Inverse limit of algebras

Summary

The inverse limit of an inverse system of algebras is the algebra of all compatible families of elements. Such a family has one element in each algebra of the system. It is a subalgebra of the direct product of the algebras. It is the limit of the system in the category of algebras of the given signature. Inverse limits build large algebras, such as the $p$-adic integers and profinite groups, out of a coherent family of smaller ones.