Free Monoids and Computing
The free monoid on a set A corresponds to 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(x1…xn) = f(x1) • … • f(xn)
- f = e
where e is the identity on M. Computationally, every such homomorphism corresponds to a map operation applying f to all the elements of a list, followed by a fold operation which combines the results using the binary operator •. This computational paradigm (which can be generalised to non-associative binary operators) has inspired the MapReduce software framework.
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