Free product

in a class $\mathfrak K$ of universal algebras $A_\alpha$, $\alpha\in\Omega$, from $\mathfrak K$

An algebra $A$ from $\mathfrak K$ that contains all the $A_\alpha$ as subalgebras and is such that any family of homomorphisms of the $A_\alpha$ into any other algebra $B$ from $\mathfrak K$ can be uniquely extended to a homomorphism of $A$ into $B$. A free product automatically exists if $\mathfrak K$ is a variety of universal algebras. Every free algebra is the free product of free algebras generated by a singleton. The free product in the class of Abelian groups coincides with the direct sum. In certain cases there is a description of the subalgebras of a free product, for example, in groups (see Free product of groups), in non-associative algebras, and in Lie algebras.

A free product in categories of universal algebras coincides with the coproduct in these categories.

Comments

Free products do not always exists in a variety of algebras: for example, the ring of integers modulo $2$ and the ring of integers modulo $3$ have no free product in the variety of rings with 1. However, coproducts (which differ from free products in not requiring the canonical homomorphisms $A_\alpha\to A$ to be injective) always exist in a variety of algebras [a1].

References

[a1]	F.E.J. Linton, "Coequalizors in categories of algebras" , Seminar on Triples and Categorical Homology Theory , Lect. notes in math. , 80 , Springer (1969) pp. 75–90