The notion of homomorphism homogeneity was introduced by Cameron and Nešetřil as a natural generalization of the classical model-theoretic notion of homogeneity. A relational structure is called homomorphism homogeneous (HH) if every homomorphism between finite substructures extends to an endomorphism. It is called polymorphism homogeneous (PH) if every finite power of the structure is homomorphism homogeneous. Despite the similarity of the definitions, the HH and PH structures lead a life quite separate from the homogeneous structures. While the classification theory of homogeneous structure is dominated by Fraïssé-theory, other methods are needed for classifying HH and PH structures. In this paper we give a complete classification of HH countable tournaments (with loops allowed). We use this result in order to derive a classification of countable PH tournaments. The method of classification is designed to be useful also for other classes of relational structures. Our results extend previous research on the classification of finite HH and PH tournaments by Ilić, Mašulović, Nenadov, and the first author.

Cameron和Nešetřil引入了同构同质性概念，作为经典的模型理论同质性概念的自然概括。如果有限子结构之间的每个同构都扩展到同构，则关系结构称为同构同构（HH）。如果结构的每个有限次幂都是同质同质的，则称为多态同质（PH）。尽管定义相似，但HH和PH结构的寿命与均质结构完全不同。虽然均质结构的分类理论主要由弗赖斯理论主导，但仍需要其他方法对HH和PH结构进行分类。在本文中，我们给出了HH可数锦标赛的完整分类（允许循环）。我们使用此结果来得出可数的PH锦标赛的分类。分类方法被设计为对于其他类别的关系结构也有用。我们的结果扩展了Ilić，Mašulović，Nenadov和第一作者对有限HH和PH锦标赛分类的先前研究。