Accept $ X $ Is a sentence $ mathcal {W} $ is a family of subsets of $ X $, and $ mu: mathcal {W} rightarrow mathcal {W} $ is an operation on these sets. There is a natural way of assigning modal logic to the tuple $ (X, mathcal {W}, mu) $ through the thinking of propositional atoms as elements of $ mathcal {W} $the modality "$ Diamond $"as a card $ mu $and the booleans as corresponding set operations. As examples, consider:

The topological semantics of Tarski / McKinsey: $ X $ is the set of points in a topological space $ mathbb {X} $, $ mathcal {W} = mathcal {P} (X) $, and $ mu (A) = cl (A) $ (with the meaning of $ mathbb {X} $). This was initially used to study S4, but is much more useful (see, for example, Bezhanishvili, Gabelaia, and LuceroBryan on modal logics of metric spaces).

Esaki's topological semantics: $ X $ and $ mathcal {W} $ are like above, but $ mu (A) $ is the set of limit points of $ A $, This is quite different from the approach of Tarski / McKinsey, and indeed there are several characterizations of the provability logic about this semantics (see, for example, this review by Beklemishev and Gabelaia).
(Actually, I'm not familiar with any approach in this direction that at least does not look like a topology, but I'm really unfamiliar with the literature.) Note that $ mathcal {W} $ has not played a role yet.
In each of the above cases $ Diamond $ is a "reasonably definable" operation on sets. Fully rely on the definability – here $ mathcal {W} $ enters the picture – we can see what happens in a rather modeltheoretical context:
Accept $ mathcal {M} $ is a first order structure and $ varphi (x, X) $ is a formula with parameters off $ mathcal {M} $ in the language of $ mathcal {M} $ along with "$ in $" (from where $ x $ is an object variable and $ X $ is a quantity variable), We let $ L_ mathcal {M} ( varphi) $ Let the logic defined along the above lines be $ X = mathcal {M} $, $ mathcal {W} $ the set of definable subsets of $ mathcal {M} $, and $$ mu: A rightarrow {m: varphi (m, A) }. $$
I'm interested in how (if anything!) The class of logic of the form $ L_ mathcal {M} ( varphi) $ reflects the model theory of $ mathcal {M} $ in the usual sense. Well, this generally seems to be a rather difficult problem, so I am currently focusing on the following simpler instance:
General problem: given a modal formula $ pi $ and a structure $ mathcal {M} $, determine whether $ pi in L_ mathcal {M} ( varphi) $ for a suitable formula $ varphi $,
(Call one of them $ pi $ "enforceable over $ mathcal {M} $, ")
At first I thought of a light starter problem, but it seems to be more difficult than I thought. The only situations that I really understand are banal or totally artificial. However, I am not sure how much of it is actually more difficult than my own inexperience. To get to the point, I would like to query the following very specific case:
Current question, Is any modal formula enforceable over the structure $ ( mathbb {N}; +, times) $?
I'm not sure which answer to expect. I am used to thinking of arithmetic, of being able to "do everything" and show maximum unpleasant behavior, but this intuition seems to cut both ways here.
And as always, I lack sources, and the wheel is likely to be reinvented – all the clues are greatly appreciated!