NNIL-formulas, introduced by Visser in 1983–1984 in a study of |$\varSigma _1$|-subsitutions in Heyting arithmetic, are intuitionistic propositional formulas that do not allow nesting of implication to the left. The first results about these formulas were obtained in a paper of 1995 by Visser et al. In particular, it was shown that NNIL-formulas are exactly the formulas preserved under taking submodels of Kripke models. Recently, Bezhanishvili and de Jongh observed that NNIL-formulas are also reflected by the colour-preserving monotonic maps of Kripke models. In the present paper, we first show how this observation leads to the conclusion that NNIL-formulas are preserved by arbitrary substructures not necessarily satisfying the topo-subframe condition. Then, we apply it to construct universal models for NNIL. It follows from the properties of these universal models that NNIL-formulas are also exactly the formulas that are reflected by colour-preserving monotonic maps. By using the method developed in constructing the universal models, we give a new direct proof that the logics axiomatized by NNIL-axioms have the finite model property.

中文翻译：

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再谈NNIL公式：通用模型和有限模型属性

NNIL-公式，由Visser于1983–1984年在| $ \ varSigma _1 $ |的研究中引入 Heyting算术中的-替换是直觉的命题公式，不允许将含义嵌套到左侧。关于这些公式的第一个结果是由Visser等人在1995年的一篇论文中获得的。尤其是，它表明NNIL公式恰好是在Kripke模型的子模型下保留的公式。最近，Bezhanishvili和de Jongh观察到NNIL公式也通过Kripke模型的保色单调图来反映。在本文中，我们首先显示该观察结果如何得出结论，即NNIL公式由不一定满足拓扑子帧条件的任意子结构保留。然后，我们将其应用于构建NNIL的通用模型。从这些通用模型的性质可以得出， NNIL公式也正是保留颜色的单调图所反映的公式。通过使用构造通用模型的方法，我们给出了新的直接证明，即由NNIL-公理公理化的逻辑具有有限模型属性。