Editing Free monoid (section)

===Map and fold===
{{unreferenced section|date=February 2015}}
The computational embodiment of a monoid morphism is a [[Map (higher-order function)|map]] followed by a [[Fold (higher-order function)|fold]]. In this setting, the free monoid on a set ''A'' corresponds to [[List (computing)|lists]] of elements from ''A'' with concatenation as the binary operation.  A monoid homomorphism from the free monoid to any other monoid (''M'',•) is a function ''f'' such that
* ''f''(''x''<sub>1</sub>...''x''<sub>''n''</sub>) = ''f''(''x''<sub>1</sub>) • ... • ''f''(''x''<sub>''n''</sub>)
* ''f''() = ''e''
where ''e'' is the identity on ''M''.  Computationally, every such homomorphism corresponds to a [[Map (higher-order function)|map]] operation applying ''f'' to all the elements of a list, followed by a [[Fold (higher-order function)|fold]] operation which combines the results using the binary operator •. This [[squiggol|computational paradigm]] (which can be generalized to non-associative binary operators) has inspired the [[MapReduce]] software framework.{{citation needed|date=February 2015}}