☯

old name of yin-yang symbol.

category theory

algebraic monoids are semigroups w/ identity, that is a set w/ associative binary operator * and

• associativity: (a * b) * c = a * (b * c)
• identity: existence of e so that e * a = a * e = a.

category theory knows abstract monoids and monads as further abstraction.

a monad on category C is an endofunctor T together with two natural transformations mu and eta and two conditions. in algebraic terms these conditions resemble associativity and identity with

• mu resembles binary operation
• eta resembles identity element.

functional programming: haskell

from http://blog.sigfpe.com/2006/08/you-could-have-invented-monads-and.html

about "extending functions".

• bind: compose extended functions, written as *
• unit: identity function regarding bind
• unit: f * unit = unit * f = f
• lift: extend "normal" function, lift f = unit . f
• lift is associative: lift f * lift g = lift(f . g)

examples:

• debuggable function w/ extra string as output
• complex square and cubic root, differently length output array
• random numbers in haskell

abstract:

• given function f : a -> m b which needs to be applied to m a instead of a
• define bind : (a -> m b) -> (m a -> m b), hence follows *
• define unit : a -> m a, hence follows lift
• unit is neutral regarding *
• lift is associative regarding *

a monad is a triple (m, unit, bind) following above laws.

usual definition

• bind is usually written as operator >>=, so bind f x equals x >>= f
• unit is usually called return
• handled by overloading type classes

in haskell debuggable example is writer monad, multivalued is list monad, randomised is state monad.