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

Hirsch and Hodkinson proved, for $3\leq m<\omega$ and any $k<\omega$, that the class S\straightNr_m\CA_{m+k+1} is strictly contained in S\straightNr_m\CA_{m+k} and if $k\geq 1$ then the former class cannot be defined by any finite set of first order formulas, within the latter class. We generalise this result to the following algebras of $m$-ary relations for which the neat reduct operator \Nr_m is meaningful: polyadic algebras with or without equality and substitution algebras. We also generalise this result to allow the case where $m$ is an infinite ordinal, using quasipolyadic algebras in place of polyadic algebras (with or without equality). \footnote{ Mathematics Subject Classification: 03G15, 03C10. {\it Key words}: algebraic logic, cylindric algebras, quasi-polyadic algebras, substitution algebras, neat reducts, neat embeddings.