Some results about neat reducts (Q992010)

This is a survey article on the concept of neat reduct of a cylindric algebra, but it contains also several new results. One of these concerns the characterization problem of the class of neat reducts \(\mathfrak{Nr}_\alpha\)\textbf{CA}\(_\beta\). The author has proved in [``The class of neat reducts is not elementary'', Log. J. IGPL 9, No.~4, 593--628 (2001; Zbl 0988.03098)] that the class is not elementary for \(1 < \alpha < \beta\); now, it is shown that no \(L_{\infty\omega}\)-sentence characterizes this class. Neat reducts are discussed also in connection with the amalgamation property (in particular, two open questions are answered) and complete representations. Reviewer's remark: For neat reducts and amalgamation see also the recent paper [\textit{J. Madarász} and \textit{T. Sayed Ahmed}, ``Neat reducts and amalgamation in retrospect, a survey of results and some methods. I: Results on neat reducts'', Log. J. IGPL 17, No. 4, 429--483 (2009; Zbl 1174.03029)].