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The Hrushovski property for hypertournaments and profinite topologies. (English) Zbl 07174669
A class $$C$$ has the Hrushovski Property if, for any structure $$M \in C$$ and any finite set of partial isomorphisms $$\varphi_1, \ldots, \varphi_n$$ from (substructures of) $$M$$, there exists an extension $$M' \in C$$ such that on $$M'$$, $$\varphi_1, \ldots, \varphi_n$$ extend to automorphisms of $$M'$$. Given a set $$L$$ of positive integers, an $$L$$-hypertournament is a relational structure such that for each $$\ell \in L$$, there exists an $$\ell$$-ary relation $$R_{\ell}$$ on that structure such that for any $$\ell$$-tuple $$\bar x$$ and any permutation $$\sigma$$ of the $$\ell$$ arguments, it is not true that $$M \models \bigwedge_{i \in [\ell]} R_{\ell} (\sigma^i(\bar x))$$. This article addresses a question of [B. Herwig and D. Lascar, Trans. Am. Math. Soc. 352, No. 5, 1985–2021 (2000; Zbl 0947.20018)]: does the class of finite tournaments have the Hrushovski Property? The primary results of this article are: if the partial isomorphisms have distinct domains and ranges, then such an extension $$M'$$ exists (and generalizes to $$L$$-tournaments, $$L$$ being a set of primes), but absent this caveat, there is a counterexample for each singleton $$L=\{p\}$$, $$p$$ being prime. This intricate and somewhat difficult article relies heavily on algebraic topology on graphs, and involves a partial answer to another question of [loc. cit.]: if $$C$$ is a cyclic subgroup of the free group on $$n$$ elements that is closed under $$\ell$$-roots for some $$\ell$$, then $$C$$ is closed in the topology generated by the profinite topology of primes not dividing into $$\ell$$. (The profinite topology of a set $$P$$ (of primes) on the free group $$F$$ is the topology generated by cosets of normal subgroups $$N$$ of finite index such that the elements of $$F/N$$ are of orders not divisible by any primes not in $$P$$.)
##### MSC:
 03C13 Model theory of finite structures 05C20 Directed graphs (digraphs), tournaments 57M05 Fundamental group, presentations, free differential calculus 57M15 Relations of low-dimensional topology with graph theory
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