The neat embedding problem for algebras other than cylindric algebras and for infinite dimensions

Abstract

Hirsch and Hodkinson proved, for 3 ≤ m < ωand anyk < ω, that the class SNrmCAm+k+1 is strictly contained in SNrmCAm+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 Nrm 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).