Some definable Galois Theory and Examples

We make explicit certain results around the Galois correspondence in the context of denable automorphism groups, and point out the relation to some recent papers dealing with the Galois theory of algebraic dierential equations when the constants are not \closed" in suitable senses. We also improve the denitions and results on generalized strongly normal extensions from [Pillay, \Dierential Galois theory I", Illinois J. Math., 42(4), 1998], using this to give a restatement of a conjecture on almost semiabelian -groups from [Bertrand and Pillay, \Galois theory, functional Lindemann-Weierstrass, and Manin maps", Pacic J. Math., 281(1), 2016].