Galois groups of first order theories (Q2772967)

Let \(T\) be a complete (possibly multi-sorted) first-order theory. The paper is concerned with two groups \(\text{Gal}_L (T)\) and \(\text{Gal}_{KP}(T)\) associated with \(T\) as invariants of its biinterpretability class, and with two corresponding equivalence relations \(E_L\) and \(E_{KP}\). In more detail, if \(\overline M\) denotes a big saturated model of \(T\), then \(E_L\) is the finest equivalence relation on \(\overline M\), or on a sort \(S\), small (in the sense that its quotient set has a power less than \(|\overline M |\)) and invariant under \(\text{Aut} (\overline M)\), while \(E_{KP}\) is the finest small type-definable (over \(\emptyset\)) equivalence relation. The Galois groups \(\text{Gal}_L (T)\) and \(\text{Gal}_{KP} (T)\) arise quite naturally from the action of \(\text{Aut} (\overline M)\) on the quotient sets of \(E_L\), \(E_{KP}\), respectively; both of them can be equipped with some additional topological structure, making, for instance, \(\text{Gal}_L (T)\) a quasi-compact group. When \(T\) is \(G\)-compact (essentially, when the two groups \(\text{Gal}_L(T)\) and \(\text{Gal}_{KP}(T)\) coincide), then \(\text{Gal}_L(T)\) is made a compact group. NEWLINENEWLINENEWLINEIn many cases, for instance when \(T\) is stable, the two groups, and the corresponding equivalence relations, coincide. But here the authors provide an example where \(E_L \not = E_{KP}\) (a non-\(G\)-compact theory \(T\)): \(T\) is essentially the theory of a product of circles with some further structure including a clockwise circular order and clockwise rotations by \(2 \pi /n\) radians when \(n\) ranges over primitive integers. NEWLINENEWLINENEWLINEHowever the author show that, for every \(T\), \(E_{KP}\) is the composition of \(E_L\) and its closure \(\overline{E_L}\) (in the Stone space of complete types); this is done by characterizing closure in the quasicompact group \(\text{Gal}_L (T)\). The previous example, and some variations on its theme (proving in particular that \(E_{KP}\) is sometimes different even from \(\overline{E_L}\)), imply that this result is the best possible.