Abstract
Hirsch and Hodkinson proved, for 3 <= m < omega and any k < omega, that the class SNr(m)CA(m+k+1) is strictly contained in SNr(m)CA(m+k) and if k >= 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 Ht(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).