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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 \(P\).)
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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