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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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