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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).
