This paper is devoted to the problem of existence of saturated models for first-order many-valued logics. We consider a general notion of type as pairs of sets of formulas in one free variable that express properties that an element of a model should, respectively, satisfy and falsify. By means of an elementary chains construction, we prove that each model can be elementarily extended to a |$\kappa $|-saturated model, i.e. a model where as many types as possible are realized. In order to prove this theorem we obtain, as by-products, some results on tableaux (understood as pairs of sets of formulas) and their consistency and satisfiability and a generalization of the Tarski–Vaught theorem on unions of elementary chains. Finally, we provide a structural characterization of |$\kappa $|-saturation in terms of the completion of a diagram representing a certain configuration of models and mappings.

中文翻译：

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一阶多值逻辑的饱和模型

本文致力于一阶多值逻辑的饱和模型的存在问题。我们将类型的一般概念视为一个自由变量中的成对的公式对，它们表示模型的元素应分别满足和伪造的特性。通过基本链构造，我们证明了每个模型都可以基本扩展为| $ \ kappa $ | -饱和模型，即实现尽可能多的类型的模型。为了证明该定理，作为副产品，我们获得了有关平稳（理解为成对的公式集）及其一致性和可满足性的一些结果，以及关于基本链结合的Tarski-Vaught定理的推广。最后，我们提供| $ \ kappa $ |的结构特征。-在表示模型和映射的某些配置的图的完成方面达到饱和。