Paragraded structures inspired by mathematical logic

We use methods from mathematical logic to give new examples of paragraded structures, showing that at certain cardinals all first order structures are paragraduaded. We introduce the notion of bi-embeddability to measure when two paragraduaded structures are basically the same. We prove that the bi-embeddability of the paragraduating system gives rise to the bi-embeddability of the limiting structures. Under certain circumstances the converse is also true, as we show here. Finally, we show that one paragraduaded structure can have many graduaded substructures, to the extent that the number of the same is not always decidable by the axioms of set theory.