Amalgamated free product

Builds on Variety of algebras Homomorphism Free product of algebras

Definition

Let $\mathcal{V}$ be a variety of algebras and let $A$ be an algebra in $\mathcal{V}$ . Let $(B_i)_{i \in I}$ be a family of algebras in $\mathcal{V}$ with homomorphisms $\alpha_i \colon A \to B_i$ . Call a family of homomorphisms $g_i \colon B_i \to D$ compatible if $g_i \circ \alpha_i = g_j \circ \alpha_j$ for all $i, j \in I$ . An amalgamated free product of the $B_i$ over $A$ is an algebra $C$ in $\mathcal{V}$ with homomorphisms $\beta_i \colon B_i \to C$ such that:

the family $(\beta_i)_{i \in I}$ is compatible;
for every algebra $D$ in $\mathcal{V}$ and every compatible family $g_i \colon B_i \to D$ , there is a unique homomorphism $g \colon C \to D$ with $g \circ \beta_i = g_i$ for all $i$ .

It is written $\mathop{\ast}_A B_i$ , or $B_1 *_A B_2$ for two factors.

Remarks

The amalgamated free product exists in every variety and is unique up to a unique compatible isomorphism. It is the quotient of the free product of the $B_i$ by the smallest congruence that identifies the images of $\alpha_i(a)$ and $\alpha_j(a)$ for all $a \in A$ and $i, j \in I$ . With $A$ the trivial group in the variety of groups, it is the free product. Usually the $\alpha_i$ are embeddings; whether the $\beta_i$ are then embeddings depends on the variety, and for groups they always are.

Example: in the variety of groups, $\Z/4\Z *_{\Z/2\Z} \Z/6\Z$ , with $\Z/2\Z$ mapped to the subgroups of order $2$ , is isomorphic to $SL_2(\Z)$ .

Non-example: in groups, the direct product $\Z \times \Z$ with the two coordinate inclusions is not $\Z *_{\set{0}} \Z$ . Clause 2 fails for $D = S_3$ and $g_1, g_2$ sending $1$ to two non-commuting transpositions, since the generators of $\Z \times \Z$ commute.

In Mathlib the construction for monoids and groups is Monoid.PushoutI.

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Tag	`FMFDEE8WN2`
URI	`m2://FMFDEE8WN2`
Kind	definition
Writing	W0 stub
MSC	08B25, 20E06
Defines	Amalgamated free product
Uses	Variety of algebras, Homomorphism, Free product of algebras
Changed	2 November 2026

Amalgamated free product

Definition

Remarks

Reading path

Further reading

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