论文标题
中间理论的晶格通过Ruitenburg定理
Lattices of Intermediate Theories via Ruitenburg's Theorem
论文作者
论文摘要
对于每个单变量公式$χ$，我们引入了一个中间理论的格子：$χ$ logics的晶格。定义卡逻辑的关键思想是将原子命题解释为公式$χ^2 $的固定点，可以使用Ruitenburg的定理进行语法表征。我们在$χ$ logics的晶格与特殊类别的Heyting代数类别之间建立了代数双重性。这种方法使我们能够构建与单变量公式可能的固定点相对应的五个不同的晶格|其中中间逻辑的负变体的晶格。我们更详细地描述了这些晶格。
For every univariate formula $χ$ we introduce a lattices of intermediate theories: the lattice of $χ$-logics. The key idea to define chi-logics is to interpret atomic propositions as fixpoints of the formula $χ^2$, which can be characterised syntactically using Ruitenburg's theorem. We develop an algebraic duality between the lattice of $χ$-logics and a special class of varieties of Heyting algebras. This approach allows us to build five distinct lattices corresponding to the possible fixpoints of univariate formulas|among which the lattice of negative variants of intermediate logics. We describe these lattices in more detail.
