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THE NEAT EMBEDDING PROBLEM FOR ALGEBRAS OTHER THAN CYLINDRIC ALGEBRASAND FOR INFINITE DIMENSIONS

Published online by Cambridge University Press:  17 April 2014

ROBIN HIRSCH  and
TAREK SAYED AHMED
ROBIN HIRSCH
Affiliation:
DEPARTMENT OF COMPUTER SCIENCE, UNIVERSITY COLLEGE LONDON, GOWER STREET LONDON, WC1E 6BT, UK E-mail: ucacrdh@live.ucl.ac.uk
TAREK SAYED AHMED
Affiliation:
DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCE, CAIRO UNIVERSITY, GIZA, EGYPT E-mail: e-mail: rutahmed@gmail.com
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Abstract

Hirsch and Hodkinson proved, for $3 \le m < \omega $ and any $k < \omega $, that the class $SNr_m {\bf{CA}}_{m + k + 1} $ is strictly contained in $SNr_m {\bf{CA}}_{m + k} $ and if $k \ge 1$ then the former class cannot be defined by any finite set offirst-order formulas, within the latter class. We generalize this result to thefollowing algebras of m-ary relations for which the neat reductoperator $_m $ is meaningful: polyadic algebras with or without equality andsubstitution algebras. We also generalize this result to allow the case wherem is an infinite ordinal, using quasipolyadic algebras inplace of polyadic algebras (with or without equality).

Keywords

03G1503C10.Algebraic logic. cylindric algebras. quasi-polyadic algebras. substitutionalgebrasneat reductsneat embeddings

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Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 1 , March 2014 , pp. 208 - 222
Copyright
Copyright © Association for Symbolic Logic 2014 

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References

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