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Publication Detail
The neat embedding problem for algebras other than cylindric algebras and for infinite dimensions
  • Publication Type:
    Journal article
  • Publication Sub Type:
    Article
  • Authors:
    hirsch R, sayed ahmed T
  • Publication date:
    2014
  • Pagination:
    208, 222
  • Journal:
    The Journal of Symbolic Logic
  • Volume:
    79
  • Issue:
    1
  • Status:
    Published
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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