Two definability results in the equational context

Abstract:

Let $\tau$ be a type bounded by an infinite regular cardinal $\alpha$, ${\mathbf {V}}$ be a variety in $\tau ,\tau ’ \subseteq \tau$ and ${\mathbf {V’}}$ the class of all $\tau ’$-reducts of the algebras in ${\mathbf {V}}$. We show that the operations in $\tau \backslash \tau ’$ are explicitely definable in ${\mathbf {V}}$ by pure formulas (i.e. existential-positive without disjunction) if and only if they are implicitely definable and ${\mathbf {V’}}$ is closed under unions of $\alpha$-chains (if and only if every $\tau ’$-homomorphisms between algebras in ${\mathbf {V}}$ are $\tau$-homomorphisms, as J. Isbell has shown). It follows that the operations in $\tau \backslash \tau ’$ are equivalent (in ${\mathbf {V}}$) to $\tau ’$-terms if and only if every algebra in the $(\tau ’ - )$ variety generated by ${\mathbf {V’}}$ has a unique $\tau$-expansion in ${\mathbf {V}}$.