Direct limit of algebras

Summary

The direct limit of a directed system of algebras is the algebra obtained by gluing the algebras of the system along its homomorphisms. It is the direct limit in the category of algebras of a fixed signature, and it is the union when the maps are inclusions. Varieties are closed under direct limits, and every algebra is the direct limit of its finitely generated subalgebras. The dual construction is the inverse limit of algebras.