A relational structure \({{\mathbb {X}}}\) is called reversible iff each bijective homomorphism from \({{\mathbb {X}}}\) onto \({{\mathbb {X}}}\) is an isomorphism, and linear orders are prototypical examples of such structures. One way to detect new reversible structures of a given relational language L is to notice that the maximal or minimal elements of isomorphism-invariant sets of interpretations of the language L on a fixed domain X determine reversible structures. We isolate certain syntactical conditions providing that a satisfiable \(L_{\infty \omega }\)-theory defines a class of interpretations having extreme elements on a fixed domain and detect several classes of reversible structures. For some of these classes, we characterize the corresponding reversible extreme interpretations. In particular, we characterize the reversible countable ultrahomogeneous graphs.

中文翻译：

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极端关系结构的可逆性

关系结构\（{{\ mathbb {X}}} \）被称为可逆，当每个双射同态从\（{{\ mathbb {X}}} \）到\（{{\ mathbb {X}}} \\ ）是同构，线性顺序是此类结构的典型示例。检测给定关系语言L的新的可逆结构的一种方法是注意到，固定域X上语言L的同构不变解释集的最大或最小元素确定了可逆结构。我们隔离某些句法条件，只要满足一个令人满意的\（L _ {\ infty \ omega} \）-理论定义了一类在固定域上具有极端元素的解释，并检测了几类可逆结构。对于其中某些类别，我们描述了相应的可逆极端解释。特别是，我们描述了可逆的可计数超均质图。