(aka "Algebraic JavaScript Specification")

This project specifies interoperability of common algebraic structures:

• Setoid
• Semigroup
• Monoid
• Functor
• Apply
• Applicative
• Foldable
• Traversable
• Chain
• Monad
• Extend
• Comonad

An algebra is a set of values, a set of operators that it is closed under and some laws it must obey.

Each Fantasy Land algebra is a separate specification. An algebra may have dependencies on other algebras which must be implemented. An algebra may also state other algebra methods which do not need to be implemented and how they can be derived from new methods.

1. "value" is any JavaScript value, including any which have the structures defined below.
2. "equivalent" is an appropriate definition of equivalence for the given value. The definition should ensure that the two values can be safely swapped out in a program that respects abstractions. For example:
   • Two lists are equivalent if they are equivalent at all indices.
   • Two plain old JavaScript objects, interpreted as dictionaries, are equivalent when they are equivalent for all keys.
   • Two promises are equivalent when they yield equivalent values.
   • Two functions are equivalent if they yield equivalent outputs for equivalent inputs.

1. a.equals(a) === true (reflexivity)
2. a.equals(b) === b.equals(a) (symmetry)
3. If a.equals(b) and b.equals(c), then a.equals(c) (transitivity)

A value which has a Setoid must provide an equals method. The equals method takes one argument:

a.equals(b)

1. b must be a value of the same Setoid

   1. If b is not the same Setoid, behaviour of equals is unspecified (returning false is recommended).

2. equals must return a boolean (true or false).

1. a.concat(b).concat(c) is equivalent to a.concat(b.concat(c)) (associativity)

A value which has a Semigroup must provide a concat method. The concat method takes one argument:

s.concat(b)

1. b must be a value of the same Semigroup

   1. If b is not the same semigroup, behaviour of concat is unspecified.

2. concat must return a value of the same Semigroup.

A value that implements the Monoid specification must also implement the Semigroup specification.

1. m.concat(m.empty()) is equivalent to m (right identity)
2. m.empty().concat(m) is equivalent to m (left identity)

A value which has a Monoid must provide an empty method on itself or its constructor object. The empty method takes no arguments:

m.empty()
m.constructor.empty()

1. empty must return a value of the same Monoid

1. u.map(a => a) is equivalent to u (identity)
2. u.map(x => f(g(x))) is equivalent to u.map(g).map(f) (composition)

A value which has a Functor must provide a map method. The map method takes one argument:

u.map(f)

1. f must be a function,

   1. If f is not a function, the behaviour of map is unspecified.
   2. f can return any value.

2. map must return a value of the same Functor

A value that implements the Apply specification must also implement the Functor specification.

1. a.map(f => g => x => f(g(x))).ap(u).ap(v) is equivalent to a.ap(u.ap(v)) (composition)

A value which has an Apply must provide an ap method. The ap method takes one argument:

a.ap(b)

1. a must be an Apply of a function,

   1. If a does not represent a function, the behaviour of ap is unspecified.

2. b must be an Apply of any value

3. ap must apply the function in Apply a to the value in Apply b

A value that implements the Applicative specification must also implement the Apply specification.

A value which satisfies the specification of an Applicative does not need to implement:

• Functor's map; derivable as function(f) { return this.of(f).ap(this); }

1. a.of(x => x).ap(v) is equivalent to v (identity)
2. a.of(f).ap(a.of(x)) is equivalent to a.of(f(x)) (homomorphism)
3. u.ap(a.of(y)) is equivalent to a.of(f => f(y)).ap(u) (interchange)

A value which has an Applicative must provide an of method on itself or its constructor object. The of method takes one argument:

a.of(b)
a.constructor.of(b)

1. of must provide a value of the same Applicative

   1. No parts of b should be checked

1. u.reduce is equivalent to u.toArray().reduce

• toArray; derivable as function() { return this.reduce((acc, x) => acc.concat(x), []); }

A value which has a Foldable must provide a reduce method. The reduce method takes two arguments:

u.reduce(f, x)

1. f must be a binary function

   1. if f is not a function, the behaviour of reduce is unspecified.
   2. The first argument to f must be the same type as x.
   3. f must return a value of the same type as x

2. x is the initial accumulator value for the reduction

A value that implements the Traversable specification must also implement the Functor specification.

1. t(u.sequence(f.of)) is equivalent to u.map(t).sequence(g.of) where t is a natural transformation from f to g (naturality)

2. u.map(x => Id(x)).sequence(Id.of) is equivalent to Id.of (identity)

3. u.map(Compose).sequence(Compose.of) is equivalent to Compose(u.sequence(f.of).map(x => x.sequence(g.of))) (composition)

• traverse; derivable as function(f, of) { return this.map(f).sequence(of); }

A value which has a Traversable must provide a sequence method. The sequence method takes one argument:

u.sequence(of) 

1. of must return the Applicative that u contains.

A value that implements the Chain specification must also implement the Apply specification.

A value which satisfies the specification of a Chain does not need to implement:

• Apply's ap; derivable as function ap(m) { return this.chain(f => m.map(f)); }

1. m.chain(f).chain(g) is equivalent to m.chain(x => f(x).chain(g)) (associativity)

A value which has a Chain must provide a chain method. The chain method takes one argument:

m.chain(f)

1. f must be a function which returns a value

   1. If f is not a function, the behaviour of chain is unspecified.
   2. f must return a value of the same Chain

2. chain must return a value of the same Chain

A value that implements the Monad specification must also implement the Applicative and Chain specifications.

A value which satisfies the specification of a Monad does not need to implement:

• Apply's ap; derivable as function(m) { return this.chain(f => m.map(f)); }
• Functor's map; derivable as function(f) { var m = this; return m.chain(a => m.of(f(a)))}

1. m.of(a).chain(f) is equivalent to f(a) (left identity)
2. m.chain(m.of) is equivalent to m (right identity)

1. w.extend(g).extend(f) is equivalent to w.extend(_w => f(_w.extend(g)))

An Extend must provide an extend method. The extend method takes one argument:

 w.extend(f)

1. f must be a function which returns a value

   1. If f is not a function, the behaviour of extend is unspecified.
   2. f must return a value of type v, for some variable v contained in w.

2. extend must return a value of the same Extend.

A value that implements the Comonad specification must also implement the Functor and Extend specifications.

1. w.extend(_w => _w.extract()) is equivalent to w
2. w.extend(f).extract() is equivalent to f(w)
3. w.extend(f) is equivalent to w.extend(x => x).map(f)

A value which has a Comonad must provide an extract method on itself. The extract method takes no arguments:

c.extract()

1. extract must return a value of type v, for some variable v contained in w.
   1. v must have the same type that f returns in extend.

1. If there's more than a single way to implement the methods and laws, the implementation should choose one and provide wrappers for other uses.
2. It's discouraged to overload the specified methods. It can easily result in broken and buggy behaviour.
3. It is recommended to throw an exception on unspecified behaviour.
4. An Id container which implements all methods is provided in id.js.