arXiv math/05084122005-08-22math.RAmath.LO

μ-代数的完备化

Completions of mu-algebras

Luigi Santocanale

备注：: 36 pages, extended abstract appears in LICS 2005 proceedings

摘要：

一个$μ$ -代数是一个一阶理论的模型，它是有界格理论的扩展，带有成对的项$(f,μ_{x}.f)$，其中$μ_{x}.f$被公理化为$f$的最小前缀点，其公理是方程或方程蕴涵。标准$μ$ -代数是完备的，这意味着它们的格约简是完全格。证明了$μ$ -代数的任何非平凡拟簇都包含一个没有嵌入到完备$μ$ -代数中的$μ$-代数.然后，我们专注于模态$μ$ -代数，即命题模态$μ$ -演算的代数模型。我们证明了自由模态$μ$ -代数满足一个条件-让人想起惠特曼的自由格条件-这使我们能够证明：（一）模态算子是自由模态$μ$ -代数的伴随，（二）$Σ_{1}$ -操作的最小前缀点满足的建设性关系$μ_{x}.f = \bigvee_{n \geq 0} f^{n}(\bot)$。这些性质意味着以下陈述：{ \em自由模态$μ$ -代数的MacNeille-Dedekind完备化是完备模态$μ$ -代数，而且规范嵌入保持了不动点交替层次的类$Comp(Σ_{1},Π_{1})$中的所有运算。

英文摘要：

A $μ$-algebra is a model of a first order theory that is an extension of the theory of bounded lattices, that comes with pairs of terms $(f,μ_{x}.f)$ where $μ_{x}.f$ is axiomatized as the least prefixed point of $f$, whose axioms are equations or equational implications. Standard $μ$-algebras are complete meaning that their lattice reduct is a complete lattice. We prove that any non trivial quasivariety of $μ$-algebras contains a $μ$-algebra that has no embedding into a complete $μ$-algebra. We focus then on modal $μ$-algebras, i.e. algebraic models of the propositional modal $μ$-calculus. We prove that free modal $μ$-algebras satisfy a condition -- reminiscent of Whitman's condition for free lattices -- which allows us to prove that (i) modal operators are adjoints on free modal $μ$-algebras, (ii) least prefixed points of $Σ_{1}$-operations satisfy the constructive relation $μ_{x}.f = \bigvee_{n \geq 0} f^{n}(\bot)$. These properties imply the following statement: {\em the MacNeille-Dedekind completion of a free modal $μ$-algebra is a complete modal $μ$-algebra and moreover the canonical embedding preserves all the operations in the class $Comp(Σ_{1},Π_{1})$ of the fixed point alternation hierarchy.}

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英文摘要：

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