing, there can also never be a case where the either operator tries to run the none predicate.

Overall the IdentityEither abstraction is a useful one to generalize over the concept of a value of a parameterized type that always fails with no information, complementing the concept of a value that always succeeds with no information from IdentityBoth.

For concrete types that have such a none value there tend to be existing operators defined on the data type that already take advantage of this this functionality, so IdentityEither tends to be less helpful when working with concrete data types.

For defining your own data types thinking about whether there is a none value can be helpful for API design. As discussed above, there is often a trade off between having an identity element, which is generally a good thing, and providing the guarantee that all failures contain some useful information.

Defining an IdentityEither instance for your own data type will also let you work with generic code in ZIO Prelude that uses this abstraction.

Finally IdentityEither can be useful when you are writing generic code to describe the concept that a data type must have this failure element with no information to provide a "default" failure value.