An imaginary sort of a first-order structure is a possible universe of an interpretable structure. The significance of imaginary sorts was established in the abstract setting of Classification Theory by Shelah, where all notions of stability theory, especially algebraic closure, were interpreted with imaginary sorts taken into account. Imaginaries were studied for specific theories by Poizat who showed that algebraically closed fields and differentially closed fields eliminate imaginaries, i.e. each imaginary sort is in definable bijection with a definable set of n-tuples. In valued fields imaginary sorts differ qualitatively from definable sets; an example is the value group. In this case it is a question of finding a useful basis for the imaginary sorts, meaning that every imaginary sort can be embedded into a product of basis elements. A basis for imaginary sorts for algebraically closed valued fields was given by Haskell-Hrushovski-Macpherson. Rideau-Hrushovski-Martin, and more recently Hils-Rideau-Kikuchi, proved that for the field of $p$-adic numbers the $p$-adic lattices provide a basis for imaginary sorts.
In this talk I will describe joint work with Ehud Hrushovski which gives a description of imaginary sorts and their basis for products and restricted products of structures in terms of a basis for imaginary sorts for the factors. In 1959 Feferman and Vaught proved that the theory of a product is determined by the theory of the factors and the theory of the Boolean algebra of subsets of the index set. This yields decidability results and descriptions of definable sets for products in terms of the factors. Our results show that imaginaries also admit a description relative to the factors. The proofs use tools from geometric stability theory including liaison groups, and descriptive set theory of smooth Borel equivalence relations, including a dichotomy theorem due to Harrington-Kechris-Louveau and Glimm-Efros. As an application, we describe a basis for the imaginary sorts for the ring of adeles of rational numbers in terms of the family of $p$-adic lattices for �almost all� $p$, and prove weak elimination of imaginaries for the adeles in these sorts.
Joint work with: Ehud Hrushovski
Zoom room information: https://us06web.zoom.us/j/81916335336?pwd=5zQT4lutMiBoY5Xp1pkFGtbqiaGozg.1
Meeting ID: 819 1633 5336
Passcode: 559618