Interpretation of varieties

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 $\mathcal{W}$.