Hugo Luiz MARIANO and Francisco MIRAGLIA PROFINITE STRUCTURES ARE RETRACTS OF ULTRAPRODUCTS OF FINITE STRUCTURES

The results presented here first appeared in Chapter 2 of [Mrn1] and were announced with proofs in [MM1]. Our motivation came from [KMS], that introduces the class of direct limits of finite abstract order spaces, a theory due to M. Marshall ([Mar1]). The theory of special groups, introduced in [DM2], is a first-order axiomatization of the algebraic theory of quadratic forms, and there is a natural categorical duality between the Received 30 November 2006 170 HUGO LUIZ MARIANO and FRANCISCO MIRAGLIA category of reduced special groups and that of abstract ordered spaces, as shown in Chapter 3 of [DM2] (first established, by a different method, in [Lim]). Let L be a first-order language with equality and let L-mod be the category of L-structures and L-morphisms. As a preliminary to the proof of our main result, Theorem 2.3, the first section recalls the notions of retract and pure morphisms in L-mod, as well as some basic material on limits and colimits in this category, together with the relation between colimits and reduced products of L-structures (Proposition 1.8). Although many of these facts are folklore, full proofs of the needed results can be found in [MM2] and in Chapter 17 of [Mir]. Our general references for Category Theory and Model Theory are [Mac] and [CK] or [BS], respectively. At the end of the paper we mention some interesting applications of our main result to the theory of special groups, some of which have already appeared in the literature.