Equivalence of varieties

Builds on Interpretation of varieties Variety of algebras

Definition

Let $\mathcal{V}$ and $\mathcal{W}$ be varieties of algebras. Let $\rho$ be an interpretation of $\mathcal{V}$ in $\mathcal{W}$ , which sends each $A \in \mathcal{W}$ to an algebra $A^{\rho} \in \mathcal{V}$ on the same set. Let $\theta$ be an interpretation of $\mathcal{W}$ in $\mathcal{V}$ , which sends each $B \in \mathcal{V}$ to an algebra $B^{\theta} \in \mathcal{W}$ on the same set. The pair $(\rho, \theta)$ is an equivalence if $(A^{\rho})^{\theta} = A$ for all $A \in \mathcal{W}$ and $(B^{\theta})^{\rho} = B$ for all $B \in \mathcal{V}$ . The varieties $\mathcal{V}$ and $\mathcal{W}$ are equivalent, or term equivalent, if such a pair exists.

Remarks

Example: Boolean algebras and Boolean rings are equivalent. One translation is $x + y \mapsto (x \wedge \neg y) \vee (\neg x \wedge y)$ , $x \cdot y \mapsto x \wedge y$ , with $0$ and $1$ kept. The other is $x \vee y \mapsto x + y + xy$ , $x \wedge y \mapsto xy$ , $\neg x \mapsto 1 + x$ . Going there and back recovers each operation, for example $x + y + xy$ becomes the join again in every Boolean algebra. Non-example: the variety of Boolean algebras is not equivalent to the variety of distributive lattices with signature $\vee, \wedge$ . There is not even an interpretation of Boolean algebras in distributive lattices, because every lattice term contains a variable, so no term can stand for the constant $0$ . Equivalent varieties have, on each set, the same subalgebras, the same congruences and the same term operations.

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Tag	`EF4M77TDF4`
URI	`m2://EF4M77TDF4`
Kind	definition
Writing	W0 stub
MSC	08B05
Defines	Equivalence of varieties
Uses	Interpretation of varieties, Variety of algebras
Changed	2 November 2026

Equivalence of varieties

Definition

Remarks

Reading path

Further reading

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