Cylindric algebras and finite polyadic algebras (Q1991591)

This survey paper is a short history of cylindric algebras and finitary polyadic algebras (a subclass of cylindric algebras with a transposition operator, which yields relativised representability). It is interesting and useful but is too brief, has a somewhat narrow point of view, is not sufficiently thorough, and is inaccurate or misleading in some cases. Several important papers are not mentioned, and there are several peculiar references. Four pages are devoted to a reiteration of technical concepts from standard textbooks, mostly not used, including the statements of just two theorems (the weak Resek-Thompson theorem and one by the author) but without citing any sources. For some concepts, already defined in the textbooks, he cites instead a technical paper published much later. The definitional equivalence of quasi-polyadic algebras and finitary polyadic algebras (and the paper proving it) is mentioned twice, in different sections. Some work of Ildiko Sain is missing from the two appropriate sections. Proofs of the weak, and later the strong, Resek-Thompson theorem were published in 1988 and 1989, respectively, but only the former is mentioned. The work of Richard Thompson is lumped together with that of Diane Resek and attributed to Resek's dissertation. The merry-go-round identities are said to be ``mysterious'', and the standard textbook explanation of them is attributed instead to a technical paper. In spite of these shortcomings, there are several pages of useful material, a page of history, a good table of comparisons between systems, and half a page of conclusions, regarding the proper algebraization of first-order logic.