(aka "Algebraic JavaScript Specification")

This project specifies interoperability of common algebraic structures:

• Semigroup
• Monoid
• Functor
• Applicative
• Chain
• Monad

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.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 specficiation.

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(function(a) { return a; })) is equivalent to u (identity)
2. u.map(function(x) { return 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 Applicative specification must also implement the Functor specification.

1. a.of(function(a) { return a; }).ap(v) is equivalent to v (identity)
2. a.of(function(f) { return function(g) { return function(x) { return f(g(x))}; }; }).ap(u).ap(v).ap(w) is equivalent to u.ap(v.ap(w)) (composition)
3. a.of(f).ap(a.of(x)) is equivalent to a.of(f(x)) (homomorphism)
4. u.ap(a.of(y)) is equivalent to a.of(function(f) { return f(y); }).ap(u) (interchange)

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

a.ap(b)

1. a must be an Applicative of a function,

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

2. b must be an Applicative of any value

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

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. m.chain(f).chain(g) is equivalent to m.chain(function(x) { return 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:

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