Reducts of random hypergraphs
Let \(M\) be a countable \(\omega\)-categorical structure on a set \(A\). The structure \(N=\langle A,R_0,\dots,R_\alpha,\dots\rangle\) is said to be a reduct of \(M\) iff every \(R_\alpha\) is definable without parameters in \(M\). For any closed permutation group \(G\) the condition \((*):\text{Aut}(M) \leq G\leq \text{Sym}(A)\) (where \(\text{Sym}(A)\) is the group of all permutations of \(A\)) is equivalent to \(G=\text{Aut}(N)\) for some reduct \(N\) of \(M\). Reducts \(N_1\) and \(N_2\) of \(M\) are said to be equivalent iff each is a reduct of the other, i.e. iff \(\text{Aut }N_1=\text{Aut }N_2\). Thus the problem of classifying the reducts of \(M\) up to equivalence is the same as that of classifying the closed permutation groups \(G\) satisfying \((*)\). The author gives the classification of the closed permutation groups \(G\) satisfying \((*)\) for the universal homogeneous \(k\)-hypergraphs \(M\).
countable \(\omega\)-categorical structure
scientific article; zbMATH DE number 933086