Structural invariance, structural definability and the Galois theory of elementarily invariant structures
Suppose A and B are two adequately described structures- can we decide whether and how A is interpretable in terms of B? The question itself is in need of interpretation, of course. In different contexts, the term interpretation admits of different readings, suggesting different kinds of operations between the alleged structures; and even the term structure, popular and precise as it may sound, is already used with somewhat divergent senses within the range of Mathematics itself- the very discipline that is supposed to focus on structure per se. The use of the term interpretation is certainly neither restricted to structures that are models of the same first-order theory, not even to such as are merely "structures for" the very same minimal set of predicates. As we all know, it is possible to envisage reductive "interpretations"- and Science is full of such- whereby the fundamental individuals and predicates in one structure are mapped on totally different types of entities, logically- entities which may be much more complex and derivative within the "interpreting" structure.