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Below, I'll use "$\approx$" for the equality symbol in an equation, as opposed to "actual" equality.

Suppose $\mathcal{V}$ is a variety (in the sense of universal algebra) in the signature $\Sigma$. Say that a $\mathcal{V}$-unpacking is a set $\mathfrak{E}$ of $\Sigma\sqcup\Pi$-equations, for some signature $\Pi$ disjoint from $\Sigma$, such that for each equation $\sigma\approx\tau\in\mathfrak{E}$, the left hand side term $\sigma$ uses only symbols from $\Sigma$. Given a $\mathcal{V}$-unpacking $\mathfrak{E}$, let $T_\mathcal{V}(\mathfrak{E})$ be the set of $\Pi$-equations which are entailed by the equational theory of $\mathcal{V}$ together with $\mathfrak{E}$.

For example, let $\mathcal{E}$ be the variety generated by the algebra $(\mathbb{N};\mathit{exp})$ (where $\mathit{exp}$ is the binary exponentiation function). The singleton $$\mathfrak{M}=\{\mathit{exp}(\mathit{exp}(x,y), z)\approx\mathit{exp}(x, y\star z)\}$$ (with $\Sigma=\{\mathit{exp}\}$ and $\Pi=\{\star\}$) is an $\mathcal{E}$-unpacking, and unless I've made a silly mistake $T_\mathcal{E}(\mathfrak{M})$ should be exactly (up to symbol-renaming) the equational theory of $(\mathbb{N};\cdot)$.

I'm generally trying to get a better sense of how unpackings behave, motivated by questions about symmetry groups associated to terms (see e.g. here) as well as general interest in anything "implicit-definability-flavored." To get things started, I'm looking for a nontrivial negative example, and the following seems like a good candidate:

Letting $\mathcal{E}$ be as above, is there an $\mathcal{E}$-unpacking $\mathfrak{X}$ such that $T_\mathcal{E}(\mathfrak{X})$ is (up to symbol-renaming) the equational theory of commutative rings?

In general, though, I'm interested in any information about this notion. For example, I expect the relation $\mathcal{V}\trianglelefteq\mathcal{W}$ = "Up to change-of-symbols, $Th_{Eq}(\mathcal{V})=T_\mathcal{W}(\mathfrak{S})$ for some $\mathcal{W}$-unpacking $\mathfrak{S}$" to be non-transitive, but I don't immediately have a counterexample for this.

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  • $\begingroup$ @MattF. Well, the thing I'm primarily interested in is the general notion of unpacking. This is really only a test question - I'm hopeful that it will provide an interesting example in one direction or the other, but the general notion is what I really care about. $\endgroup$
    – Noah Schweber
    Commented Oct 15, 2022 at 17:58
  • $\begingroup$ How about “Given languages $L$ and $L’$, define an unpacking of the $L$-model $M$ into the $L’$-theory $T$ as an $L+L’$-theory $U$ whose restriction to $L$ is the theory of $M$, whose restriction to $L’$ is the theory $T$, and which is axiomatized by equations whose left sides are contained in $L$. Then, considering the languages $\wedge$ and $\{\oplus, \otimes\}$, is there an unpacking of $\mathbb{N}$ (under exponentiation) into the theory of commutative rings?” $\endgroup$
    – user44143
    Commented Oct 15, 2022 at 18:40
  • $\begingroup$ It's a bit odd that your definition of a $\mathcal V$-unpacking does no really depend on $\mathcal V$ (only on $\Sigma$). Or am I missing something? $\endgroup$
    – tomasz
    Commented Oct 16, 2022 at 10:46
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    $\begingroup$ @tomasz That's a fair point, but $\mathcal{V}$ comes into play when we interpret the unpack (as $T_\mathcal{V}(\mathfrak{E})$). $\endgroup$
    – Noah Schweber
    Commented Oct 16, 2022 at 10:48

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