In Universal Algebra, it is possible to say that two presentations denote the "same" kind of algebraic structures, if the two corresponding varieties are "rationally equivalent" (Mal'cev 1958). In Model Theory, such "synonymity" is defined by various non-equivalent equivalence relations, such that definitional equivalence, mutual interpretability or bi-interpretability.

If I consider the universal closures of two equational presentations, what is the model-theoretical counterpart of the rational equivalence?

If there is a standard reference for a proof of this correspondance, many thanks, in advance!

PS: given a universal algebra $ \mathcal{A} = <A, \sigma>$ of signature $\sigma$, denote by $Ter(\sigma)$ the collection of all terms of signature $\sigma$ and by $Ter(\mathcal{A})$ the collection of all termal functions of $\mathcal{A}$. Two algebras $\mathcal{A} = <A, \sigma_{1}>$ and $\mathcal{B} = <B, \sigma_{2}>$ are *rationally equivalent* whenever there exists a bijection $\phi$ from the set $A$ to the set $B$ such that:

$\phi f ( \phi^{-1}(x_{1},...,\phi^{-1}(x_{n})) \in Ter(\mathcal{B})$ for every signature function $f$ of $\mathcal{A}$

and $\phi^{-1} g (\phi(x_{1},...,\phi(x_{n})) \in Ter(\mathcal{A})$ for every signature function $g$ of $\mathcal{B}$