Interpretation of varieties

Summary

An interpretation of a variety $\mathcal{V}$ in a variety $\mathcal{W}$ translates each basic operation of $\mathcal{V}$ into a term of $\mathcal{W}$. The translation must turn every algebra of $\mathcal{W}$ into an algebra of $\mathcal{V}$ on the same set. Interpretations order varieties by strength: if $\mathcal{V}$ is interpretable in $\mathcal{W}$, then $\mathcal{W}$ is at least as rich as $\mathcal{V}$. Suppose $\mathcal{V}$ is interpretable in $\mathcal{W}$ and $\mathcal{V}$ has terms that satisfy some identities. Then the translated terms of $\mathcal{W}$ satisfy the same identities, so such properties pass from $\mathcal{V}$ to $\mathcal{W}$.