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Abstract

A structure is called homomorphism-homogeneous if every homomorphism between finitely generated substructures of the structure extends to an endomorphism of the structure (P. J. Cameron and J. Nešetřil, 2006). In this paper we introduce oligomorphic transformation monoids in full analogy to oligomorphic permutation groups and use this notion to propose a solution to a problem, posed by Cameron and Nešetřil in 2006, to characterize endomorphism monoids of homomorphism-homogeneous relational structures over finite signatures. However, the main goal of this paper is to provide more evidence that the concept of homomorphism-homogeneity is analogous to that of ultrahomogeneity. It turns out that many results that hold for ultrahomogeneous or $\omega$-categorical structures have their analogues in the class of countable homomorphism-homogeneous structures, or countable weakly oligomorphic structures (these are structures whose endomorphism monoids are oligomorphic). For example, we characterize countable weakly oligomorphic structures in terms of the Ryll-Nardzewski property with respect to positive formulas; we prove that for countable weakly oligomorphic structures homomorphism-homogeneity is equivalent to quantifier elimination for positive formulas; finally, we prove that an $\omega$-categorical structure is both ultrahomogeneous and homomorphism-homogeneous if and only if it has quantifier elimination where positive formulas reduce to positive quantifier-free formulas