The first part of this paper provides a comprehensive and self-contained account of the interrelationships between algebraic properties of varieties and properties of their free algebras and equational consequence relations. In particular, proofs are given of known equivalences between the amalgamation property and the Robinson property, the congruence extension property and the extension prope...

We present definitions of homology groups Hn(p), n ≥ 0, associated to a complete type p. We show that if the generalized amalgamation properties hold, then the homology groups are trivial. We compute the group H2(p) for strong types in stable theories and show that any profinite abelian group can occur as the group H2(p). The work described in this paper was originally inspired by Hrushovski’s ...

The aim of this paper is to present a detailed explanation of three models of Shelah. We show the rule of the amalgamation in the construction of models in which all deenable sets of reals have Baire property or are Lebesgue measurable. Next we construct a model in which every projective set of reals has Baire property, a model with the Uniformization Property and a model of ZF + DC in which al...

38 and the rich amount of interesting relationships between them, are now well-understood (e.g., Co83, Ma88]). We believe that free amalgamation, rational amalgamation and a further construction called \innnite amalgamation" (still to be investigated) reeect this role on the higher level of amalgamation constructions. Many of the results that we have obtained for free and rational amalgamation ...

We prove a downward transfer from categoricity in a successor in tame abstract elementary classes (AECs). This complements the upward transfer of Grossberg and VanDieren and improves the Hanf number in Shelah’s downward transfer (provided the class is tame). Theorem 0.1. Let K be an AEC with amalgamation. If K is LS(K)-weakly tame and categorical in a successor λ ≥ i(2LS(K))+ , then K is catego...

We observe how a classical model-theoretic fact proves the existence of many strong amalgamation bases for the class ofR-embeddable von Neumann algebras, where R is the hyperfinite II1 factor. In particular, we shows that R itself is a strong amalgamation base.

This is a sequel to the paper [6] ‘Homology of types in model theory I: Basic concepts and connections with type amalgamation’. We compute the group H2 for strong types in stable theories and show that any profinite abelian group can occur as the group H2 in the model-theoretic context. The work described in this paper was originally inspired by Hrushovski’s discovery [8] of striking connection...