Let p be a strong type of an algebraically closed tuple over B = acl eq (B) in any theory T . Depending on a ternary relation ⌣ | * satisfying some basic axioms (there is at least one such, namely the trivial independence in T ), the first homology group H * 1 (p) can be introduced, similarly to [3]. We show that there is a canonical surjective homomorphism from the Lascar group over B to H * 1 (p). We also notice that the map factors naturally via a surjection from the 'relativised' Lascar group of the type (which we define in analogy with the Lascar group of the theory) onto the homology group, and we give an explicit description of its kernel. Due to this characterization, it follows that the first homology group of p is independent from the choice of ⌣ | * , and can be written simply as H 1 (p).As consequences, in any T , we show that |H 1 (p)| ≥ 2 ℵ0 unless H 1 (p) is trivial, and we give a criterion for the equality of stp and Lstp of algebraically closed tuples using the notions of the first homology group and a relativised Lascar group.We also argue how any abelian connected compact group can appear as the first homology group of the type of a model.In this paper we study the first homology group of a strong type in any theory.Originally, in [3] and [4], a homology theory only for rosy theories is developed. Namely, given a strong type p in a rosy theory T , the notion of the nth homology group H n (p) depending on thorn-forking independence relation is introduced. Although the homology groups are defined analogously as in singular homology theory in algebraic topology, the (n + 1)th homology group for n > 0 in the rosy theory context has to do with the nth homology group in algebraic topology. For example as in [3], H 2 (p) in stable theories has to do with the fundamental group in topology. 1 2 JAN DOBROWOLSKI, BYUNGHAN KIM, AND JUNGUK LEE theories, H 1 (p) is detecting somewhat endemic properties of p existing only in model theory context.Indeed, in every known rosy example, H n (p) for n ≥ 2 is a profinite abelian group. In [5], it is proved to be so when T is stable under a canonical condition, and conversely, every profinite abelian group can arise in this form. On the other hand, we show in this paper that the first homology groups appear to have distinct features as follows.Let p = tp(a/B) be a strong type over B = acl eq (B) in any theory T . Fix a ternary invariant independence relation ⌣ | * among small sets satisfying finite character, normality, symmetry, transitivity and extension. (There is at least one such relation, by putting A ⌣ | C D for any sets A, C, D.) Then we can analogously define H * 1 (p) depending on ⌣ | * , (which of course is the same as H 1 (p) when ⌣ | * is thorn-independence in rosy T ). In this note, a canonical epimorphism from the Lascar group over B of T to H * 1 (p) is constructed. Indeed, we also introduce the notion of the relativised Lascar group of a type which is proved to be independent from the choice of the monster model of T , and the homomorphism factors th...

We classify, in a nontrivial amenable collection of functors, all 2-chains up to the relation of having the same 1-shell boundary. In particular, we prove that in a rosy theory, every 1-shell of a Lascar strong type is the boundary of some 2-chain, hence making the 1st homology group trivial. We also show that, unlike in simple theories, in rosy theories there is no upper bound on the minimal lengths of 2-chains whose boundary is a 1-shell.

In this paper, we investigate a new model theoretical tree property (TP), called the antichain tree property (ATP). We develop combinatorial techniques for ATP. First, we show that ATP is always witnessed by a formula in a single free variable, and for formulas, not having ATP is closed under disjunction. Second, we show the equivalence of ATP and [Formula: see text]-ATP, and provide a criterion for theories to have not ATP (being NATP). Using these combinatorial observations, we find algebraic examples of ATP and NATP, including pure groups, pure fields and valued fields. More precisely, we prove Mekler’s construction for groups, Chatzidakis’ style criterion for pseudo-algebraically closed (PAC) fields, and the AKE-style principle for valued fields preserving NATP. We give a construction of an antichain tree in the Skolem arithmetic and atomless Boolean algebras.

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We achieve several results. First, we develop a variant of the theory of absolute Galois groups in the context of many sorted structures. Second, we provide a method for coding absolute Galois groups of structures, so they can be interpreted in some monster model with an additional predicate. Third, we prove the “Weak Independence Theorem” for pseudo-algebraically closed (PAC) substructures of an ambient structure with no finite cover property (nfcp) and the property [Formula: see text]. Fourth, we describe Kim-dividing in these PAC substructures and show several results related to the SOPn hierarchy. Fifth, we characterize the algebraic closure in PAC structures.