Completions of μ-algebras

Annals of Pure and Applied Logic 154 (1):27-50 (2008)
  Copy   BIBTEX

Abstract

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 where μx.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 nontrivial quasivariety of μ-algebras contains a μ-algebra that has no embedding into a complete μ-algebra.We then focus 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 modal operators are adjoints on free modal μ-algebras, least prefixed points of Σ1-operations satisfy the constructive relation μx.f=logical and operatorn≥0fn. These properties imply the following statement: 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 image of the fixed point alternation hierarchy

Other Versions

No versions found

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 103,343

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

The variable hierarchy for the games μ-calculus.Walid Belkhir & Luigi Santocanale - 2010 - Annals of Pure and Applied Logic 161 (5):690-707.
Power Set Modulo Small, the Singular of Uncountable Cofinality.Saharon Shelah - 2007 - Journal of Symbolic Logic 72 (1):226 - 242.
A decision procedure for alternation-free modal μ-calculi.Yoshinori Tannabe, Koichi Takahashi & Masami Hagiya - 1998 - In Marcus Kracht, Maarten de Rijke, Heinrich Wansing & Michael Zakharyaschev (eds.), Advances in Modal Logic. CSLI Publications. pp. 341-362.
A decision procedure for alternation-free modal μ-calculi.Yoshinori Tannabe, Koichi Takahashi & Masami Hagiya - 1998 - In Marcus Kracht, Maarten de Rijke, Heinrich Wansing & Michael Zakharyaschev (eds.), Advances in Modal Logic. CSLI Publications. pp. 341-362.
Fallen cardinals.Menachem Kojman & Saharon Shelah - 2001 - Annals of Pure and Applied Logic 109 (1-2):117-129.
Recursive inseparability for residual Bounds of finite algebras.Ralph Mckenzie - 2000 - Journal of Symbolic Logic 65 (4):1863-1880.
μ-complete Souslin trees on μ+.Menachem Kojman & Saharon Shelah - 1993 - Archive for Mathematical Logic 32 (3):195-201.
Complete Axiomatization of the Sutter-invariant Fragment of the Linear Time μ-calculus.Amélie Gheerbrant - 1998 - In Marcus Kracht, Maarten de Rijke, Heinrich Wansing & Michael Zakharyaschev (eds.), Advances in Modal Logic. CSLI Publications. pp. 140-155.

Analytics

Added to PP
2013-12-26

Downloads
28 (#837,946)

6 months
6 (#572,300)

Historical graph of downloads
How can I increase my downloads?

Citations of this work

Completeness for μ-calculi: A coalgebraic approach.Sebastian Enqvist, Fatemeh Seifan & Yde Venema - 2019 - Annals of Pure and Applied Logic 170 (5):578-641.
Completeness for flat modal fixpoint logics.Luigi Santocanale & Yde Venema - 2010 - Annals of Pure and Applied Logic 162 (1):55-82.
Proof systems for the coalgebraic cover modality.Marta Bílková, Alessandra Palmigiano & Yde Venema - 1998 - In Marcus Kracht, Maarten de Rijke, Heinrich Wansing & Michael Zakharyaschev, Advances in Modal Logic. CSLI Publications. pp. 1-21.

Add more citations

References found in this work

Add more references