Amalgamated free product

Summary

An amalgamated free product glues several algebras of a variety along a common algebra that maps into each of them. It is the most general algebra of the variety that receives maps from all the pieces. In it, the images of the common part are identified. In category terms it is the colimit of the diagram formed by the maps out of the common algebra. For two pieces it is the pushout. For groups, amalgamated free products are central in combinatorial group theory and in the Seifert–van Kampen theorem.