Name for $X^\infty=\bigcup\limits_{k=0}^\infty X^k$ I'm making structures associated with groups, rings and so on in OCaml and in order to do so I started by defining sets and a few operations (intersection, union, difference, carthesian product, carthesian power) and now I want to define a set $X^\infty=\bigcup\limits_{k=0}^\infty X^k\subsetneq X^\Bbb N$ which I would represent with arrays but I couldn't find a proper name for it. Is there one?

This is the free monoid on the set $X$. A monoid is a set together with a binary operation which is associative and has a unit. The monoid structure on the set you call $X^{\infty }$ is concatenation and the unit is the unique $0$-tuple (the only element in the set $X^0$). The more common notation for the free monoid is $X^*$. 
