---
tag: 48AR35RY9R
kind: definition
name: Interpretation of varieties
msc: [08B05]
defines: 32PDB3WQ4D
uses: [AJNZJD8442, N9M2N27END]
sources:
  - url: https://en.wikipedia.org/wiki/Interpretation_(model_theory)
---

## Definition

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] of signature $\sigma$ and let $\mathcal{W}$ be a variety of signature $\tau$.
An *interpretation* of $\mathcal{V}$ in $\mathcal{W}$ is a map $f \mapsto t_f$ with the following properties.
It assigns to each $n$-ary operation symbol $f$ of $\sigma$ a $\tau$-term $t_f(x_1, \dots, x_n)$.
For $A$ in $\mathcal{W}$, let $A^{*}$ be the algebra on the universe of $A$ where each $f$ acts as the [[term operation|N9M2N27END]] $t_f^{A}$.
Then $A^{*}$ belongs to $\mathcal{V}$ for every $A$ in $\mathcal{W}$.
The variety $\mathcal{V}$ is *interpretable* in $\mathcal{W}$ if an interpretation of $\mathcal{V}$ in $\mathcal{W}$ exists.

## Remarks

Example: let $\mathcal{V}$ be the variety of Boolean algebras, with operations $\vee, \wedge, \neg, 0, 1$, and let $\mathcal{W}$ be the variety of Boolean rings.
The assignment $x \vee y \mapsto x + y + xy$, $x \wedge y \mapsto xy$, $\neg x \mapsto 1 + x$, $0 \mapsto 0$, $1 \mapsto 1$ is an interpretation.
In every Boolean ring these term operations satisfy the Boolean algebra identities.
Non-example: let $\mathcal{V}$ be the variety of semigroups and $\mathcal{W}$ the variety of groups, and assign to the semigroup product the group term $x y^{-1}$.
This is not an interpretation, since in the group $(\Z, +)$ the operation $a - b$ is not associative: $(1 - 1) - 1 = -1$ but $1 - (1 - 1) = 1$.
It suffices to check the defining identities of $\mathcal{V}$ on the free algebras of $\mathcal{W}$.

## Explanations

### L1 human

An interpretation writes each operation of one variety as a formula in the operations of another, in a way that keeps all the laws true.