[Gehinnn]

A nice property of monoids is associativity, which allows for some interesting incremental algorithms, e.g. by using balanced trees: If the fold of × on a list is computed in clever way, the fold of × on a list where one element is modified can be computed in logarithmic time, given the computation of the first list.

A good example for a monoid are strings with the string concatenation operation! This is used a lot in text editors. Homomorphisms are also of practical relevance here.

If × is a commutative group operation (i.e. inverse elements exist and a×b=b×a), that can even be done in constant time in a trivial way.