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 ω-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 ω-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.

A relational structure is homomorphism-homogeneous if every homomorphism between finite substructures extends to an endomorphism of the structure. This notion was introduced recently by Cameron and Ne\v{s}et\v{r}il. In this paper we consider a strengthening of homomorphism-homogeneity --- we call a relational structure polymorphism-homogeneous if every partial polymorphism with a finite domain extends to a global polymorphism of the structure. It turns out that this notion (under various names and in completely different contexts) has been existing in algebraic literature for at least 30 years. Motivated by this observation, we dedicate this paper to the topic of polymorphism-homogeneous structures. We study polymorphism-homogeneity from a model-theoretic, an algebraic, and a combinatorial point of view. E.g., we study structures that have quantifier elimination for positive primitive formulae, and show that this notion is equivalent to polymorphism-homogeneity for weakly oligomorphic structures. We demonstrate how the Baker-Pixley theorem can be used to show that polymorphism-homogeneity is a decidable property for finite relational structures. Eventually, we completely characterize the countable polymorphism-homogeneous graphs, the polymorphism-homogeneous posets of arbitrary size, and the countable polymorphism-homogeneous strict posets.Comment: 31 page

It was a long journey, and at the end of the day it is time to sum up the impressions. The work on this thesis began in the spring 2005 during my stay at the Institute of Algebra in Dresden, Germany, in the frame of the programme GK 334: Specification of discrete processes and systems of processes by operational models and logics, and it continued at the same place in the next school year, thanks to the financial support from DAAD. My host and scientific father there was Professor Reinhard Pöschel, whom I owe my deepest gratitude for the influence that he made on this work, with many discussions, suggestions and useful ideas. The Institute of Algebra in Dresden was always a place where I was welcome, and I use this opportunity to thank all its members on scientific collaboration and many reinvitations to visit them and present my current work. There is no place like home, and all these years for me it was the Department of Mathematics and Informatics in Novi Sad, which provide me the opportunity to interrupt my teaching assistant work from time to time and to make many scientific visits abroad. One of them, that was of the great importance for this thesis, I made in the summer 2008. During two months at the Department of Mathematics at the Ben-Gurion University, Be'er Sheva, Israel, my host was Professor Mikhail Klin, whom I thank on several very interesting discussions and directions for the future work. Despite to many changes of workplaces and work conditions, one factor has stayed stable. Professor Dragan Mašulović is my supervisor since my very first scientific step, and I thank him on all advices, support and critics that improved my work. A few people made all this possible, although they had nothing to do with the scientific part of this thesis. My mother-in-law was on many occasions taking the role of the Hotel Mamma, allowing me to concentrate on mathematics and to forget about everyday housework. Without any doubt, a great role played the quartet of dear people, my parents and my grandparents on my mother's side. They gave me all their love, support and the opportunity to freely choose my way in life. As a small sign of my gratitude, this thesis is dedicated to the memory of them. My last and the greatest thanks go to my husband, for being both my retreat and the first line of the battle during the fight to obtain the results that are written on the pages that follow.

We study the problem of characterizing all relations that can be defined from the fundamental relations of a given relational structure using positive existential formulae. The notion of k-endolocality is introduced in order to measure the complexity of relational structures with respect to this task. The hierarchy of kendolocal structures is thoroughly analysed in algebraic and model-theoretic ways. Interesting cross-connections with homomorphism-homogeneous relational structures are revealed. The interrelations between endolocal relational structures and several model-theoretic notions are collected in the Main Theorem. This Main Theorem is demonstrated to be a useful tool for studying relational algebras and, in particular, weak Krasner algebras. For example, a short proof of F. Börners characterization of weak Krasner clones on a countable set is given.