Towards a Ryll-Nardzewski-type theorem for weakly oligomorphic structures (Q2793904)

A permutation group \(G \leq\mathrm{Sym}(A)\) is called oligomorphic if it has only finitely many orbits on \(n\)-tupels for every \(n\). The Ryll-Nardzewski theorem states that a countable structure has an oligomorphic automorphism group if and only if its elementary theory is \(\omega\)-categotrical. \textit{P. J. Cameron} and \textit{J. Nešetřil} [Comb. Probab. Comput. 15, No. 1--2, 91--103 (2006; Zbl 1091.08001)] introduced several variations of homogeneity. So, a structure is called homomorphism-homogeneous if every homomorphism between finitely generated substructures of the given structure can be extended to an endomorphism of the structure. The notion of a weakly oligomorphic structure is closely related with this notion; a structure is weakly oligomorphic if its endomorphism monoid is oligomorphic.NEWLINENEWLINEIn this paper, the authors investigate homomorphism-homogeneous structures. They show that the notions of homomorphism-homogeneity and weak oligomorphy are closely related. They prove a Fraïssé-type theorem for homomorphism-homogeneous structures and they show that countable models of the theories of countable weakly oligomorphic structures are mutually homomorphism-equivalent. Further on, they show that every countable weakly oligomorphic homomorphism-homogeneous structure has a unique (up to isomorphism) homomorphism-equivalent substructure that is oligomorphic, homogeneous, and a core. Combined with a result of \textit{M. Bodirsky} [Log. Methods Comput. Sci. 3, No. 1, Paper 2, 16 p. (2007; Zbl 1128.03021)] they show that the positive existential theories of countable weakly oligomorphic structures are exactly the positive existential parts of \(\omega\)-categorical theories.