In mathematics, a shuffle algebra is a Hopf algebra with a basis corresponding to words on some set, whose product is given by the shuffle product X ⧢ Y of two words X, Y: the sum of all ways of interlacing them.
The shuffle product occurs in generic settings in non-commutative algebras; this is because it is able to preserve the relative order of factors being multiplied together - the riffle shuffle permutation.
This can be held in contrast to the divided power structure, which becomes appropriate when factors are commutative.
The closely related infiltration product was introduced by Chen, Fox & Lyndon (1958).
It is defined inductively on words over an alphabet A by For example: The infiltration product is also commutative and associative.