Infinitary operation

Summary

An infinitary operation on a set $A$ takes an infinite family of elements of $A$ and returns an element of $A$. Its arity is an infinite cardinal or ordinal, in contrast to a finitary operation. The supremum of an arbitrary subset of a complete lattice and countable unions in a $\sigma$-algebra of sets are typical infinitary operations. Algebras with infinitary operations need care, because some results of finitary universal algebra fail for them.