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FIRST-ORDER AXIOMATISATIONS OF REPRESENTABLE RELATION ALGEBRAS NEED FORMULAS OF UNBOUNDED QUANTIFIER DEPTH

Part of: Graph theory Algebraic logic

Published online by Cambridge University Press:  29 October 2021

ROB EGROT Open the ORCID record for ROB EGROT [Opens in a new window]  and
ROBIN HIRSCH
ROB EGROT
Affiliation:
FACULTY OF INFORMATION AND COMMUNICATION TECHNOLOGY MAHIDOL UNIVERSITY, SALAYA73170, THAILANDE-mail:[email protected]
ROBIN HIRSCH
Affiliation:
DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY COLLEGE LONDON LONDON WC1E 6BT, UKE-mail:[email protected]
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Abstract

Using a variation of the rainbow construction and various pebble and colouring games, we prove that RRA, the class of all representable relation algebras, cannot be axiomatised by any first-order relation algebra theory of bounded quantifier depth. We also prove that the class At(RRA) of atom structures of representable, atomic relation algebras cannot be defined by any set of sentences in the language of RA atom structures that uses only a finite number of variables.

Keywords

RRArepresentable relation algebrasfinite variable axiomatisationbounded quantifier depth axiomatisation

MSC classification

Primary: 03G15: Cylindric and polyadic algebras; relation algebras
Secondary: 05C90: Applications
Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 3 , September 2022 , pp. 1283 - 1300
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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