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THE NEAT EMBEDDING PROBLEM FOR ALGEBRAS OTHER THAN CYLINDRIC ALGEBRAS AND FOR INFINITE DIMENSIONS
Published online by Cambridge University Press: 17 April 2014
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 of
first-order formulas, within the latter class. We generalize this result to the
following algebras of m-ary relations for which the neat reduct
operator
$_m $ is meaningful: polyadic algebras with or without equality and
substitution algebras. We also generalize this result to allow the case where
m is an infinite ordinal, using quasipolyadic algebras in
place of polyadic algebras (with or without equality).
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- Copyright © Association for Symbolic Logic 2014
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