Given a binary operation $\star$ on $\mathbb{N}$, we can naturally extend $\star$ to a semicontinuous operation $\widehat{\star}$ on the set $\beta\mathbb{N}$ of ultrafilters on $\mathbb{N}$ as follows:
$$\mathcal{U}\mathbin{\widehat \star}\mathcal{W}:=\{A:\{k: \{a: a\star k\in A\}\in\mathcal{U}\}\in\mathcal{W}\}.$$
It's a standard (and quite useful!) fact that associativity is preserved under this transformation: if $\star$ is associative, then so is $\widehat{\star}$. On the other hand, commutativity is not preserved. Say that an equational sentence $\sigma$ involving a single binary function symbol is a $\beta$-property iff we have $$(\mathbb{N};\star)\models\sigma\quad\implies\quad (\beta\mathbb{N};\widehat{\star})\models\sigma$$ for every binary operation $\star$ on $\mathbb{N}$.
Question: Which equational sentences are $\beta$-properties?
Of course there is no need to restrict attention to equational properties per se (which is why I've used "$\implies$" rather than "$\iff$" in the definition of $\beta$-property), and we can similarly transform arbitrarily many arbitrary-arity operations on $\mathbb{N}$; however, already this question seems difficult. I recall seeing a partial(?) answer to this question which involved the way variables were/were not reordered in the left vs. right hand sides of $\sigma$, but at the moment I can't track it down.
Incidentally, the following question is at least of a related flavor, and may involve relevant ideas (and be easier to tackle at first): what is $(\beta\mathbb{N};\widehat{+})$ (or $(\beta\mathbb{Z};\widehat{+})$ using the analogous definition) like from a universal-algebraic perspective? For example, is it congruence modular?
