There exist two known types of ultrafilter extensions of first-order models, both in a certain sense canonical. One of them (Goranko in Filter and ultrafilter extensions of structures: universal-algebraic aspects, preprint, 2007) comes from modal logic and universal algebra, and in fact goes back to Jónsson and Tarski (Am J Math 73(4):891–939, 1951; 74(1):127–162, 1952). Another one (Saveliev in Lect Notes Comput Sci 6521:162–177, 2011; Saveliev in: Friedman, Koerwien, Müller (eds) The infinity project proceeding, Barcelona, 2012) comes from model theory and algebra of ultrafilters, with ultrafilter extensions of semigroups (Hindman and Strauss in Algebra in the Stone–Čech Compactification, W. de Gruyter, Berlin, 2012) as its main precursor. By a classical fact of general topology, the space of ultrafilters over a discrete space is its largest compactification. The main result of Saveliev (Lect Notes Comput Sci 6521:162–177, 2011; in: Friedman, Koerwien, Müller (eds) The infinity project proceeding, Barcelona, 2012), which confirms a canonicity of this extension, generalizes this fact to discrete spaces endowed with an arbitrary first-order structure. An analogous result for the former type of ultrafilter extensions was obtained in Saveliev (in On two types of ultrafilter extensions of binary relations. arXiv:2001.02456). Results of such kind are referred to as extension theorems. After a brief introduction, we offer a uniform approach to both types of extensions based on the idea to extend the extension procedure itself. We propose a generalization of the standard concept of first-order interpretations in which functional and relational symbols are interpreted rather by ultrafilters over sets of functions and relations than by functions and relations themselves, and define ultrafilter models with an appropriate semantics for them. We provide two specific operations which turn ultrafilter models into ordinary models, establish necessary and sufficient conditions under which the latter are the two canonical ultrafilter extensions of some ordinary models, and obtain a topological characterization of ultrafilter models. We generalize a restricted version of the extension theorem to ultrafilter models. To formulate the full version, we propose a wider concept of ultrafilter models with their semantics based on limits of ultrafilters, and show that the former concept can be identified, in a certain way, with a particular case of the latter; moreover, the new concept absorbs the ordinary concept of models. We provide two more specific operations which turn ultrafilter models in the narrow sense into ones in the wide sense, and establish necessary and sufficient conditions under which ultrafilter models in the wide sense are the images of ones in the narrow sense under these operations, and also are two canonical ultrafilter extensions of some ordinary models. Finally, we establish three full versions of the extension theorem for ultrafilter models in the wide sense. The results of the first three sections of this paper were partially announced in Poliakov and Saveliev (in: Kennedy, de Queiroz (eds) On two concepts of ultrafilter extensions of first-order models and their generalizations, Springer, Berlin, 2017).

存在两种已知类型的一阶模型的超滤波器扩展，在某种意义上都是规范的。其中之一（Goranko in Filter and ultrafilter extensions of structure：通用代数方面，预印本，2007）来自模态逻辑和通用代数，实际上可以追溯到 Jónsson 和 Tarski（Am J Math 73(4):891– 939, 1951; 74(1):127–162, 1952)。另一个（Saveliev in Lect Notes Comput Sci 6521:162–177, 2011；Saveliev in: Friedman, Koerwien, Müller (eds) The infinity projectprocedure, Barcelona,​​ 2012）来自超滤器的模型理论和代数，超滤器扩展为半群（Hindman and Strauss in Algebra in the Stone–Čech Compactification, W. de Gruyter, Berlin, 2012）作为其主要前驱。根据一般拓扑学的经典事实，超滤器在离散空间上的空间是其最大的紧凑化。Saveliev (Lect Notes Comput Sci 6521:162–177, 2011; in: Friedman, Koerwien, Müller (eds) The infinity projectprocedure, Barcelona,​​ 2012) 的主要结果证实了这个扩展的规范性，将这一事实概括为离散空间具有任意的一阶结构。在 Saveliev 中获得了前一种类型的超滤扩展的类似结果（在 On two types of ultrafilter extensions of binary关系中。arXiv:2001.02456）。这种结果被称为扩展定理。在简要介绍之后，我们基于扩展扩展程序本身的想法为这两种类型的扩展提供了统一的方法。我们提出了一阶解释的标准概念的概括，其中函数和关系符号由超滤器对函数和关系集而不是由函数和关系本身进行解释，并为它们定义具有适当语义的超滤器模型。我们提供了两个将超滤模型转化为普通模型的具体操作，建立了后者是一些普通模型的两个典型超滤扩展的充要条件，并获得了超滤模型的拓扑表征。我们将扩展定理的受限版本推广到超滤模型。为了制定完整版本，我们提出了一个更广泛的超滤模型概念，其语义基于超滤器的限制，并表明前一个概念可以以某种方式与后者的特定情况相同；而且，新概念吸收了模型的普通概念。我们提供了两个更具体的将狭义超滤模型转化为广义超滤模型的操作，并建立了广义超滤模型在这些操作下成为狭义超滤模型的图像的充要条件，以及是一些普通模型的两个规范超滤扩展。最后，我们建立了广义超滤模型的扩展定理的三个完整版本。本文前三部分的结果在 Poliakov 和 Saveliev（在：Kennedy,