Gadget construction and structural convergence

Ne\v{s}et\v{r}il and Ossona de Mendez recently proposed a new definition of graph convergence called structural convergence. The structural convergence framework is based on the probability of satisfaction of logical formulas from a fixed fragment of first-order formulas. The flexibility of choosing the fragment allows to unify the classical notions of convergence for sparse and dense graphs. Since the field is relatively young, the range of examples of convergent sequences is limited and only a few methods of construction are known. Our aim to extend the variety of constructions by considering the gadget construction that appears, e.g., in studies of homomorphisms. We show that, when restricting to the set of sentences, the application of gadget construction on an elementarily convergent sequence and elementarily convergent gadgets results in an elementarily convergent sequence. For the general case, we show counterexamples witnessing that a generalization to the full first-order convergence is not possible without additional assumptions. Moreover, we give several different sufficient conditions to ensure the convergence, one of them states that the resulting sequence is first-order convergent if the replaced edges are dense in the original sequence of structures.