General algebraic systems Free algebras Theorem

Universal property of free algebras

Builds on Free algebra in a variety Variety of algebras Term algebra Quotient algebra Identity Birkhoff's HSP theorem

Statement

Let $\mathcal{V}$ be a variety of signature $\sigma$ , and let $X$ be a nonempty set of variables. Let $T(X)$ be the term algebra of $\sigma$ over $X$ . Let $\theta$ be the set of pairs $(s, t)$ of terms such that the identity $s \approx t$ holds in every algebra of $\mathcal{V}$ . Then:

$\theta$ is a congruence of $T(X)$ , and the quotient algebra $F = T(X) / \theta$ belongs to $\mathcal{V}$ .
With $\eta(x) = x / \theta$ , the algebra $F$ is a free algebra of $\mathcal{V}$ on $X$ . So for every $A \in \mathcal{V}$ and every map $f \colon X \to A$ , there is exactly one homomorphism $h \colon F \to A$ with $h \circ \eta = f$ .
For terms $s, t$ over $X$ , the elements $s / \theta$ and $t / \theta$ are equal if and only if $\mathcal{V}$ satisfies $s \approx t$ .

Remarks

The map $f$ extends to a homomorphism $T(X) \to A$ by evaluating terms, and its kernel contains $\theta$ because $A$ satisfies every identity of $\mathcal{V}$ ; this gives $h$ . To see $F \in \mathcal{V}$ , choose, for each pair $(s, t) \notin \theta$ , an algebra of $\mathcal{V}$ and an assignment that separate $s$ and $t$ . The induced maps embed $F$ in the product of these algebras, and $\mathcal{V}$ is closed under subalgebras and products. Example: for $\mathcal{V}$ the variety of commutative rings with identity, $F$ is the polynomial ring $\Z[X]$ , and the unique homomorphism is evaluation of polynomials. The variable set must be nonempty only when $\sigma$ has no constants and empty algebras are not allowed; otherwise $X = \emptyset$ is fine and gives the initial algebra of $\mathcal{V}$ .

Going deeper

For a student

The theorem gives free groups, free modules, polynomial rings and free lattices by one construction. Part 3 says that the free algebra on countably many generators encodes all identities of $\mathcal{V}$ , which links free algebras with equational logic. Every algebra $A \in \mathcal{V}$ is a quotient of $F_{\mathcal{V}}(A)$ , by extending the identity map of $A$ ; presentations by generators and relations come from this. In categorical terms, the construction is a left adjoint to the forgetful functor from $\mathcal{V}$ to sets.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Class operators H S P concept
Free algebra in a variety concept
Identity concept
Quotient algebra concept
Term algebra concept
Variety of algebras concept
Birkhoff's HSP theorem statement
Universal property of free algebras statement

Tag	`3F2W916W64`
URI	`m2://3F2W916W64`
Kind	statement, theorem
Writing	W0 stub
MSC	08B20
Status	This copy of the site has no build data. Run `m2 status 3F2W916W64`.
About	Free algebra in a variety
Uses	Free algebra in a variety, Variety of algebras, Term algebra, Quotient algebra, Identity, Birkhoff's HSP theorem
Changed	2 November 2026