PP-2003-13:
Hodkinson, Ian and Venema, Yde
(2003)
*Canonical varieties with no canonical axiomatisation.*
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## Abstract

We give a simple example of a variety V of modal algebras that is

canonical but cannot be axiomatised by canonical equations or

first-order sentences. We then show that the variety RRA of

representable relation algebras, although canonical, has no

canonical axiomatisation. Indeed, we show that every axiomatisation

of these varieties involves infinitely many non-canonical sentences.

Using probabilistic methods of Erdos, we construct an infinite

sequence G_0,G_1,... of finite graphs with arbitrarily large chromatic

number, such that each G_n is a bounded morphic image of G_{n+1} and

has no odd cycles of length at most n. The inverse limit of the

sequence is a graph with no odd cycles and hence is 2-colourable.

It follows that a modal algebra (respectively, a relation algebra)

obtained from the G_n satisfies arbitrarily many axioms from a

certain axiomatisation of V (RRA), while its canonical extension

satisfies only a bounded number of them. It now follows by

compactness that V (RRA) has no canonical axiomatisation.

A variant of this argument shows that there is no axiomatisation

using finitely many non-canonical sentences.

Item Type: | Report |
---|---|

Report Nr: | PP-2003-13 |

Series Name: | Prepublication (PP) Series |

Year: | 2003 |

Uncontrolled Keywords: | Canonical axiomatisation, canonical equation, canonical modal logic, canonical variety, game, inverse system, random graph, relation algebra |

Subjects: | Logic |

Date Deposited: | 12 Oct 2016 14:36 |

Last Modified: | 12 Oct 2016 14:36 |

URI: | https://eprints.illc.uva.nl/id/eprint/97 |

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