Oligomorphic transformation monoids and homomorphism-homogeneous structures

Dragan Mašulović¹ ; Maja Pech¹

¹ Department of Mathematics and Informatics University of Novi Sad Trg Dositeja Obradovića 4 21000 Novi Sad, Serbia

Fundamenta Mathematicae, Tome 212 (2011) no. 1, pp. 17-34.

Résumé

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

Détail

DOI : 10.4064/fm212-1-2

Keywords: structure called homomorphism homogeneous every homomorphism between finitely generated substructures structure extends endomorphism structure nbsp nbsp cameron nbsp paper introduce oligomorphic transformation monoids full analogy oligomorphic permutation groups notion propose solution problem posed cameron nbsp characterize endomorphism monoids homomorphism homogeneous relational structures finite signatures however main paper provide evidence concept homomorphism homogeneity analogous ultrahomogeneity turns out many results ultrahomogeneous omega categorical structures have their analogues class countable homomorphism homogeneous structures countable weakly oligomorphic structures these structures whose endomorphism monoids oligomorphic example characterize countable weakly oligomorphic structures terms ryll nardzewski property respect positive formulas prove countable weakly oligomorphic structures homomorphism homogeneity equivalent quantifier elimination positive formulas finally prove omega categorical structure ultrahomogeneous homomorphism homogeneous only has quantifier elimination where positive formulas reduce positive quantifier free formulas