Hirsch, R; Sayed Ahmed, T; (2014) The neat embedding problem for algebras other than cylindric algebras and for infinite dimensions. The Journal of Symbolic Logic , 79 (1) pp. 208-222. 10.1017/jsl.2013.20.

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Abstract

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. }

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